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Some New Classes of Orthogonal Polynomials and Special Functions:

A Symmetric Generalization of Sturm-Liouville Problems and its Consequences

by

Mohammad MASJED-JAMEI

University of Kassel Department of Mathematics, Kassel, Germany

May 2006

Ph.D thesis supervised by

Professor Wolfram KOEPF

Department of Mathematics, University of Kassel, Kassel, Germany

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Contents 1 Outline of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 A symmetric generalization of Sturm - Liouville Problems . . . . . . . . . . . . . . . . . 9 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2. Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1. Usual Sturm-Liouville problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3. A symmetric generalization of Sturm-Liouville problems . . . . . . . . . . . . . . . . 11 2.3.1. A symmetric generalization of the associated Legendre functions . . . . . 13 2.3.1.1. The special case 2/2)( xxE −= in the generalized associated Legendre equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 A main class of symmetric orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 19 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 3.2. A main class of symmetric orthogonal polynomials using the extended Sturm- Liouville theorem (MCSOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3. An analogue of Pearson distributions family . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.1. A generalization of dual symmetric distributions family . . . . . . . . . . . . . 23 3.4. A direct relationship between first and second kind of classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5. Some further standard properties of MCSOP . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6. First subclass, Generalized ultraspherical polynomials (GUP) . . . . . . . . . . . . . 28 3.6.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6.2. Recurrence relation of monic polynomials . . . . . . . . . . . . . . . . . . . . . . . 29 3.6.3. Orthogonality relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.6.4. Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.7. Fifth and Sixth kind of Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . 30 3.8. Second subclass, Generalized Hermite polynomials (GHP) . . . . . . . . . . . . . . . 34 3.8.1. Initial vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.8.2. Explicit form of monic GHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.8.3. Recurrence relation of monic polynomials . . . . . . . . . . . . . . . . . . . . . . . 34 3.8.4. Orthogonality relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.8.5. Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.9. Third subclass, A finite class of symmetric orthogonal polynomials with weight function ba xx −− + )1( 22 on ),( ∞−∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.10. Fourth subclass, A finite class of symmetric orthogonal polynomials with weight function

212 xaex /−− on ),( ∞−∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.10.1. Initial vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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3.10.2. Explicit form of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.10.3. Recurrence relation of monic polynomials . . . . . . . . . . . . . . . . . . . . . . 37 3.10.4. Orthogonality relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.10.5. Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.11. A unified approach for MCSOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.11.1. How to find the parameters if a special case of the main weight function of MCSOP is given?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.11.2. How to find the parameters if a special case of the main recurrence equation of MCSOP is given? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Finite classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2. First finite class of hypergeometric orthogonal polynomials . . . . . . . . . . . . . . . 43 4.3. Second finite class of hypergeometric orthogonal polynomials: If 2)( xx =σ and 1)2()( +−= xpxτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3.1 A direct relationship between Bessel polynomials and Laguerre polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4. Third finite class, Hypergeometric orthogonal polynomials with weight function ))/()(arctanexp())()(( 22 dcxbaxqdcxbax p +++++ − on ),( ∞−∞ . . 49 4.5. Application of finite orthogonal polynomials in functions approximation and numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6. A connection between infinite and finite classical orthogonal polynomials . . . 56 5 A generalization of Student’s t-distribution from the viewpoint of special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2. A generalization of the t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3. Some particular sub-cases of the generalized t (and F) distribution . . . . . . . . . 64 5.3.1. A symmetric generalization of the t-distribution, the case q ib= and 1 2 1/ 2λ λ= = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3.2. An asymmetric generalization of the t-distribution, the case 02 =λ . . . . 67 6 A generic polynomial solution for the differential equation of hypergeometric type and six sequences of classical orthogonal polynomials related to it . . . . . . 71 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.1.1. An algebraic identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2. The main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.3. A special case of the generic obtained polynomials . . . . . . . . . . . . . . . . . . . . . 77 6.4. Some further properties of the generic obtained polynomials . . . . . . . . . . . . . . 78 6.4.1. A linear change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.4.2. A generic three-term recurrence equation . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.4.3. A generic formula for the norm square value of the polynomials . . . . . . 79

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6.5. Six special cases of the generic obtained polynomials as classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.5.1. Jacobi orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.5.2. Laguerre orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.5.3. Hermite orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.5.4. Finite classical orthogonal polynomials with weight function )(

1 )1(),,( qpq xxqpxW +−+= on ),0[ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.5.5. Finite classical orthogonal polynomials with weight function xpexpxW /1

2 ),( −−= on ),0[ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.6. How to find the parameters if a special case of the main weight function is given? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.7. A generic formula for the values at the boundary points of monic classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.7.1. Values of the classical orthogonal polynomials at the boundary points. . 88 7 Application of rational classical orthogonal polynomials for explicit computation of inverse Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2. Evaluation of inverse Laplace transform using rational classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2.1. Inverse Laplace transform using rational Jacobi orthogonal polynomials )(),( xP n

βα− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2.2. Inverse Laplace transform using rational Laguerre orthogonal polynomials )()( xL n

α− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3. Special examples of section 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8 Some further special functions and their applications . . . . . . . . . . . . . . . . . . . . 99 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.2. Application of zero eigenvalue for solving the potential, heat and wave equations using a sequence of special functions . . . . . . . . . . . . . . . . . . . . . . . . 99 8.2.1. Application of defined functions in the solution of potential, heat and wave equations in spherical coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.3. Two classes of special functions using Fourier transforms of finite classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.3.1. Evaluating a complicated integral and a conjecture . . . . . . . . . . . . . . . . 111 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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Chapter 1 Outline of dissertation This thesis includes 8 chapters. The chapter 2 specifies how to generalize the usual Sturm-Liouville problems with symmetric solutions to a larger class. In other words, if the symmetric sequence )(xy nΦ= satisfies a usual Sturm-Liouville problem as

,0)(,0)(;0))()(()( >>=−+⎟⎠⎞

⎜⎝⎛ xxkyxqx

dxdyxk

dxd

n ρρλ (1.1)

with the boundary conditions

1 1

2 2

( ) ( ) 0;

( ) ( ) 0y a y a

a x by b y b

α βα β

′+ =⎧< <⎨ ′+ =⎩

, (1.1.1)

then, under some specific conditions, it can be generalized to a more extensive problem as

0))(2

)1(1)()(()( =−−

+−+⎟⎠⎞

⎜⎝⎛ yxExqx

dxdyxk

dxd n

n ρλ , (1.2)

with the individual condition 3 3( ) ( ) 0 ;y y xα θ β θ θ θ′+ = − < < . (1.2.1) It should be noted that the main advantage of this extension is that the generalized solutions preserve the orthogonality property. In this sense, by using the above theorem, the well-known symmetric associated Legendre functions (having extensive applications in physics and engineering) are generalized and it is shown that the orthogonality interval is the same as [-1,1]. Another important consequence of the extension of usual Sturm-Liouville problems with symmetric solutions is given in the chapter 3. In this chapter, by using the mentioned theme, a main class of symmetric orthogonal polynomials (MCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation of the form (1.3)

( ) 0)(2/))1(1())1(()()()()( 2222 =Φ−−+−+−Φ′++Φ ′′+ xsxpnrnxsrxxxqpxx nn

nn , together with its explicit polynomial solution in the form

6

( )

( )∑ ∏=

−+−

=+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛++−+++−+

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ]2/[

0

2)1(]2/[

01

1

2)1(2]2/[2)1(2]2/[n

k

knkn

in

n

n xsqi

rpnik

nx

qpsr

S , (1.3.1)

a generic orthogonality relation, a generic three term recurrence relation and so on are obtained. Moreover, it is shown that four main sub-classes of symmetric orthogonal polynomials, i.e. the generalized ultraspherical polynomials, generalized Hermite polynomials and two new sequences of finite symmetric orthogonal polynomials are all special sub-cases of MCSOP. In this way, two half-trigonometric sequences of MCSOP are introduced. But, usually the finite orthogonal polynomials are less known (and discussed) in the literature. In chapter 4, a comprehensive treatment about finite classical orthogonal polynomials that are specific solutions of the well-known differential equation of Sturm-Liouville type ( ) ( ) ( ) ( ) ( ) 0 ,n n n nx y x x y x y xσ τ λ′′ ′+ − = (1.4) in which cbxaxx ++= 2)(σ , edxx +=)(τ and ndannn +−= )1(λ is given. In this chapter, three sequences of hypergeometric polynomials, which are finitely orthogonal with respect to the F, inverse gamma and generalized-T distribution functions are reintroduced in detail and their application in functions approximation and numerical integration are investigated. It is interesting to know that the weight function of the third class of finite orthogonal polynomials corresponds to a generalization of T distribution, which as far as we know has not appeared in the mathematical statistics branches. For this reason, chapter 5 is allocated to a large generalization of Student’s t-distribution with four free parameters and a comprehensive discussion of mathematical properties of this new distribution is investigated from special functions point of view. It is also shown that, similar to usual normal distribution, the generalized t-distribution converges to the normal distribution again when the number of samples tends to infinity. In this sense, since the Fisher F-distribution has a close relationship with the t-distribution, a generalization of the F-distribution is also introduced and shown that it similarly converges to the chi-square distribution as the number of samples tends to infinity. At the end of the chapter, some special cases of the generalized distributions are studied. But as we observe, the introduced equation (1.3) has a generic polynomial solution as (1.3.1). Now, is it possible to find a generic polynomial solution for the differential equation (1.4) similarly? Chapter 6 replies this question in detail. In chapter 6, it is shown that the equation (1.4) has a generic monic polynomial solution as

kn

k

nknn xedcbaG

kn

xcbaed

Pxy ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

0

)( ),,,,()( , (1.4.1)

where

7

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−+

−

−−

−−+−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

−

acbbacb

nad

na

d

acba

bdaenkF

acbbaG

nk

nk

442

2/22

142

2

42

2

22122

)( .

Then, it is shown that all six sequences of infinite and finite classical orthogonal polynomials described in chapter 4 can be indicated by the general derived formula. Some general properties of the formula (1.4.1), such as the affection of a linear change of variables, a generic three-term recurrence equation, a generic formula for the norm square value of the polynomials and so on are also given. At the end of the chapter, it is explained how to derive a generic formula for the values at the boundary points of monic classical orthogonal polynomials. Here is interesting to know that classical orthogonal polynomials have a direct use in the explicit computation of inverse Laplace transforms. In other words, if the Mobius transform Rqpqpzx ∈≠+= − ,0,1 is employed in each six sequences of classical orthogonal polynomials, then the generated rational orthogonal polynomials can be used to compute the inverse Laplace transform directly, with no additional calculation for finding their roots. In chapter 7, by achieving this purpose, three basic examples are given to explicitly obtain the inverse Laplace transforms. Finally, chapter 8 introduces some new sequences of special functions that have application in the solutions of classical potential, heat and wave equations in spherical coordinate. In other words, it is known that in the solution of Sturm-Liouville problems, usually zero eigenvalue is ignored. Now, if the foresaid eigenvalue is considered to be zero, one of the solutions will be pre-assigned. This approach causes to appear new sequences of special functions in the solutions of the classical equations of potential, heat and wave in spherical coordinate. By noting these comments, a class of special functions is introduced in the first part of chapter 8 and is applied in the mentioned classical equations. In this way, some applied examples are given. And eventually, in the second part of last chapter, two new classes of orthogonal hypergeometric functions are introduced and it is shown that using Fourier transforms and Paraseval identity they are finitely orthogonal with respect to two specific functions of Gamma type. Moreover, as the third new hypergeometric class, first a complicated integral is explicitly evaluated and then it is conjectured that the integrand of this integral may be finitely orthogonal with respect to the Ramanujan weight function on the real line.

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Chapter 2 A symmetric generalization of Sturm - Liouville Problems 2.1. Introduction In this chapter, we present some conditions under which the usual Sturm-Liouville problems with symmetric solutions can be extended to a larger class. The main advantage of this extension is that the corresponding solutions preserve the orthogonality property. As a sample, we investigate a basic example of generalized Sturm-Liouville problems and obtain their orthogonal solutions. The foresaid example generalizes the well-known associated Legendre functions (having extensive applications in physics and engineering) and preserves the orthogonality interval [-1,1] for the generalized functions. 2.2. Boundary value problems When partial differential equations are solved by the method of separation of variables (see also chapter 8 section 8.2), the problem reduces to the solution of ordinary differential equations. The solutions of these equations can, in many interesting problems of mathematical physics, be expressed in terms of special functions. In order to obtain such solutions of the partial differential equations in specific cases, we have to impose additional conditions on the solutions such that the problems will have unique solutions. These conditions in turn lead to conditions on the solutions of the ordinary differential equations and thus to boundary value problems. We here intend to first have a survey on the solution of boundary value problems by the method of separation of variables. 2.2.1. Usual Sturm-Liouville problems It has been proved that the method of separation of variables can extensively be applied for solving partial differential equations of the form

,)()(),,( *2

2* Lu

tutB

tutAzyx =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

ρ (2.1)

where =Lu div ),,(( zyxk grad uzyxqu ),,() − , (2.1.1)

FvectorofDivergenceFdiv =→

, (2.1.2)

A symmetric generalization of Sturm-Liouville problems

10

→→→

∂∂

+∂∂

+∂∂

= kzFj

yFi

xFFgrad . (2.1.3)

The equation (2.1) describes the propagation of a vibration such as electromagnetic or acoustic waves if 1=)(* tA and 0)(* =tB and describes transfer processes such as heat

transfer or the diffusion of particles in a medium when 0)(* =tA and 1=)(* tB .

Finally it describes the corresponding time-independent processes if 0)(* =tA and 0)(* =tB [56, p. 299].

To have a unique solution of equation (2.1), which corresponds to an actual physical problem, some supplementary conditions must be imposed. The most typical conditions are initial or boundary conditions. The initial conditions for equation (2.1) are usually the values of ),,,( tzyxu and ttzyxu ∂∂ /),,,( , while the simplest boundary condition is in the form

0),,(),,( =⎥⎦

⎤⎢⎣

⎡∂∂

+S

uzyxuzyxη

βα , (2.2)

where ),,( zyxα and ),,( zyxβ are given functions; S is the surface bounding the domain where (2.1) is to be solved; η∂∂ /u is the derivative in the direction of the outward normal to S. But the particular solutions of (2.1) under the boundary conditions (2.2) will be found if one looks for a solution of the form ),,()(),,,( zyxvtTtzyxu = . (2.3) By substituting (2.3) into the main equation (2.1) one respectively gets ,0)()( ** =+′+′′ TTtBTtA λ (2.4) ,0=+ vvL ρλ (2.5) where λ is a constant. Clearly (2.4) can be solved for typical problems in mathematical physics. However, to solve (2.5) we should use a boundary condition that follows from (2.2), namely ( ( , , ) ( , , ) ) 0

S

vx y z v x y zα βη∂

+ =∂

. The described problem is a known as

multidimensional boundary value problem. Nevertheless, it can be simplified to a one-dimensional problem if (2.5) is reduced to an equation of the form ,0)( =+ yxyL ρλ (2.6) where

.0)(,0)(,)()( >>−⎟⎠⎞

⎜⎝⎛= xxkyxq

dxdyxk

dxdyL ρ (2.6.1)


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The equation (2.6) should be considered on an open interval, say ),( ba , with boundary conditions in the form

,0)()(,0)()(

22

11

=′+=′+

bybyayay

βαβα

(2.7)

where 21 , αα and 21 , ββ are given constants and )(,)(,)( xqxkxk ′ and )(xρ in (2.6) and (2.6.1) are to be assumed continuous for ],[ bax∈ . The simplified boundary value problem (2.6)-(2.7) is called a regular Sturm-Lioville problem. Moreover, if one of the boundary points a and b is singular (i.e. 0)( =ak or

0)( =bk ), the problem will be transformed to a singular Sturm-Liouville problem. In this case, one can ignore boundary conditions (2.7) and obtain the orthogonality property directly. Now, let )(xyn and )(xym be two solutions (eigenfunctions) of equation (2.6). According to the Sturm-Liouville theory [18, 56], these functions are orthogonal with respect to the weight function )(xρ on ),( ba under the given conditions (2.7), i.e.

mn

b

an

b

amn dxxyxdxxyxyx ,

2 )()()()()( δρρ ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫∫ if

⎩⎨⎧

=≠

=.1,0

, mnifmnif

mnδ (2.8)

Many important special functions in theoretical and mathematical physics are the solutions of regular or singular Sturm-Liouville problems that satisfy the orthogonality condition (2.8). For instance, the associated Legendre functions [18], Bessel functions [18, 56], trigonometric sequences related to Fourier analysis [13], Ultraspherical functions [27, 70], Hermite functions [27, 70] and so on are particular solutions of some Sturm-Liouville problems. Fortunately, most of these mentioned functions have the symmetry property, namely )()1()( xx n

nn Φ−=−Φ , and because of this they have

found various applications in physics and engineering, see e.g. [18, 56] for more details. Now, if one can extend the mentioned examples symmetrically and preserve their orthogonality property, it seems that one will be able to find some new applications in physics and engineering that logically extend the previous applications. In the next sections, by achieving this matter, we generalize some classical symmetric orthogonal functions and obtain their orthogonality property. 2.3. A symmetric generalization of Sturm-Liouville problems [12] Without loss of generality, let )(xy nΦ= be a sequence of symmetric functions that satisfies the following differential equation

0)()(2

)1(1)()()()()()( =Φ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+++Φ′+Φ ′′ xxExDxCxxBxxA n

n

nnn λ , (2.9)

where )(,)(,)(,)( xDxCxBxA and )(xE are independent functions and }{ nλ is a sequence of constants. Clearly choosing 0)( =xE in (2.9) is equivalent to the one-


12

dimensional Sturm-Liouville equation (2.6). Since )(xnΦ is a symmetric sequence, by substituting the symmetry property )()1()( xx n

nn Φ−=−Φ into (2.9), one can

immediately conclude that )(,)(,)( xDxCxA and )(xE are even functions, while )(xB must be an odd function. This note will frequently be used in this section. To prove the orthogonality property of the sequence )(xnΦ , first both sides of equation (2.9) should be multiplied by the positive function

))()(exp(

)(1)

)()()(exp()( ∫∫ =

′−= dx

xAxB

xAdx

xAxAxBxR , (2.10)

in order to convert in the form of a self-adjoint differential equation. Note that )(xR in (2.10) is an even function, because )(xB is odd and )(xA is even. Therefore, the self-adjoint form of equation (2.9) becomes (2.11)

.)()()(2

)1(1)()())()(())(

)()(( xxRxExxRxDxCdx

xdxRxA

dxd

n

n

nnn Φ

−−−Φ+−=

Φλ

Since )()( xRxA is an even function, the orthogonality interval corresponding to equation (2.11) should be considered symmetric, say ],[ θθ− . Hence, by assuming that

θ=x is a root of the function )()( xRxA and applying the Sturm-Liouville theorem on (2.11) we have (2.12)

( )[ ]

.)()()()()2

)1()1(()()()()()(

)()()()()()(

∫∫−−

−

ΦΦ−−−

−ΦΦ−

=ΦΦ′−ΦΦ′θ

θ

θ

θ

θθ

λλ dxxxxRxEdxxxxRxC

xxxxxRxA

mn

nm

mnnm

nmmn

Obviously the left-hand side of (2.12) is zero. So, to prove the orthogonality property, it is enough to show that the value

∫θ

θ−

ΦΦ−−−

= dxxxxRxEmnF mn

nm

)()()()(2

)1()1(),( , (2.13)

is always equal to zero for every +∈ Znm, . For this purpose, four cases should generally be considered for m and n:

a) If both m and n are even (or odd), then 0),( =mnF , because we have 0)12,12()2,2( =++= jiFjiF .

b) If one of the two mentioned values is odd and the other one is even (or

conversely) then

∫θ

θ−+ΦΦ−=+ dxxxxRxEjiF ji )()()()()12,2( 122 . (2.14)


13

But in above relation )(,)( xRxE and )(2 xiΦ are even functions and )(12 xj+Φ is odd. Thus the integrand of (2.14) is an odd function and consequently 0)12,2( =+jiF . This issue also holds for the case 12 += in and jm 2= , i.e. 0)2,12( =+ jiF . By noting the above comments, the main theorem of section (2.3) can here be expressed. Theorem 1. [12] The symmetric sequence ( ) ( 1) ( )n

n nx xΦ = − Φ − , as a specific solution of differential equation (2.9), satisfies the orthogonality relation

mnnmn dxxxWdxxxxW ,2** )()()()()( δ⎟⎟

⎠

⎞⎜⎜⎝

⎛Φ=ΦΦ ∫∫

θ

θ−

θ

θ−

, (2.15)

where

))(

)()(exp()()()()(* ∫′−

== dxxA

xAxBxCxRxCxW , (2.16)

denotes the corresponding weight function and is a positive and even function on [ , ]θ θ− . It is now a good position to propound a practical example that generalizes the well-known symmetric associated Legendre functions and preserves their orthogonality property. 2.3.1. A symmetric generalization of the associated Legendre functions [12] When Laplace’s equation 02 =∇ u (or Helmholtz’s equation uu λ=∇ 2 ) is solved in spherical coordinates ϕθ ,,r , the following results appear [56]

ur

uu r ϕθ ,22 1

Δ+Δ=∇ , (2.17)

where

,)(1 22 r

urrr

ur ∂∂

∂∂

=Δ (2.17.1)

.sin

1)(sinsin

12

2

2, ϕθθθ

θθϕθ ∂∂

+∂∂

∂∂

=Δuuu (2.17.2)

As was explained in the section (2.2), by separating )()()( ϕθ ΩΨ= rRu and substituting it into Laplace’s equation (Potential equation), three following ordinary differential equations are derived

2

2

( ) ,0 ,

sin (sin ) ( sin ) 0 ,

r R R

d dd d

μυ

θ θ μ θ υθ θ

′ ′ =′′Ω + Ω =

Ψ+ − Ψ =

(2.18)


14

where μ and υ are two constant values. For θcos=x , the last equation of (2.18) changes to

0)1

(2)1( 22

22 =Ψ

−−+

Ψ−

Ψ−

xdxdx

dxdx υμ , (2.19)

which is called the associated Legendre differential equation [18] and its solutions are naturally known as the associated Legendre functions. These functions play a key role in the theory of potential. In general, there are three types of solution for equation (2.19) depending on different values of μ and υ : i) If )1)(( +++= ααμ nn and 1,;2 −>∈= + ααυ Zn , then the solution of (2.19) is indicated by

)()1()()( ),(22)( xPxxUx nnαα

αα −==Ψ , (2.20)

where )(),( xPn

αα , known as Ultraspherical polynomials, is a special case of Jacobi polynomials [27, 70]

kn

kn

xkn

nk

knxP )

21()(

0

),( −⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ +++= ∑

=

αβαβα , (2.21)

for βα = . In this way, the solution (2.20) satisfies the orthogonality relation

mnmnnmn nnnndxxUdxxUxU ,

212

,

1

1

2)(1

1

)()(

)12()122(!)1(2)))((()()( δαα

αδα

ααα

++Γ++++Γ

==+

−−∫∫ . (2.22)

ii) If )1( += nnμ and +∈= Znmm ,;2υ , (2.19) has a solution in the form (discovered by Legendre)

mn

mmm

n dxxPd

xxPx))((

)1()()( 22−==Ψ , (2.23)

where )()( )0,0( xPxP nn = denotes the Legendre polynomials [56]. Moreover, according to [18], (2.23) satisfies the orthogonality relation

jijim

imj

mi mii

midxxPdxxPxP ,,

1

1

21

1 )!)(12()!(2)))((()()( δδ−+

+== ∫∫

−−

. (2.24)

iii) To obtain the third type of solution, let us first consider the Jacobi polynomials differential equation [27, 70]

( ) )(0)1()()2()1( ),(2 xPyynnyxyx nβαβαβαβα =⇔=++++′−+++−′′− , (2.25)


15

and suppose )()1()1()( ),(2/2/ xPxxxA nnβαβα +−= . After some computations in hand,

the differential equation )(xAn is derived as

.0)()]1(4)()(2))((2

))1(4)2)([((41)()1(2)()1(

2

2222

=+++−−++−−++

+++++++−′−−′′−

xAnnx

xnnxAxxxAx

n

nn

βαβαβαβαβα

βαβαβα (2.26)

Now suppose 0=+ βα in (2.26) that reduces it to the differential equation

0)1

)1((2)1( 2

22 =

−−++′−′′− A

xnnAxAx α . (2.27)

The condition 11 <<− α is necessary for the orthogonality property of the polynomials )(),( xPn

αα − , because we must have 1,and0 −>=+ βαβα . Therefore, by considering )1( += nnμ and 11;2 <<−= ααυ in (2.19), the related solution takes the form

)()11()()( ),(2)( xP

xxxVx nn

ααα

α −

+−

==Ψ , (2.28)

satisfying the orthogonality relation

mnmnnmn nnnndxxVdxxVxV ,2,

1

1

2)(1

1

)()(

)12()!()1()1(2)))((()()( δ

ααδααα

+−+Γ++Γ

== ∫∫−−

. (2.29)

To extend the associated Legendre functions, it is enough in (2.9) to choose

2

2

( ) 1 ( ) ; ( ) 2 ( ) ,

( ) 1 ( ) ; ( ) ( ) ,1

; ( ) ( ) Arbitrary ,n n

A x x A x B x x B x

C x C x D x D xx

E x E x

υ

λ μ

= − = − = − = − −

= = − = − = −−

= = −

(2.30)

and apply the theorem 1 to establish the orthogonality property on the same interval [-1,1]. To do this task, let ))(,;()( xExQx nn υ=Φ be a known and predetermined solution of the differential equation

0)())(2

)1(11

()(2)()1( 22 =Φ

−−+

−−+Φ′−Φ ′′− xxE

xxxxx n

n

nnnυμ . (2.31)

According to the main theorem 1, we should have

mnnmn dxxExQdxxExQxExQ ,

1

1

21

1

))))(,;((())(,;())(,;( δυυυ ∫∫−−

= , (2.32)


16

provided that the arbitrary function )(xE is even. The sequence ))(,;( xExQn υ (as a known solution of equation (2.31)) is now orthogonal under the pre-assigned condition )()( xExE =− . So it is applicable in the theory of the expansion of functions, as many boundary value problems of mathematical physics are solved by using the expansion of functions in terms of the eigenfunctions of a usual Sturm-Liouville problem. Hence, if we assume

∑∞

=

υ=0

))(,;()(n

nn xExQqxf , (2.33)

then according to the property (2.32), the unknown coefficients nq are found by

∫∫−−

υυ=1

1

21

1

)))(,;(())(,;()( dxxExQdxxExQxfq nnn . (2.34)

Clearly various options can be selected for )(xE , that are directly related to the orthogonal sequence ))(,;( xExQn υ . For example, choosing the even function 0)( =xE gives one of the three cases of usual associated Legendre functions, stated before. A further special example is when 2/2)()( xxExE −=−= . Let us study this case in the next section in detail. 2.3.1.1. The special case 2/2)( xxE −= in the generalized equation (2.31) If 2/2)( xxE −= in (2.31) it reduces to the equation

0)()1)1(1

()(2)()1( 222 =Φ

−−+

−−+Φ′−Φ ′′− x

xxxxxx n

n

nnnυμ . (2.35)

To obtain the solution of this equation for specific values of nμ and υ , one should refer to the generalized ultraspherical polynomials (GUP), which were first investigated by Chihara [27]. On the other hand, according to the main theorem 1 if in the general equation (2.9) one chooses

),122(,;)(,;0)(,;)(,;))1((2)(,;)1()(

2

2

22

+++=−=

==

−++−=

−=

nqpnfunctionevenAnpxEfunctionevenAnxDfunctionevenAnxxCfunctionoddAnqxqpxxBfunctionevenAnxxxA

nλ

(2.36)

then the symmetric differential equation


17

( ) ,0)()1)1(()122(

)())1((2)()1(2

222

=Φ−−+++++

Φ′−++−Φ ′′−

xpxnqpn

xpxqpxxxx

nn

nn (2.37)

is the generalized ultraspherical equation having the monic polynomial solution (2.38)

.)1(222

])2/[2)1(222(2]2/[])2/[2)1(222(2

)1(222)(

]2/[

0

2)1(]2/[

0

1]2/[

0

),(

∑ ∏

∏

=

−+−

=

−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−++

+−−++−−⎟⎟⎠

⎞⎜⎜⎝

⎛×

+−−++−−−−++

=

n

k

knkn

in

n

n

in

nqp

n

xpi

npqik

nnpqi

pixU

The monic GUP are orthogonal with respect to the weight function qp xx )1( 22 − on [-1,1] and satisfies the following orthogonality relation

.)2/3(

)1()2/1())1222)(1222(

)2))1(1()())1(1(((

)()()1(

,1

1

1

),(),(22

mn

n

i

ii

qpm

qpn

qp

qpqp

qpiqpiqpipi

dxxUxUxx

δ++Γ

+Γ+Γ+++−+++−−+−−+

=

−

∏

∫

=

− (2.39)

Now, by defining

,1,21;)()1(),;( ),(22 −>−>−= baxUxxbaxG ba

n

ba

n (2.40)

and replacing it into equation (2.37), one gets (2.41)

.0)())1((1

)1)(()(2)()1( 22

22 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−+

−−++++++′−′′− xG

xaa

xbbanbanxGxxGx n

n

nn

If the above equation is compared with (2.35), then it can be concluded that

22

2( ; , ) ( ;1, ) ; 1 ,n nQ x b G x b bx

− = > − (2.42)

is a general solution of (2.31) for 2,)2)(1( bbnbnn =++++= υμ and 2/2)( xxE −= . Moreover, substituting these values into (2.32) and noting (2.39) yields (2.43)

.)2/5(2)1()

)322)(122()2)1(1)()1(1(()2,;()2,;( ,

1

1

12

22

2mn

n

i

ii

mn bb

bibibiidx

xbxQ

xbxQ δπ

+Γ+Γ

+++++−−+−−+

=−− ∏∫=−

Remark 1. Although some special functions such as Bessel functions [18], Fourier trigonometric sequences and so on are symmetric and satisfy a differential equation whose coefficients are alternatively even and odd, it is anyway important to note that


18

they do not satisfy the conditions of the main theorem 1. For instance, if in the generic equation (2.9) we choose

2

2 2

( ) ; ( ) ; ( ) 1,( ) ; ( ) 0 ; ,n

A x x B x x C xD x x E x nλ

= = =

= = = − (2.44)

we get to the Bessel differential equation 0)()()()( 222 =Φ−+Φ′+Φ ′′ xnxxxxx nnn , (2.45) with the symmetric solution as

2

0

( 1)( 1) ( ) ( ) ( ) ,!( )! 2

kn n k

n nk

xJ x J x nk n k

∞+ +

=

−− − = = ∈

+∑ Z , (2.46)

but the orthogonality interval of Bessel functions is not symmetric (i.e. ]1,0[ ). Hence, theorem 1 cannot be applied for )(xJ n , unless there exists a specific even function

)(xE for equation (2.9) such that the corresponding solution has infinity zeros (see e.g. [18, 56] for more details), exactly similar to the case of usual Bessel functions of order n . Therefore the conclusion of this chapter can be summarized as follows: the usual Sturm-Liouville problem

,0)(,0)(;0))()(()( >>=−+⎟⎠⎞

⎜⎝⎛ xxkyxqx

dxdyxk

dxd

n ρρλ

under the boundary conditions

1 1

2 2

( ) ( ) 0;

( ) ( ) 0y a y a

a x by b y b

α βα β

′+ =⎧< <⎨ ′+ =⎩

can be extended to the following problem

0))(2

)1(1)()(()( =−−

+−+⎟⎠⎞

⎜⎝⎛ yxExqx

dxdyxk

dxd n

n ρλ ,

with the boundary condition 3 3( ) ( ) 0 ;y y xα θ β θ θ θ′+ = − < < , provided that the solution of latter differential equation has the symmetry property, i.e.

)()1()( xyxy nn

n −=− .

Chapter 3 A main class of symmetric orthogonal polynomials 3.1. Introduction In the previous chapter, we determined how to extend the usual Sturm-Liouville problems with symmetric solutions. In this chapter, by using the extended Sturm-Liouville theorem for symmetric functions, we are going to introduce a main class of symmetric orthogonal polynomials (MCSOP) with four free parameters and obtain all its standard properties, such as a generic second order differential equation together with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on. Then, we show that essentially four main sequences of symmetric orthogonal polynomials can be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on the real line and can be expressed in terms of the mentioned class directly. In this way, we also introduce two half-trigonometric sequences of orthogonal polynomials as special sub-cases of MCSOP. 3.2. A Main Class of Symmetric Orthogonal Polynomials (MCSOP) using the extended Sturm-Liouville theorem [1] By referring to the main theorem of chapter 2 and generic differential equation (2.9) choose the following options

2 2

2

2

( ) ( ) ; ,( ) ( ) ; ,( ) ; ,( ) 0 ; ,( ) ; ,

A x x px q An even functionB x x rx s An odd functionC x x An even functionD x An even functionE x s An even function

= +

= +

=== −

(3.1)

where rqp ,, and s are real free parameters and ( ( 1) )n n r n pλ = − + − . Therefore, one deals with a second order differential equation of the form (3.2)

( ) 0)(2/))1(1())1(()()()()( 2222 =Φ−−+−+−Φ′++Φ ′′+ xsxpnrnxsrxxxqpxx nn

nn . To obtain the polynomial solution of above equation, let us first suppose that mn 2= , i.e.

A Main Class of Symmetric Orthogonal Polynomials

20

0)())12((2)()()()( 222

22 =Φ−+−Φ′++Φ ′′+ xxpmrmxsrxxqpxx mmm . (3.3)

After doing some calculations in hand, one can get the solution of equation (3.3) as

,)12(

)212(0

22)1(

02 ∑ ∏

=

−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

++++−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ m

k

kmkm

jm x

sqjrpmj

km

xqpsr

S (3.4)

where 1

1

0

=∏−

=rra . Similarly, if 12 += mn is considered in (3.2), the simplified equation

(3.5) 0)())2()12(()()()()( 12

212

212

22 =Φ+++−Φ′++Φ ′′+ +++ xsxmprmxsrxxxqpxx mmm , has the polynomial solution

∑ ∏=

−++−

=+ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+++++

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ m

k

kmkm

jm x

sqjrpmj

km

xqpsr

S0

212)1(

012 )32(

)212( . (3.6)

Consequently, combining (3.4) with (3.6) gives

( )

( )∑ ∏=

−+−

=+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛++−+

++−+⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ]2/[

0

2)1(]2/[

01

1

2)1(2]2/[2)1(2]2/[n

k

knkn

in

n

n xsqi

rpnik

nx

qpsr

S , (3.7)

as the most common source of classical symmetric orthogonal polynomials with four free parameters rqp ,, and s where neither both values q and s nor both values p and r can vanish together. Here let us point out that almost all known symmetric orthogonal polynomials, such as Legendre polynomials, first and second kind of Chebyshev polynomials, ultraspherical polynomials, generalized ultraspherical polynomials (GUP), Hermite polynomials and generalized Hermite polynomials (GHP) are special sub-cases of (3.7) and can be expressed in terms of it directly. Because of this matter, we call these polynomials, “The second kind of classical orthogonal polynomials”. Furthermore, there are two other symmetric sequences of finite orthogonal polynomials that are special sub-cases of general representation (3.7) and we will introduce them in the sections 3.9 and 3.10. The symbol

⎟⎟⎠

⎞⎜⎜⎝

⎛x

qpsr

Sn is used only because of Symmetry property, however, to shorten

this notation in the text, we will show it as );,,,( xsrqpSn . A straightforward result from (3.7) is that

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++=⎟⎟

⎠

⎞⎜⎜⎝

⎛+ x

qpqspr

Sxxqpsr

S nn

22212 . (3.8)

Moreover, since the monic type of orthogonal polynomials (i.e. with leading coefficient 1) is often required, we define


21

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

++−+++−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛∏

−

=+

+

xqpsr

Srpni

sqixqpsr

S n

n

in

n

n

1]2/[

01

1

]2/[2)1(22)1(2 . (3.9)

For instance, if 5...,,0=n in (3.9) we have

.)5)(7()3)(5(

752

,)3)(5(

))(3(532

,33

,

,

,1

355

244

33

22

1

0

xrprpsqsqx

rpsqxx

qpsr

S

rprpsqsqx

rpsqxx

qpsr

S

xrpsqxx

qpsr

S

rpsqxx

qpsr

S

xxqpsr

S

xqpsr

S

++++

+++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

++++

+++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛

(3.9.1)

The explicit representation of (3.9) helps us now obtain a three-term recurrence relation for the polynomials. Hence, if we assume that they satisfy the relation

N∈==⎟⎟⎠

⎞⎜⎜⎝

⎛+= −+ nxxSxSxS

qpsr

CxSxxS nnnn ,)(,1)(;)()()( 1011 , (3.10)

then after doing a series of computations in hand, we obtain

( ))32)(2(

2/))1(1()2()1()2(2

prpnprpnsprnpsqprnpq

qpsr

Cnn

n −+−+−−−+−−−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛, (3.10.1)

which reveals the explicit form of (3.10). On the other hand, since the recurrence relation (3.10) has explicitly been specified, to determine the norm square value of the polynomials one can use Favard's theorem [27] by noting that there is orthogonality with respect to a weight function. According to this theorem, if ∞

=0)}({ nn xP is defined by ,...2,1,0;)()()()( 11 =++= −+ nxPCxPBxPAxPx nnnnnnn , (3.11) where realCBAxPxP nnn ,,,1)(,0)( 01 ==− and 01 >+nnCA for ,...1,0=n , then there exists a weight function )(xW so that


22

mn

n

i i

imn dxxW

AC

dxxPxPxW ,

1

0

1 )()()()( δ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫∏∫

∞

∞−

−

=

+∞

∞−

. (3.12)

Moreover, if the positive condition 01 >+nnCA only holds for Nn ,...,1,0= then the orthogonality relation (3.12) only holds for a finite number of m, n. The latter note will help us in the next sections to obtain two new sub-classes of );,,,( xsrqpS n that are finitely orthogonal on the interval ),( ∞−∞ . It is clear that the Favard theorem is also valid for the recurrence relation (3.10) in which 0,1 == nn BA and

),,,( srqpCC nn −= . Furthermore, the condition 0),,,(1 >− + srqpCn must always be satisfied if one demands to apply the Favard theorem for (3.10). By noting this subject and (3.12), the generic form of the orthogonality relation of MCSOP can be designed as (3.13)

mn

ni

ii

nmn dxx

qpsr

Wqpsr

Cdxxqpsr

Sxqpsr

Sxqpsr

W ,1

)1( δα

α

α

α⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∫∏∫−

=

=−

where the weight function, by referring to (2.16) and (3.1), is defined as

))(

)2(exp())(

)2()4(exp( 2

2

2

22 dx

qpxxsxprdx

qpxxqsxprxx

qpsr

W ∫∫ ++−

=+

−+−=⎟⎟

⎠

⎞⎜⎜⎝

⎛, (3.14)

and α takes the standard values ∞,1 . Note that the function );,,,()( 2 xsrqpWqpx + must vanish at α=x in order to establish the main orthogonality relation (3.13). 3.3. An analogue of Pearson distributions family [1] The positive function (3.14) can also be investigated from statistical point of view. In fact, this function is an analogue of Pearson distributions family having the general form

))()2(exp( 2∫ ++−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛dx

cbxaxbexadx

cbaed

ρ , (3.15)

and satisfying the first order differential equation

)()())()(( 2 xedxxcbxaxdxd ρρ +=++ . (3.15.1)

Therefore, we would like to point out that, similar to equation (3.15.1), the weight function (3.14) satisfies a first order differential equation as

)()())()(( 22 xWsrxxWqpxdxdx +=+ , (3.16)

which is equivalent to


23

⎪⎩

⎪⎨⎧

+=

+=+=+

qssprr

tsxWsxrxxWqpxxdxd

22

..)()())()((*

**2*22 (3.16.1)

From (3.16) or (3.16.1) it can be deduced that );,,,( xsrqpW is an analytic integrable function and since it is also a positive function, its probability density function (pdf) must be available. In general, there are four main sub-classes of distributions family (3.14) (and consequently sub-solutions of equation (3.16)) whose explicit pdfs are respectively as follows

.01;02/1

;11;)1()1()2/1(

)2/3(1,1

2,222 221

>+>+

≤≤−−+Γ+Γ

++Γ=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−−

ba

xxxba

baxaba

WK ba

(3.17)

.02/1;;)2/1(

11,0

2,2 222 >+∞<<∞−

+Γ=⎟⎟

⎠

⎞⎜⎜⎝

⎛− − axexa

xa

WK xa (3.18)

(3.19) 2

3 2

2 2 2, 2 ( ) ; ;1, 1 ( 1/ 2) ( 1/ 2) (1 )

1/ 2 ; 1/ 2 ; 0.

a

b

a b a b xK W x xb a a x

b a a b

−⎛ − − + − ⎞ Γ= −∞ < < ∞⎜ ⎟ Γ + − Γ − + +⎝ ⎠

+ > < >

.2/1;;)2/1(

10,12,22 2

12

4 >∞<<∞−−Γ

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +− −− axex

ax

aWK xa (3.20)

The values 4,3,2,1; =iKi play the normalizing constant role in above distributions. Moreover, as it is observed, the value of distribution vanishes at 0=x in each four cases, i.e. 0)0;,,,( =srqpW for 0≠s . Hence, let us call the positive function (3.14) “The dual symmetric distributions family”. 3.3.1. A generalization of dual symmetric distributions family First it is important to remember that if 0=s in (3.16), the foresaid equation will be reduced to a special sub-case of Pearson differential equation (3.15.1). Hence, we hereafter suppose that 0≠s . Since the explicit forms of );,,,( xsrqpSn in (3.7),

),,,( srqpCn in (3.10.1) and );,,,( xsrqpW in (3.14) are all known, a further main pdf can directly be defined by referring to the orthogonality relation (3.13), so that we have

2

1

)1(⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛×⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

∫∏−

=

=

xqpsr

Sxqpsr

Wdxx

qpsr

Wqpsr

Cx

qpsr

D mmi

ii

m

m α

α

. (3.21)

Clearly choosing 0=m in this definition gives the same as dual symmetric distributions family. Moreover, the Fisher information [36] and Boltzmann-Shannon


24

information entropy [28] are two important factors in statistical estimation theory that can be investigated for the generalized distribution (3.21). 3.4. A direct relationship between first and second kind of classical orthogonal polynomials One can verify that there is a direct relation between first kind of classical orthogonal polynomials (including Jacobi, Laguerre and Hermite polynomials as well as three finite classes of orthogonal polynomials, see chapter 4) and the explicit polynomials

);,,,( xsrqpSn indicated in (3.7). To find this relationship, we start with the following differential equation 0)())1(()()()()( 2 =+−−′++′′++ xydannxyedxxycbxax nnn . (3.22) According to [56], the monic polynomial solution of (3.22) can be shown by a Rodrigues-type formula as (3.23)

))(())2((

1 2

1

⎟⎟⎠

⎞⎜⎜⎝

⎛++×

⎟⎟⎠

⎞⎜⎜⎝

⎛−++

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∏=

xcbaed

cbxaxdxd

xcbaed

akndx

cbaed

P nn

n

n

k

n ρρ

,

where );,,,,( xedcbaρ is defined as the same form as (3.15). But, since );,,,(2 xsrqpS n is generally an even function, taking vwtx += 2 in (3.22) gives (3.24)

.0)())1((4)())(

))2()(2()2(()())2((422

24222422

=−+−′++−

−+−+−+′′+++++

tytandnwtycbvav

tbewadwvtwadttycbvavtbavwtawt

nn

n

If (3.24) is equated with (3.3), we should have

a

acbbvorcbvav2

402

2 −±−==++ . (3.24.1)

The condition (3.24.1) simplifies the equation (3.24) to (3.24.2)

.2

4)(0)())1((4

)()2)4)(1()2(()()4(

22

2222

⎟⎟⎠

⎞⎜⎜⎝

⎛ −±−+=⇔=−+−

′−+−±−−+−+′′−±

aacbbwt

cbaed

Ptytytandwn

tybeacbbadwtadtyacbawtt

nnn

nn

The equation (3.24.2) is clearly a special case of (3.3). This means that (3.25)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−±

−+−±−−−=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −±−+ t

acbaw

beacbbadwad

SKa

acbbwtcbaed

P nn

4,

2)4)(1(,)2(2

42

2

2

22


25

where K is the leading coefficient of the left-hand side polynomial of relation (3.25) divided to leading coefficient of the right-hand side polynomial. As we observe in the above relation, there exist 5 free parameters a, b, c, d, e in the left-hand side of (3.25). So, one of them must be pre-assigned in order that one can get the explicit form of );,,,(2 xsrqpS n in terms of );,,,,( 2 vwtedcbaPn + similarly. For this purpose, if for instance 0=c is considered in (3.25), two following cases appear

0;0,,/

2,

22

2 ≠⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ ++=⎟⎟

⎠

⎞⎜⎜⎝

⎛ − wwtqwp

qsw

prPwt

qpsr

S nn

n , (3.26)

0,;0,,/

2,

2 22 ≠

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

−

−+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ − wppqwwt

qwpprqsp

wpr

Pwtqpsr

S nn

n . (3.27)

Furthermore, if (3.8) is applied for two latter relations we respectively get

0;0,,/

23,

23

212 ≠

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ ++=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+ wwt

qwp

qsw

prPtwt

qpsr

S nn

n , (3.28)

0,;0,,/

2,

23

212 ≠

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

−

−+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+ wp

pqwwt

qwpprqsp

wpr

Ptwtqpsr

S nn

n . (3.29)

3.5. Some further standard properties of MCSOP The relations (3.26)-(3.29) are useful tool to get a generating function for MCSOP. Usually, a generating function for a system of polynomials )(zPn is defined by a function like ),( tzG whose expansion in powers of t has, for sufficiently small || t , the form

∑∞

=

=0 !

)(),(n

n

n ntzPtzG . (3.30)

If )(zPn has the Rodrigues-type formula [56], Cauchy’s integral formula

∫Ω

+−= du

zuuf

inzf n

n1

)(

)()(

2!)(π

, (3.31)

where Ω is a closed contour surrounding the point zu = , is employed to obtain

),( tzG explicitly, see e.g. [56, p. 27]. This means that by considering the Rodrigues-type representation (3.23) for 0=c and applying Cauchy’s integral theorem on it we have


26

atzbtzba

ed

uba

ed

ntz

baed

Pakndn

n

n

n

k 4)1(1

0,,

0,,!0

))2((2

0 1 −−×

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛−++∑ ∏

∞

= = ρ

ρ. (3.32)

atatzbtbt

u2

4)1(1:where

2 −−+−= .

Now if 2

and2

,,,, 22 qseprdqbpattzz +=

+===→→ are substituted into

(3.32), by noting the relations (3.26)-(3.29) we obtain (3.32.1)

22222

*

0

2

21 4)1(

1

0,,2/)(2/)(

0,,2/)(2/)(

!))2(

2(

zptqtzqp

qspr

uqp

qspr

ntz

qpsr

Spknprn

n

n

n

k −−⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

=⎟⎟⎠

⎞⎜⎜⎝

⎛−++

+∑∏∞

= = ρ

ρ

2

22222*

24)1(1

:wherept

zptqtqtu

−−+−= .

Similarly by multiplying both sides of (3.32) by tz and using (3.28) we get (3.32.2)

22222

*

0

12

121 4)1(

0,,2/)3(2/)3(

0,,2/)3(2/)3(

!))2(

23(

zptqt

zt

zqp

qspr

uqp

qspr

ntz

qpsr

Spknprn

n

n

n

k −−⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

=⎟⎟⎠

⎞⎜⎜⎝

⎛−++

+∑ ∏∞

=

+

+= ρ

ρ

Finally let us define

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ∏=

+

xqpsr

Sn

pknrx

qpsr

S n

n

k

n

n ]!2/[

)4/))1(52((2/]2/[

1

1

* . (3.33)

Therefore, by noting (3.32.1) and (3.32.2) a kind of generating function for MCSOP is derived as (3.34)

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

+

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∑∞

= 2

*

2

*

22220

*

0,,2/)3(,2/)3(

0,,2/)3(,2/)3(

)(

0,,2/)(,2/)(

0,,2/)(,2/)(

4)1(1

zqp

qspr

uqp

qsprtz

zqp

qspr

uqp

qspr

zptqttz

qpsr

Sn

nn

ρ

ρ

ρ

ρ

where *u is the same form as (3.32.1). The explicit form of polynomials (3.7) can also be applied to obtain a generic hypergeometric representation for the polynomials, because by using it one can easily


27

indicate the coefficients of polynomials (3.7) in terms of the Pochhammer symbol: )1)...(1()( −++= maaaa m . For instance, we have

( )k

nnkn

kn

i

n nprprpni−

+−+−

=

+ −++=++−+∏]

2[

1]2/[)1(]2/[

0

1 2/)1(]2/[2/)2/(])2/[2)1(2( . (3.35)

Hence, after some calculations, we eventually find that

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−+−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛212 2/))32((

]2/)1[(2/)(,]2/[pxq

ppnrnqsqn

Fxxqpsr

S nn , (3.36)

where !)(

)()(0

12 kxxF

k

k k

kk∑∞

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛γβα

γβα denotes the hypergeometric function of order

(2,1) [43]. Furthermore, since (3.37)

1||;0ReRe)1()1()()(

)( 1

0

1112 <>>⇔−−

−ΓΓΓ

=⎟⎟⎠

⎞⎜⎜⎝

⎛∫ −−−− zbcdttztt

bcbcz

cba

F abcb ,

the integral representation of MCSOP can easily be derived by referring to (3.36). In this way, by applying the Gauss identity [43]

)()()()(112 bcac

baccc

baF

−Γ−Γ−−ΓΓ

=⎟⎟⎠

⎞⎜⎜⎝

⎛, (3.38)

one can also determine the value of polynomials );,,,( xsrqpSn at a specific boundary

point, i.e. pqx /−= . To reach this goal, it is sufficient to put pqx /−= in (3.36) and use (3.38) to get

])

2[

221()]

2[

223()

223(

)22

1()22

3()( 2

np

rqsnn

prn

pr

pr

qs

pr

pq

pq

qpsr

Sn

n

n

−−+Γ−+−Γ−−

−+Γ−Γ−=⎟⎟⎠

⎞⎜⎜⎝

⎛− . (3.39)

Note that to derive the above identity, the following equalities have been used

.)1()1()(,

4)1(1

2]

2[

,]2

[]2

1[,2

)1(1]2

[]2

1[

nn

n

n

n

naann

nnnnn

−−−=−−

−=

=++−−

=−+

(3.40)

The standard properties of MCSOP have been found now. So, it is a good position to introduce four special sub-cases of main polynomials (3.7) in detail.


28

3.6. First subclass, Generalized ultraspherical polynomials (GUP) These polynomials were first investigated by Chihara in detail [see 27]. He obtained the main properties of GUP via a direct relation between them and Jacobi orthogonal polynomials. The asymptotic behaviors of foresaid polynomials were also studied by Konoplev [48]. On the other hand, Charris and Ismail in [24] (see also [25, 41]) introduced the Sieved random walk polynomials to show that the generalized ultraspherical polynomials are a special case of them. Of course, there are some other generalizations of ultraspherical polynomials. For instance, Askey in [19] introduced two classes of orthogonal polynomials as a limiting case of the q-Wilson polynomials on ]1,1[− with the weight functions

λα−

−α−=λα |)(||)(|)1(),,( 2

121

21 xTxUxxW kk

and λα

−

+α−=λα |)(||)(|)1(),,( 2

121

22 xTxUxxW kk

in which Nk ∈ is a fixed integer and )(xTk and )(1 xU k− are respectively the Chebyshev polynomials of the first and second kind, to generalize GUP for 1=k and

0=λ . For the case 0=λ the polynomials were introduced by Al-Salam, Allaway and Askey [15] as a limiting case of the q-Ultraspherical polynomials of Rogers [64]. Anyway, we intend in this section to show that GUP can directly be represented in terms of );,,,( xsrqpSn and consequently all its standard properties will be obtained. For this purpose, it is only enough to have the initial vector corresponding to these polynomials and replace it into the standard properties of MCSOP. 3.6.1. Definition Choose the initial vector )2,222,1,1(),,,( abasrqp −−−−= and substitute it into (3.7) to get (3.41)

,)1(222

])2/[2)1(222(2]2/[

])2/[2)1(222(2)1(222

1,12,222

]2/[

0

2

)1(]2

[

0

1]2/[

0

∑ ∏

∏

=

−

+−

=

−

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−+++−−++−−

⎟⎟⎠

⎞⎜⎜⎝

⎛×

+−−++−−−−++

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−

n

k

kn

kn

in

n

n

in

n

n

xai

nabik

n

nabiaix

abaS

as the explicit form of monic GUP. Moreover, since the ultraspherical (Gegenbauer), Legendre, and Chebyshev polynomials of the first and second kind are all special cases of GUP, they can be expressed in terms of );,,,( xsrqpSn directly and we have Ultraspherical polynomials:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

= xa

Sna

xC nn

nan 11

012!)(2

)( . (3.41.1)

Legendre polynomials:


29

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

= xSn

nxP nnn 1102

2)!()!2()( 2 . (3.41.2)

Chebyshev polynomials of the first kind:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

= − xSxT nn

n 1101

2)( 1 . (3.41.3)

Chebyshev polynomials of the second kind:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

= xSxU nn

n 1103

2)( . (3.41.4)

3.6.2. Recurrence relation of monic polynomials By replacing the initial vector (3.41) into the explicit expression ),,,( srqpC n , given in (3.10.1), the recurrence relation of monic GUP takes the form (3.42)

,,)(,1)(;)(1,1

2,222)()( 1011 N∈==⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−−+= −+ nxxSxSxS

abaCxSxxS nnnn

where according to (3.10.1) (3.42.1)

.

)1222)(1222()2))1(1()())1(1((

)1222)(1222())1(1)((2)))1(1(22(

1,12,222 2

+++−+++−−+−−+−

=

+++−++−−+−−−+−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−

banbanbanan

banbanbaanabnaba

C

nn

nn

n

3.6.3. Orthogonality relation Clearly the weight function of GUP is the same distribution as (3.17) without considering its normalizing constant, i.e. ba xx )1( 22 − . Also, since this function must be even and positive, the condition 1)1( 2 =− a is essential. Hence, the mentioned weight function can also be considered as ]1,1[;)1(|| 22 −∈− xxx ba . By noting this comment and generic relation (3.13) for 1=α , the orthogonality relation of first sub-class takes the form (3.43)

,)1(1,1

2,222)1(

1,12,222

1,12,222

)1(

,

1

1

22

1

1

1

22

mnba

ni

ii

n

mnba

dxxxaba

C

dxxaba

Sxaba

Sxx

δ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−−

∫∏

∫

−

=

=

−

where

)2/3(

)1()2/1()1,21()1(

1

1

22

++Γ+Γ+Γ

=++=−∫− ba

babaBdxxx ba . (3.43.1)


30

From (3.43.1) one can conclude that the constraints on the parameters a and b should be 02/1 >+a , 1)1( 2 =− a 01and >+b . Note that ),( 21 λλB in (3.43.1) denotes the Beta integral [18] having various definitions in the form

).,(

)()()(cossin2

)1()1()1(),(

1221

212/

0

)12()12(

0

11

1

12121

0

1121

21

21

12121

λλλλλλ

λλ

πλλ

λλ

λλλλλ

Bdxxx

dxx

xdxxxdxxxB

=+ΓΓΓ

==

+=−=−=

∫

∫∫∫

−−

∞

+

−

−

−−−−

(3.44)

3.6.4. Differential equation: );2,222,1,1()( xabaSx nn −−−−=Φ . To derive the differential equation of GUP it is enough to substitute its initial vector into the main differential equation (3.2) to get to (3.45)

( ) 0)()1)1(()122()())1((2)()1( 2222 =Φ−−+++++Φ′−++−Φ ′′+− xaxnbanxaxbaxxxx nn

nn

3.7. Fifth and Sixth kind of Chebyshev polynomials [1] As we know, four kinds of trigonometric orthogonal polynomials, known as first, second, third and fourth kind of Chebyshev polynomials, have been investigated in the literature up to now, see e.g. [63, 37, 27, 70]. The explicit definitions of them are respectively as

θθπ cos;)cos()2

)12(cos(2)(1

1 ==−

−= ∏=

− xnn

kxxTn

k

nn , (3.46)

θθ

θπ cos;sin

))1sin(()1

cos(2)(1

=+

=+

−= ∏=

xnnkxxU

n

k

nn , (3.47)

)2cos(;)cos(

))12cos(()12)12(cos(2)(

1

θθ

θπ=

+=

+−

−= ∏=

xnn

kxxVn

k

nn , (3.48)

)2cos(;)sin(

))12sin(()12

2cos(2)(1

θθ

θπ=

+=

+−= ∏

=

xnnkxxW

n

k

nn . (3.49)

Now, we would like to add here that there exist two further kinds of Half-trigonometric orthogonal polynomials, which are particular sub-cases of );,,,( xsrqpSn . Since they are generated by using the first and second kind of Chebyshev polynomials and have the half-trigonometric forms, let us call them the fifth and sixth kind of Chebyshev polynomials. To generate these two sequences, we should refer to the important relation (3.8). According to (3.41.3), the initial vector of first kind Chebyshev polynomials is:

)0,1,1,1(),,,( −−=srqp . So, if this vector is replaced into (3.8) then we have


31

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

==⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++ xSxxTxS nnn 1123

)(1101

21212 . (3.50)

By means of (3.50), the secondary vector )2,3,1,1(),,,( −−=srqp , as a special case of the set of vectors ),,,( srqp , appears. By using this new vector first we define the half-trigonometric polynomials

⎪⎩

⎪⎨⎧

+=−−

=+

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=,12);2,3,1,1(

,2cos

))1cos((1

)1(

1123

)(

2/

mnifxS

mnifnnxSxX

n

n

nn θθ

(3.51)

where x=θcos . According to (3.51) and (3.10), )(xX n satisfies the recurrence relation

,,)(,1)(;)(1123

)()( 1011 N∈==⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+= −+ nxxXxXxXCxXxxX nnnn (3.52)

in which

)1(4

))1(1)()1((1123

+−−+−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

nnnnC

nn

n . (3.52.1)

Consequently, substituting the secondary vector (-1,1,-3,2) into the generic relation (3.13) gives the orthogonality relation of the first kind of half-trigonometric orthogonal polynomials as

mn

n

ii

nmn dxxWCdxxXxXxW ,

1

11

1

1 1123

1123

)1()()(1123

δ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

∫∏∫−=−

. (3.53)

On the other hand since

2

)21,

23(

11123 1

12

21

1

π==

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

∫∫−−

Bdxx

xdxxW , (3.54)

(3.54) is simplified to

mn

n

ii

nmn CdxxXxX

x

x,

1

1

12

2

21123

)1()()(1

δπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=−

∏∫=−

. (3.55)

Clearly a half of polynomials )(xX n is decomposable and we have

))12(2

)12(cos()(2

12 ∏

= +−

−=n

kn n

kxxX π . (3.56)

So, if one can find the roots of )(12 xX n+ too, these polynomials will find many applications in numerical analysis such as Gaussian quadrature rules [27, 70].


32

Similarly, the subject holds for the initial vector of the second kind Chebyshev polynomials, i.e. )0,3,1,1(),,,( −−=srqp . Again, if this vector is substituted into (3.8) then

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

==⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++ xSxxUxS nnn 1125

)(1103

21212 , (3.57)

and subsequently the secondary vector is obtained in the form )2,5,1,1(),,,( −−=srqp . Now, by assuming that x=θcos let us define the polynomials

⎪⎩

⎪⎨

⎧

+=−−

=+

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=,12);2,5,1,1(

,2sincos

))2sin((2

)1(

1125

)(

2/

mnifxS

mnifnnxSxY

n

n

nn θθθ

(3.58)

satisfying the recurrence relation

,,)(,1)(;)(1125

)()( 1011 N∈==⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+= −+ nxxYxYxYCxYxxY nnnn (3.59)

where

)2()1(4

))1(2)()1(1(1125

++−−+−−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

nnnnC

nn

n , (3.59.1)

and having the orthogonality relation

mn

n

ii

nmn dxxWCdxxYxYxW ,

1

11

1

1 1125

1125

)1()()(1125

δ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

∫∏∫−=−

, (3.60)

where

8

)23,

23(1

1125 1

1

221

1

π==−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

∫∫−−

BdxxxdxxW . (3.61)

The relation (3.61) simplifies (3.60) as

mn

n

ii

nmn CdxxYxYxx ,

1

1

1

22

81125

)1()()(1 δπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=− ∏∫=−

. (3.62)

Similar to previous case, )(2 xY n is decomposable as

)22

cos()(2

12 ∏

= +−=

n

kn n

kxxY π . (3.63)

Here it should be added that there are two other sequences of half-trigonometric polynomials that are not orthogonal, but can be shown in terms of MCSOP. These sequences are defined as


33

⎪⎩

⎪⎨⎧

+=

=−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−

− .12)(,2);2,1,1,1(

1121

1 mnifxTxmnifxS

xSn

nn (3.64)

⎪⎩

⎪⎨⎧

+=

=−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−

− .12)(,2);2,1,1,1(

1121

1 mnifxUxmnifxS

xSn

nn (3.65)

However since we have

+∞=−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−∫∫−−

1

122

1

1 11121

xxdxdxxW , (3.66)

+∞=−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−∫∫−−

1

12

21

1

11121

dxx

xdxxW , (3.67)

they cannot fall into the half-trigonometric orthogonal polynomials category. Thus we can generally consider the following table showing some properties of the first kind to sixth kind of monic Chebyshev polynomials orthogonal on ]1,1[− . Table 1: Representations of Chebyshev polynomials by );,,,( xsrqpSn

Type Notation Definition Weight First Kind

)(xTn ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

xSn 1101

211

x−

Second Kind

)(xU n ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

xSn 1103

21 x−

Third Kind

)(xVn ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−−

21

1123

2 2xS n

n

xx

−+

11

Fourth Kind

)(xWn ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−−

21

1123

2 2xS n

n

xx

+−

11

Fifth Kind

)(xX n ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

xSn 1123

2

2

1 xx−

Sixth Kind

)(xYn ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

xSn 1125

22 1 xx −

Remark 1. According to (3.41.4), the initial vector of the monic Chebyshev polynomials )(xU n is )0,3,1,1(),,,( −−=srqp . If this vector is replaced in (3.10.1), a very simple case of three-term recurrence relation (3.10) with 4/1)0,3,1,1( −=−−nC is derived. A system of monic orthogonal polynomials that satisfies the relation


34

,Z,1)(;)()()()( 011+

−+ ∈=−−= nxPxPxPxxP nnnnn βα (3.68) and has the property

2

lim and lim 0 ; , ,4n nn n

ba a bα β→∞ →∞

= = > ∈R (3.69)

is said to belong to the class ),( baN . The monic polynomials corresponding to the

conditions (3.69) are perturbations of )(b

axUbx nn −

→ . Now since

3 2 5 21 1lim and lim ,1 1 1 14 4n nn n

C C→∞ →∞

− −⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

(3.70)

the defined polynomials )(xX n and )(xYn belong to the class )1,0(N . 3.8. Second subclass, Generalized Hermite polynomials (GHP) The GHP were first introduced by Szego who gave a second order differential equation for these polynomials [70, problem 25] as almost the same form as we will give in this section. These polynomials can be characterized by using a direct relationship between them and Laguerre orthogonal polynomials [27]. Of course, there are some other approaches for this matter, see e.g. [32]. Because of this, it is better to only point to the main properties of GHP in terms of the obtained properties of );,,,( xsrqpSn . 3.8.1. Initial vector )2,2,1,0(),,,( asrqp −= . (3.71) 3.8.2. Explicit form of monic GHP (3.72)

∑ ∏=

−+−

=+ ⎟⎟

⎠

⎞⎜⎜⎝

⎛++−+

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−+−=⎟⎟

⎠

⎞⎜⎜⎝

⎛− ]2/[

0

2)1(]2/[

01]

2[

]2

[

22)1(22]2/[

)2)1(1()1(

1022 n

k

knkn

inn

nn

n xaik

nax

aS

3.8.3. Recurrence relation of monic GHP

,,)(,1)(;)(10

22)()( 1011 N∈==⎟⎟

⎠

⎞⎜⎜⎝

⎛−+= −+ nxxSxSxS

aCxSxxS nnnn (3.73)

where

ana

Cn

n 2)1(1

21

1022 −−

−−=⎟⎟⎠

⎞⎜⎜⎝

⎛− . (3.73.1)

3.8.4. Orthogonality relation (3.74)

mn

n

i

inmn

xa aiadxxa

Sxa

Sex ,1

2 )21())1(1(

21

1022

10222

δ+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛−∏∫=

∞

∞−

− .


35

The above relation shows that orthogonality is valid for 021 >+ /a and 1)1( 2 =− a . 3.8.5. Differential equation: );2,2,1,0()( xaSx nn −=Φ . ( ) 0)()1)1((2)()(2)( 222 =Φ−−++Φ′−−Φ ′′ xaxnxaxxxx n

nnn . (3.75)

Finally note that since the leading coefficient of Hermite polynomials )(xH n is n2 , the following equality holds

⎟⎟⎠

⎞⎜⎜⎝

⎛−= xSxH n

nn 10

022)( . (3.76)

3.9. Third subclass, A finite class of symmetric orthogonal polynomials with weight function ba xx −− + )1( 22 on ),( ∞−∞ [1] According to Favard theorem, if the condition 0),,,(1 >− + srqpCn holds only for a finite number of positive integers, i.e. for Nn ,...,1,0= then the related polynomials class would be finitely orthogonal. This note helps us obtain some new classes of finite symmetric orthogonal polynomials, which are special sub-cases of );,,,( xsrqpSn and can be indicated by it directly. To derive the first finite sub-class, we should first compute the logarithmic derivative of the weight function ba xxxW −− += )1()( 22 as

)(

)()()(

2

2

122

xxaxba

xWxW

+−+−

=′

. (3.77)

If the above fraction is compared with the logarithmic derivative of the main weight function );,,,( xsrqpW then we get ),,,(),,,( abasrqp 222211 −+−−= , (3.78) which is in fact the initial vector corresponding to the first finite sub-class of symmetric orthogonal polynomials. Hence, if (3.78) is replaced in (3.7) the explicit form of polynomials is derived as (3.79)

∑ ∏=

−

+−

=+

+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−+−+−−+−++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+−− ]2/[

0

2

)1(]2

[

01

1

22)1(2222)1(]2/[22]

2[

1,12,222 n

k

kn

kn

in

n

n xai

bani

k

nx

abaS

Replacing (3.78) in the main recurrence relation (3.10) also gives (3.80)

,,)(,1)(;)(1,12,222

)()( 1011 N∈==⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−+= −+ nxxSxSxS

abaCxSxxS nnnn

where


36

)1222)(1222(

)2))1(1(()))1(1((1,12,222

−−−+−−−−−−−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−banban

bananabaC

nn

n . (3.80.1)

Therefore, the orthogonality relation

,)(

)2/1()2/1(1,12,222

)1(

1,12,222

1,12,222

)1(

,1

2

2

mn

n

ii

n

mnb

a

baababa

C

dxxaba

Sxaba

Sx

x

δΓ

+−Γ−+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−

+

∏

∫

=

∞

∞−

−

(3.81)

is valid iff 0and2/1;2/1;;0

112222

1 ><>+∈∀>⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−− +

+ baabZnaba

Cn.

Here is a good position to explain how we can determine the parameters conditions to be established the orthogonality property (3.81). For this purpose, there is an interesting technique. Let us consider the differential equation of polynomials (3.79) using the initial vector (3.78) and subsequently the main equation (3.2) as (3.82)

( ) 011122121 2222 =Φ−−++−++Φ′+−+−Φ ′′+ )())(())(()())(()()( xaxnbanxaxbaxxxx nn

nn

If the above equation is written in self-adjoint form, then according to theorem 1 the following term must vanish, i.e. 0))]()()()(()1([ 122 =ΦΦ′−ΦΦ′+ ∞

∞−+−− xxxxxx nmmn

ba . (3.83) On the other hand, since )(xnΦ is a polynomial of degree n, so 1))()()()(deg(max −+=ΦΦ′−ΦΦ′ mnxxxx nmmn . (3.84) Consequently from (3.83) and (3.84) we must have 01)1(22 ≤−+++−+− mnba , (3.85) which gives the following result

⎩⎨⎧

><≤+++−−

.0,2/1,0122

bamnba

(3.86)

In other words, (3.81) holds if and only if 21,...,1,0 /baNm,n −+≤= in which

},max{ nmN = , 2/1<a and 0>b .


37

Corollary 1. The finite polynomial set { } Nnnn xabaS ==−+−− 0);2,222,1,1( is orthogonal

with respect to the weight function ba xx −− + )1( 22 on ),( ∞−∞ if and only if 2/1−+≤ baN , 2/1<a and 0>b .

We add that the explained technique can similarly be applied for the first and second sub-classes of );,,,( xsrqpSn i.e. GUP and GHP. 3.10. Fourth subclass, A finite class of symmetric orthogonal polynomials with weight function

212 xaex /−− on ),( ∞−∞ [1] Similar to the first finite sub-class, one can compute the logarithmic derivative of the given weight function to get respectively 3.10.1. Initial vector )2,22,0,1(),,,( +−= asrqp . (3.87) 3.10.2. Explicit form of polynomials

∑ ∏=

−

+−

=

+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+−++

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ +− ]2/[

0

2

)1(]2

[

0

1

222)1(]2/[22]2/[

01222 n

k

kn

kn

i

n

n xanik

nx

aS . (3.88)

3.10.3. Recurrence relation of monic polynomials

,,)(,1)(;)(01222

)()( 1011 N∈==⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+= −+ nxxSxSxS

aCxSxxS nnnn (3.89)

where

)122)(122(

2)()1(201222

−−+−−−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +−anan

aanaC

n

n . (3.89.1)

3.10.4. Orthogonality relation (3.90)

.)21(

01222

)1(

01222

01222

,1

12 2

mn

n

ii

n

mnxa

aa

C

dxxa

Sxa

Sex

δ−Γ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

∏

∫

=

∞

∞−

−−

But (3.90) is valid if

0))]()()()()(1exp([ 222 =ΦΦ′−ΦΦ′− ∞

∞−− xxxx

xx nmmn

a , (3.91)

or equivalently

},max{;210122 nmNaNmna =−≤⇔≤−++− . (3.92)


38

Corollary 2. The finite polynomial set { } Nnnn xaS ==+− 0);2,22,0,1( is orthogonal with

respect to the weight function 212 xaex /−− on ),( ∞−∞ if and only if 2/1−≤ aN .

3.10.5. Differential equation: );2,22,0,1()( xaSx nn +−=Φ .

( ) 0)()1(1)21()()1)1((2)( 224 =Φ−−+−+−Φ′+−+Φ ′′ xxannxxaxxx nn

nn . (3.93) 3.11. A unified approach for the classification of MCSOP As we observed in the previous sections, each four introduced sub-classes of symmetric orthogonal polynomials were determined by );,,,( xsrqpSn directly and it was only sufficient to obtain the initial vector corresponding to them. On the other hand, it is clear that the orthogonality interval of the sub-classes, other than first one (GUP), are all infinite, i.e. ),( ∞−∞ . Hence, applying a linear transformation, say vwtx += , preserves the orthogonality interval. For example, if wvwtx /+= in (3.74), the orthogonality interval will not change and consequently a more extensive class with weight function )2exp()( 2222 vttwvtw a −−+ will be derived on the interval ),( ∞−∞ . However, it is important to know that the latter weight corresponds to the class of orthogonal polynomials )/;,,,( wvwxsrqpSn + . Therefore, only by having the initial vector we can have access to all other standard properties and design a unified approach for the cases that may occur. In other words, if one can obtain the parameters

),,,( srqp by referring to the initial data, such as a given three term recurrence relation, a given weight function and so on, then all other properties will be derived straightforwardly. Here, we consider two special sub-cases of this approach. 3.11.1. How to find the parameters srqp ,,, if a special case of the main weight function );,,,( xsrqpW is given? By referring to third and fourth orthogonal sub-classes, it is easy to find out that the best way for deriving srqp ,,, is to compute the logarithmic derivative )(/)( xWxW ′ and then equate the pattern with (3.14). The following examples will clarify this matter. Example 1. The weight functions

∞<<∞−+++=

∞<<∞−−+−=

≤≤−+−=

− xxxxxWiii

xxxxxxWii

xxxxWi

;)122()12()()

;))21(2exp()1816()()

22;4)()

5223

22

461

and their orthogonality intervals are given. Find other standard properties such as explicit form of polynomials, orthogonality relation and …. To solve the problem, it is only sufficient to find the initial vector corresponding to each given weight functions. For this purpose, if the logarithmic derivative of the first weight is computed, then we have


39

)16,8,4,1(),,,()(

)2()4(

166)()() 2

2

2

2

1

1 −−=⇒+

+−=

−−

=′

srqpqpxx

sxprxxx

xWxWi .

Hence the related monic polynomials are { }∞=−− 0);16,8,4,1( nn xS . Note that these polynomials are orthogonal with respect to )4( 24 xx − on ]2,2[− for every n and it is not necessary to know that they are the same as shifted GUP on ]2,2[− , because they can explicitly and independently be expressed by );,,,( xsrqpSn . But, for the weight function )(2 xW it differs somewhat and we have

)2,2,1,0(),,,(22)()(

)()4

12(41))

212(exp()

212(4)()

2

*2

*2

*2

22

41

2241

2

2

−=⇒+−

=′

→

==+

⇒−−−= −−

srqpt

ttWtW

tWettWexxexWii t

Hence the related orthogonal polynomials are as ∞

=⎭⎬⎫

⎩⎨⎧ −−

0

)212;2,2,1,0(

nn xS .

)2,6,1,1(),,,()1(28

)()(

)()1(

)2

1(2))12(1(

)12(2)()

2

2

*3

*3

*352

2

35

52

25

3

−=⇒++−

=′

→

=+

=−

⇒+++

= −

srqpttt

tWtW

tWt

ttWx

xxWiii

Consequently, by noting the orthogonality relation of the third sub-class of MCSOP, the finite set { } 3

0)12;2,6,1,1( ==+− n

nn xS is orthogonal with respect to )(3 xW on ),( ∞−∞ and the upper bound of this set has been determined based on the condition

2/1−+≤ baN for 5=b and 1=a . 3.11.2. How to find the parameters srqp ,,, if a special case of the main three-term recurrence equation (18) is given? In general, there are two ways to determine the special case of );,,,( xsrqpS n corresponding to a given three-term recurrence equation. The first way is to directly compare the given recurrence equation with (3.10). This leads to a system of polynomial equations in terms of the four parameters srqp ,,, . In [46] a similar method is applied for the first kind of classical orthogonal polynomials. The second way is to equate the first four terms of each two recurrence equations together, which leads to a polynomial system with 4 equations and 4 unknowns srqp and,, respectively. The following example will better illustrate these methods. Example 2. If the recurrence equation

xxSxSxSnnnxSxxS n

n

nn ==−−−−+

−= −+ )(and1)(;)()132)(112(

)6()1(62)()( 1011 ,


40

is given, then find its explicit polynomial solution, differential equation of polynomials, the related weight function and finally orthogonality relation of polynomials. Solution. If the above recurrence equation is directly compared with the main equation (3.10) and subsequently (3.10.1), then one can obtain the values

)2,10,0,1(),,,( −=srqp . Hence, the explicit solution of above recurrence equation is the polynomials );2,10,0,1( xSn − and therefore their differential equation is found as ( ) 0)()1(1)11()()210()( 224 =Φ−−+−−Φ′+−+Φ ′′ xxnnxxxxx n

nnn .

Moreover, by replacing the initial vector in the main weight function (3.14) as

21

123

2

)212exp(01210

xexdxxxxW

−−=

+−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−∫ ,

one can find out that the related polynomials are a particular case of the fourth introduced sub-class. Hence we have

5,)2

11(01210

)1(01210

01210

,1

112 2 ≤⇔Γ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛−∏∫=

=

∞

∞−

−− nmCdxxSxSex mn

ni

ii

nmn

x δ

Second method. If the given recurrence relation is only expanded for n = 2, 3, 4, 5 and then equated with (3.9.1), the following system will be derived

2 2 32 3 2 1

4 24 3 2

2 2 4 3( ) ; ( ) ( ) ( )9 9 63 3

18 3 (3 )( ) 5 4( ) ( ) ( ) 2 and 2 .35 5 (5 )(3 ) 7 3

q s q s q sS x x x S x x S x S x x xp r p r p r

q s q s q s q sS x x S x S x x xp r p r p r p r

+ + += − = + ⇒ = − = − = +

+ + ++ + + +

= − = + + = −+ + + +

Solving this system again results that )2,10,0,1(),,,( −=srqp . In conclusion, by using the extended Sturm-Liouville theorem for symmetric functions explained in chapter 2, one can define a generic second order differential equation having a main polynomial solution with four free parameters. This solution satisfies a generic orthogonality relation whose weight function corresponds to an analogue of Pearson distributions. In other words, there are four special cases of the dual symmetric distributions family that can respectively be considered as the weight functions of four introduced sub-classes of MCSOP. In this way, the following table shows the explicit forms of the mentioned sub-classes in terms of );,,,( xsrqpSn as well as their weight functions, kind of polynomials (finite or infinite), orthogonality interval and finally constraint on the parameters.


41

Table 2: Four special sub-cases of );,,,( xsrqpSn Definition Weight

function Interval & Kind Parameters

Constraint

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−x

abaSn 1,1

2,222

ba xx )1( 22 −

]1,1[− , Infinite 1

2/1

−>

−>

b

a

⎟⎟⎠

⎞⎜⎜⎝

⎛−x

aSn 1,0

2,2

22 xaex −

),( ∞−∞ ,Infinite 2

1−>a

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−x

abaSn 1,1

2,222 b

a

xx

)1( 2

2

+

−

),( ∞−∞ , Finite

⎩⎨⎧

><−+≤

0,2/12/1

babaN

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−x

aSn 0,1

2,22 2

12 xaex

−−

),( ∞−∞ , Finite 2

1−≤ aN

Finally we repeat that since all weights in above table are even functions, the condition

1)1( 2 =− a must always be satisfied by noting the constraint of parameters for each introduced weight functions. Therefore, they can also be considered in the forms

ba xx )1(|| 22 − , 22|| xa ex − , ba xx −− + )1(|| 22 and

2/12|| xa ex −− respectively.


42

Chapter 4 Finite classical orthogonal polynomials 4.1. Introduction It is well known that the classical orthogonal polynomials of Jacobi, Laguerre and Hermite are infinitely orthogonal and satisfy a second order differential equation of the form 0)())1(()()()()( 2 =+−−′++′′++ xyDAnnxyEDxxyCBxAx nnn in which EDCBA and,,, are parameters independent of n. In this chapter, we intend to study three other sequences of hypergeometric polynomials in detail, which are special solutions of above equation and are finitely orthogonal with respect to three specific weight functions. These classes have respectively relation with the Jacobi and Laguerre polynomials. In particular, the second class is directly related to the generalized Bessel polynomials and consequently Laguerre polynomials. Let us start with the first finite case. 4.2. First finite class of hypergeometric orthogonal polynomials Consider the differential equation of Sturm-Liouville type ( ) ( ) ( ) ( ) ( ) 0 ,n n n nx y x x y x y xσ τ λ′′ ′+ − = (4.1) where CBxAxx ++= 2)(σ and EDxx +=)(τ are polynomials independent of n and

nDAnnn +−= )1(λ is the eigenvalue parameter depending on ,...2,1,0=n . The Jacobi orthogonal polynomials for 21)( xx −=σ , )()2()( αββατ −+++−= xx , Laguerre for xx =)(σ , xx −+= 1)( ατ and finally Hermit for 1)( =xσ , xx 2)( −=τ are three known types of polynomial solutions of equation (4.1). But, there are three other types of polynomial solutions that are finitely orthogonal. The first finite class is defined when xxx += 2)(σ , )1()2()( qxpx ++−=τ in (4.1). So, substituting these values in (4.1) gives the following differential equation

0)()1(()())2(()()( 2 =−+−′+−+′′+ xypnnxyqxpxyxx nnn . (4.2) By applying the Frobenius method, an explicit polynomial solution for the equation (4.2) will be derived as

Finite classical orthogonal polynomials

44

∏∑−

=

=

=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛⇔−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−=

1

00

),( )(!

1)()1(

!)1()(k

i

nk

k

knqpn ia

kka

xknnq

knp

nxM . (4.3)

Moreover, one can prove that these polynomials are finitely orthogonal with respect to the weight function )(

1 )1(),;( qpq xxqpxW +−+= on ),0[ ∞ if and only if 1}{max2 +> np and 1−>q . To prove this claim, one should first write the self-

adjoint form of the equation (4.2) as

).()(;)()1()1())()1((

),()1()1())()1((),()(11

)(11

xMxyxyxxpmmxyxx

xyxxpnnxyxxqp

nnmqpq

mqpq

nqpq

nqpq

=+−+=′′+

+−+=′′++−−−+

+−−−+

(4.4)

Then using the Sturm-Liouville theorem, one gets (4.5)

∫∞

+

∞

−+

+

+−=⎥

⎦

⎤⎢⎣

⎡′−′

+ 0

),(),(

01

1

)()()1(

)())()()()(()1(

dxxMxMxxxyxyxyxy

xx qp

mqp

nqp

q

mnnmmnqp

q

λλ

where )1( pnnn −+=λ . As we applied in chapter 3, since max deg 1)}()()()({ −+=′−′ nmxyxyxyxy nmmn so if 1−>q , },max{,12 nmNNp =+> , the left side of (4.5) tends to zero and we will have

⎩⎨⎧

=−>+>≠

⇔=+∫

∞

+ }.,max{,1,12,

0)()()1(0

),(),(

nmNqNpnm

dxxMxMx

x qpm

qpnqp

q

(4.6)

Corollary 1. The finite set 2/)1(

0),(

0)1,12( )}({)}({ −<

==−>+> = pN

nqp

nNn

qNpn xMxM must be

orthogonal with respect to the weight function )(1 )1(),;( qpq xxqpxW +−+= on ),0[ ∞ .

As the differential equation (4.2) shows, the polynomials (4.3) have a direct relation with the Jacobi polynomials. Hence, by referring to the Rodrigues representation of Jacobi polynomials [16], the Rodrigues formula of the defined polynomials (4.3) can be indicated as

,...2,1,0;))1(()1()1()(),( =++

−=−−++

ndx

xxdxxxM n

qpnqnn

q

qpnqp

n (4.7)

One of the advantages of above representation is to calculate the norm square value of the polynomials. For this purpose, if (4.7) is replaced in the norm 2 relation

∫∫∞ −−+∞

+

+−=

+ 0

),(

0

2),( ))1(()()1())(()1(

dxdx

xxdxMdxxMx

xn

qpnqnnqp

nnqp

nqp

q

, (4.8)

then integration by parts yields


45

∫∫∞

−−+∞ −−+

++−+−

=+

−00

),( )1())!12(())!1((!))1(()()1( dxxx

npnpndx

dxxxdxM qpnqn

n

qpnqnnqp

nn . (4.9)

On the other hand

))!1(()!())!22(()1(

0 +−+++−

=+∫∞

−−+

nqpnqnpdxxx qpnqn . (4.10)

So, we have

)!1()12(

)!()!1(!))(()1(0

2),(

−−+−−+−−

=+∫

∞

+ nqpnpnqnpndxxM

xx qp

nqp

q

. (4.11)

The foresaid relation shows that 12 +> np is a necessary condition for the orthogonality of the polynomials )(),( xM qp

n . Therefore Corollary 2 (Orthogonality relation)

mnqp

mqp

nqp

q

nqpnpnqnpndxxMxM

xx

,0

),(),(

)!1()12()!()!1(!)()(

)1(δ

−−+−−+−−

=+∫

∞

+

if and only if 1,2

1,...,2,1,0, −>−

<= qpNnm .

For instance, the polynomial set 1000

0

1000

)0,202( })(201

!)1{()}({ =

=

== −⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −−= ∑ n

knk

k

nnn x

kn

kn

nxM is

finitely orthogonal with respect to the weight function 2021 )1()0,202,( −+= xxW on

),0[ ∞ and

100,2201

)!()()()1( ,

2

0

)0,202()0,202(202 ≤⇔−

=+∫∞

− nmn

ndxxMxMx mnmn δ . (4.12)

4.3. Second finite class of hypergeometric orthogonal polynomials: If 2)( xx =σ and

1)2()( +−= xpxτ in (4.1) The second case is directly related to the generalized Bessel polynomials. The Bessel polynomials for 2)( xx =σ , 22)( += xxτ were first studied by [51] in 1949. They established the complex orthogonality of Bessel polynomials on the unit circle ( the real orthogonalizing weights of these polynomials have recently given in [35] ). Then, in 1973, the generalized Bessel polynomials were reviewed by [38]. They are special solutions of equation (4.1) for 2)( xx =σ , 2)2()( ++= xx ατ ; ,...3,2 −−≠α and are indicated by

∑= ++Γ

+++Γ⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

k

knn

xnkn

kn

xB0

)( )2

()12()1(2)(

ααα , (4.13)

where )()( xBn

α denotes the monic Bessel polynomial. Now, without loss of generality, let us consider the following differential equation


46

0)()1(()()1)2(()(2 =−+−′+−+′′ xypnnxyxpxyx nnn . (4.14) By applying the Frobenius method one can show that

∑=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−−=

n

k

knpn x

knn

knp

kxN0

)( ,)()1(

!)1()( (4.15)

is a polynomial solution of (4.14). Clearly these polynomials are related to the generalized Bessel polynomials as

)2(

12

!)( )()( xBn

npnxN pnn

pn

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−= . (4.16)

If the equation (4.14) is written in self-adjoint form (4.17)

),()(;)()/1exp()1())()/1exp((

),()/1exp()1())()/1exp((),(2

2

xMxyxyxxpmmxyxx

xyxxpnnxyxxqp

nnmp

mp

np

np

=−−+=′′−

−−+=′′−−+−

−+−

then multiplying by )(xym , )(xyn in (4.17) respectively and subtracting them, we get

∫∞ −−

∞−+− −=⎥

⎦

⎤⎢⎣

⎡′−′

0

)()(1

0

12 )()()())()()()(( dxxNxNexxyxyxyxyex p

mp

nxp

mnnmmnxp λλ , (4.18)

where )1( pnnn −+=λ . Again, since max deg 1)}()()()({ −+=′−′ nmxyxyxyxy nmmn the condition },max{,12 nmNNp =+> causes the left side of (4.18) to tend to zero and therefore

⎩⎨⎧

=+>≠

⇔=∫∞

−−

},max{12,

0)()(0

)()(1

nmNNpnm

dxxNxNex pm

pn

xp (4.19)

Corollary 3. The finite set 2/)1(

0)(

0)12( )}({)}({ −<

==+> = pN

np

nNn

Npn xNxN must be an

orthogonal one with respect to the weight function )/1exp();(2 xxpxW p −= − on ),0[ ∞ . For instance, the polynomial set 100

0)202( )}({ =nn xN must be finitely orthogonal with

respect to the weight function )/1exp()202,( 2022 xxxW −= − on ),0[ ∞ .

But the Rodrigues representation of classical orthogonal polynomials can be denoted by

n

nnn

dxxWxPd

xWC ))()((

)( where )(xW is a weight function, nC is a constant value and

)(xP is an appropriate polynomial. So, if nn

p CxxxW )1(,)/1exp()( −=−= − and 2)( xxP = are supposed in the mentioned formula, then


47

,...2,1,0;)()1()(/121

)( =−=−+−

ndx

exdexxN n

xnpnxpnp

n (4.20)

To complete the orthogonality relation of )()( xN p

n , one can use the above formula and compute the norm square value of the polynomials. Accordingly, if (4.20) is replaced in the relation

∫∫∞ −+−∞ −− −=0

12

)(

0

2)(1 )()()1())(( dx

dxexdxNdxxNex n

xnpnp

nnp

nxp , (4.21)

then integration by parts yields

∫∫∞ −+−

∞ −+−

+−+−

=−0

12

0

12

)(

))!12(())!1((!)()()1( dxex

npnpndx

dxexdxN xnp

n

xnpnp

nn . (4.22)

Here one should note that if 1>p then

)1();(

0

1

02 −Γ== ∫∫

∞ −−∞

pdxexdxpxW xp . (4.23)

Therefore, (4.22) is simplified as

)12())!1((!

))!12(())!1((!

0

12

+−+−

=+−+−

∫∞ −+−

npnpndxex

npnpn xnp (4.24)

and we finally get

)21())!1((!))((

0

2)(1

npnpndxxNex p

nxp

−−+−

=∫∞ −− . (4.25)

Corollary 4. (Orthogonality Relation)

21,...,2,1,0,

)12(())!1((!)()( ,

0

)()(1 −

<=⇔⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−

=∫∞ −− pNnm

npnpndxxNxNex mn

pm

pn

xp δ

4.3.1 A direct relationship between Bessel polynomials and Laguerre polynomials [8] It is interesting to know that there is a direct relationship between Bessel polynomials and Laguerre polynomials. To find this relation, we first consider the generalized Bessel equation

0,0)()()()( 23212 ≠=+′++′′ rxyrxyrxrxyx (4.26)

and suppose that )/1()( xFxxy r= . Therefore )/1()( xyxxF r= and (4.26) is transformed to


48

0)())1(()()22()( 312

2122 =+−+++′−−+−+′′ xFrrrrxrrxFrrxrxxFx . (4.27)

Now if in (4.27)

2

4)1(10)1( 3

211

312 rrr

rrrrr−−±−

=⇒=+−+ , (4.27.1)

then it changes to

0)(2

4)1(1)()4)1(1()( 3

211

2232

1 =−−±−

+′−−−+′′ xFrrr

rxFxrrrxFx ∓ . (4.27.2)

On the other hand, it is known that the general solution of 0)()()()( =−′−+′′ xgbaxgbxcxgx , (4.28) can be indicated by the confluent hypergeometric functions [42]

∑∞

=

==0

11 !)(

)()(

);,()(k

k

k

k

kbx

ca

bxcaFxg . (4.28.1)

So, by comparing the equations (4.27.2) and (4.28) it is concluded that the function

);4)1(1,2

4)1(1()( 2

32

13

211

112

4)1(1 32

11

xrrr

rrrFxxy

rrr

−−−−−

=−−±−

∓∓

(4.29)

satisfies the equation (4.26) if and only if 02 ≠r . Subsequently the monic Bessel polynomials are representable in terms of the Laguerre polynomials and we have

)2(

)!2()!(!)1()( )12()(

xLx

npnpnxB np

nn

np

n−−−

++−

= . (4.30)

In this way, )()( xN p

n can also be represented by the Laguerre polynomials so that

)1(!)( ))12(()(

xLxnxN np

nnp

n+−= . (4.31)

The latter relation is useful to generate a new definite integral for the Laguerre polynomials, because substituting (4.31) into (4.25) yields

( )

!)!(1)(

0

2)(1

npn

pdxxLex p

nxp +

=∫∞

−− . (4.32)


49

4.4. Third finite class: Classical hypergeometric orthogonal polynomials with

weight function )arctanexp())()(( 22

dcxbaxqdcxbax p

++

+++ − on ),( ∞−∞ [9]

As was expressed up to now, from the main equation +∈=+−−′++′′++ ZnxyDAnnxyEDxxyCBxAx nnn ;0)())1(()()()()( 2 one can extract six “infinite” and “finite” sequences of classical orthogonal polynomials. In this part, we study the last finite class of hypergeometric polynomials, which is “finitely” orthogonal with respect to the well-behaved weight function

))/()(arctanexp())()(( 22 dcxbaxqdcxbax p +++++ − on ),( ∞−∞ . The foresaid function can be considered as an important statistical distribution too, because by having its explicit criterion, one can generalize the T sampling distribution and prove that it tends to the Normal distribution just like T- student distribution. Of course, the next chapter is devoted to this subject with more details. However, before studying the last finite case, we should introduce a special sub-case of the equation (4.1) for

21)( xx +=σ and xpx )23()( −=τ and show that the polynomials generated from this

case are finitely orthogonal with respect to the positive measure )

21(2 )1(),(

−−+=

pxpxρ

on the real interval ),( ∞−∞ . Hence, similar to previous cases, we see that the differential equation ,0)()22(()()23()()1( 2 =−+−′−+′′+ xypnnxyxpxyx nnn (4.33) has a polynomial solution in the form [10]

∑=

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=]2/[

0

2)( )2(1

)1(!)(n

k

knkpn x

kkn

knp

nxI . (4.34)

By transforming (4.33) as a Sturm-Liouville equation and applying the technique that was applied for the first and second kind of finite classical orthogonal polynomials we arrive at

⎩⎨⎧

=+>≠

⇔=+∫∞

∞−

−−

},max{1,

0)()()1( )()()21

(2

nmNNpnm

dxxIxIx pm

pn

p (4.35)

On the other hand, since (see [10])

,...2,1,0;))1(()1()122()()2(

)()

21(2

21

2)( =+

+−−−−

=−−

−n

dxxdx

npnp

xI n

pnnp

n

nn

pn (4.36)

we finally get Corollary 5. (Orthogonality Relation)


50

mn

np

mp

n npnpnpnpnppn

dxxIxIpx ,

212)()( )

)12()2/1()()1()22()(2!

()()(),( δπ

ρ−−Γ+−Γ−Γ−−

−ΓΓ=

−∞

∞−∫

If and only if 1,...,2,1,0, −<= pNnm and )2/1(2 )1(),( −−+= pxpxρ . It is important to point out that the weight function of above orthogonality relation corresponds to the usual T student distribution so that we have

.,;)12

;())2/()2/)1((();( N∈∞<<∞−+

Γ+Γ

= nxnn

xnn

nnxT ρπ

(4.37)

Now, we here wish to say that it is not the end of the story of polynomials )()( xI p

n , rather, there is a much more extensive polynomial class which is finitely orthogonal on

),( ∞−∞ and generalizes the sequence )()( xI pn . For this purpose, first we consider the

following polynomials

( )

( ),

))arctanexp()()((

)arctanexp()()()1(),,,;(

22

22),(

n

pnn

pnqpn

dxdcxbaxqdcxbaxd

dcxbaxqdcxbaxdcbaxJ

++

+++×

++

−+++−=

− (4.38)

such that 0det >−=⎥⎦

⎤⎢⎣

⎡bcad

dcba

. After doing some computations on (4.38), the

following differential equation will be derived for the polynomials (4.39)

( ).0)())(21(

)())(1(2)())(1(2)())()((22

2222

=+−+−

′+−+−++−+′′+++

xycapnn

xycdabpbcadqxcapxydcxbax

n

nn

On the other hand, the equation (4.39) can be written in terms of the differential equation of hypergeometric function );,,(12 xcbaF [56]. So, after applying an appropriate change of variable one will reach this result that (4.40)

knk

k

kn

nnqpn

xcdabibcad

bcadnp

iqnpnkF

bcadicdabca

kn

pnbcadicdabdcbaxJ

∑=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

−−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛−++

+⎟⎟⎠

⎞⎜⎜⎝

⎛

×−+−++−=

012

22

),(

)()()(2

222/

)()(

)21())()(()1(),,,;(

in which )(/)()( akaa k Γ+Γ= . This formula is an explicit representation for the defined polynomials (4.38). Furthermore, the polynomials (4.40) have an important linear property. Using the Rodrigues representation (4.38) it can be obtained easily, because if vwtx += is considered in (4.38), then (4.41)


51

( ) ( )

( )

.),,,;(

))arctanexp()()((

)arctanexp()()(/1),,,;(

),(

22

22),(

dcvcwbavawtJwdt

dcvcwtbavawtqdcvcwtbavawtd

dcvcwtbavawtqdcvcwtbavawtwdcbavwtJ

qpn

n

n

pnn

pnqpn

++=

++++

+++++×

++++

−+++++−=+

−

−

For instance, if 0and1 =−= vw in (4.41) then ),,,;()1(),,,;( ),(),( dcbatJdcbatJ qp

nnqp

n −−−=− . (4.41.1) Now, let us consider the differential equation (4.39) in the form of a self-adjoint equation

( )

( ) )()arctanexp()()())(21(

))()arctanexp()()((

2222

122

xydcxbaxqdcxbaxcapnn

xydcxbaxqdcxbax

np

np

++

++++−+=

′′++

+++

−

−

(4.42)

and apply the Sturm-Liouville theorem for it on ),( ∞−∞ to reach (4.43)

( ) ( )

( )∫∞

∞−

−

∞∞−

−

++

+++×

−=′−′++

+++

dxdcbaxJdcbaxJdcxbaxqdcxbax

xyxyxyxydcxbaxqdcxbax

qpm

qpn

p

mnnmmnp

),,,;(),,,;()arctanexp()()(

))]()()()()(arctanexp()()([

),(),(22

122 λλ

where )21( pnnn −+=λ . Since ( ) 1)()()()(degmax −+=′−′ mnxyxyxyxy nmmn is valid in (4.43), if the conditions R∈=+> qdcbanmNNp ,,,,and},max{,2/1 hold, the left side of (4.43) tends to zero and consequently (4.44)

( )

⎩⎨⎧

>−∈=+>≠

⇔

=++

+++∫∞

∞−

−

0,,,,,},max{,2/1,

0),,,;(),,,;()arctanexp()()( ),(),(22

bcadqdcbanmNNpnm

dxdcbaxJdcbaxJdcxbaxqdcxbax qp

mqp

np

RIt just remains to compute the norm square value of the polynomials. To compute the norm 2, let us first replace the Rodrigues representation (4.38) in

( )

( ).

))arctanexp()()((),,,;()1(

)),,,;(()arctanexp()()(

22

),(

2),(22

∫

∫

∞

∞−

−

∞

∞−

−

++

+++−=

++

+++

n

pnn

qpn

n

qpn

pn

dxdcxbaxqdcxbaxd

dcbaxJ

dxdcbaxJdcxbaxqdcxbax

(4.45)


52

Then, by noting that ),,,;(),( dcbaxJ qpn has the orthogonality property, integration by

parts from (4.45) yields

( )

( ) .)arctanexp()()()22(

)2()(!

))arctanexp()()((),,,;()1(

2222

22

),(

∫

∫∞

∞−

−

∞

∞−

−

++

+++−Γ

−Γ+=

++

+++−

dxdcxbaxqdcxbax

npnpcan

dxdcxbaxqdcxbaxd

dcbaxJ

pnn

n

pnn

qpn

n

(4.46)

Now, suppose tdcxbax=

++ in the right hand side of (4.46). This simplifies it as (4.47)

( )

.)sincos()(arctan)exp()1()(

)()arctanexp()()(

2/

2/

2222122222

21222

∫∫

∫

−

−−−+∞

∞−

−−−

−+∞

∞−

−

−−=+−

×−=++

+++

π

π

θ θθθ decabcaddtqtcta

bcaddcxbaxqdcxbax

qnppnpnnp

pnpn

Therefore, the orthogonality property of defined polynomials can be expressed as:

Corollary 6. If ( ) )arctanexp()()(),,,;( 22),(

dcxbaxqdcxbaxdcbaxW pqp

++

+++=− is the

weight function of the polynomials ),,,;(),( dcbaxJ qpn , then we have

.0&,2/1,...,2,1,0,ifonlyandIf

))sincos()22()(

)2()(!(

),,,;(),,,;(),,,;(

,

2/

2/

222122

22

),(),(),(

>−∈−<=

−−Γ−−Γ+

= ∫

∫

−

−−−−

∞

∞−

bcadqpNnm

decanpbcad

npcan

dxdcbaxJdcbaxJdcbaxW

mnqnp

np

n

qpm

qpn

qp

R

δθθθπ

π

θ (4.48)

Note that (4.48) will be simplified more if one takes p2 as a natural number, because it is known that ∫ − θθθ θ deca qm)sincos( is analytically integrable if N∈m . Hence, by knowing that

θθθ θθθθθ iii eiciseicaeicaca =+=−

++

=− − sincos&22

sincos , (4.49)

we have (4.50)

)()2(

)()(2)sincos( 2))2((

2))2((

0

2/

2/

πππ

π

θ θθθqikmqikmmk

k

kkmmqm ee

qikmicaica

km

deca+−−+−=

=

−−

−

−+−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛=− ∑∫

The equality (4.50) implies that (4.50.1)


53

∑∫=

−+−

− +−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=−

m

k

kkmkmmqm

qikmicaica

kmqdeca

2

0

212

2/

2/

2

)22()()(2

)1()2

sinh()1(2)sincos( πθθθπ

π

θ

and (4.50.2)

∑∫+

=

−+−

−

+

+−+−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−=−

12

0

122

2/

2/

12

)212()()(12

)1()2

cosh()1(2)sincos(m

k

kkmkmmqm

qikmicaica

kmqideca πθθθ

π

π

θ

Thus, if p2 is a natural number in (4.48), the norm square value of the polynomials

will explicitly be determined. For instance, set 2

,1,0 mpdc === and N∈m to get

.

))222()(12(

))1(()2/)1)()1(1(()!1(!

))1,0,,;())(arctan(exp())(1(

)1(]2

[

0

22

2212

2),2

(22

∏

∫

+−

=

−−

∞

∞−

−

−−−+−−

−−−−−−−−=

+++

nm

k

qm

qmn

qm

n

m

knmqnm

eeqqnmna

dxbaxJbaxqbax

ππ

(4.51)

Here is a good position to propound a complete example of the above orthogonality property. Suppose the finite set 5

0)2,6( )}1,0,1,2;({ =

=− nnn xJ is given. This set is orthogonal

with respect to the measure ))12arctan(2exp()122()1,0,1,2;( 62)2,6( −+−=− − xxxxW on ),( ∞−∞ and satisfies the orthogonality relation (4.51.1)

mnn

k

n

mn

knn

nndxxJxJxx

x,5

0

2

54)2,6()2,6(

62

))5(1)(211(

)(sinh)!11(!2)1,0,1,2;()1,0,1,2;()122(

))12arctan(2exp( δπ

∏∫ −

=

−∞

∞− −−+−

−=−−

+−−

if and only if 5, ≤nm . 4.5. Application of defined polynomials in functions approximation and numerical integration Usually, the finite sets of orthogonal polynomials are applied to the discrete orthogonal polynomials rather than continuous cases. In other words, there is a main difference equation in the form 0)())()(())()(( =−Δ+∇Δ xyxyxxyx nλτσ , (4.52) with ;)()1()( xyxyxy −+=Δ ;)1()()( −−=∇ xyxyxy CBxAxx ++= 2)(σ ;

EDxx +=)(τ and nDAnnn +−= )1(λ , that contains classical orthogonal polynomials of a discrete variable [57]. For instance, Hahn discrete polynomials [42] NnNxnnFNxQn ,...,2,1,0),1;,1,,1,(),,,( 23 =−+−+++−= αβαβα (4.53)


54

are finitely orthogonal with the known conditions 1, −>βα or βα −<−< NN , and satisfy

.)12()1(!)(

)1()1(!)1()()( ,

1

0mn

nn

NnnN

xmn nNN

nnxQxQ

xNxN

xx

δβαα

βαββα++++−

++++−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ + +

=∑ (4.53.1)

The Krawtchouk polynomials [57], Racah polynomials [57], Dual Hahn discrete polynomials [42] are some further cases that have been classified in the category of finite discrete hypergeometric orthogonal polynomials. According to orthogonal polynomials theory, it is obvious that any arbitrary function )(xf can be expanded in terms of each mentioned polynomials, provided that )(xf satisfies the Dirichlet conditions. Now, we would like to point out that the finite polynomial set

2/1;)},,,;({ 0),( −<=

= pNdcbaxJ Nnn

qpn can similarly be applied for approximating the

function )(xf . For this purpose, it is enough to consider the following finite approximation

0,,2/1;),,,;()(0

),( >−∈−<≅ ∑=

bcadqpNdcbaxJCxfN

n

qpnn R . (4.54)

According to Corollary 6

.)(),,,;()arctanexp())()((

)sincos()2()(!

)22()(

),(22

2/

2/

22222

122

∫

∫∞

∞−

−

−

−−

−−

++

+++×

−−Γ+

−Γ−=

dxxfdcbaxJdcxbaxqdcxbax

decanpcan

npbcadC

qpn

p

qnpn

np

n π

π

θ θθθ(4.55)

Note that the integer 2/1−< pN is the maximum precision degree of the approximation (4.54). For example, in the polynomial set )}1,0,0,1;({ )1,4( xJ n , the maximum precision degree is at most 3. The following example clarifies this matter by applying the Gram-Schmidt orthogonalization process [16]. Example 1. Let us compute )1,0,0,1;(),( xJ qp

n for 1,4 == qp and 3≤n . Therefore }7123642,31020,16,1{)}1,0,0,1;({ 2323

0)1,4( +−−−−−== =

= xxxxxxxJS nnn is a finite

orthogonal polynomial set with respect to the weight function )exp()1()1,0,0,1;( 42)1,4( ArctgxxxW −+= on ),( ∞−∞ and

3,))226(1()27(

)2

sinh2()!7(!)1,0,0,1;()1,0,0,1;(

)1()exp(arctan

,3

0

2

)1,4()1,4(42 ≤⇔

−−+−

−=

+ ∏∫ −

=

∞

∞−

nmknn

nndxxJxJ

xx

mnn

k

mn δ

π

For example by referring to the set S , 2== nm in this relation gives


55

2

sinh32)31020()1(

)exp(arctan 2242

π=−−

+∫∞

∞−

dxxxx

x.

Now, by noting the above orthogonality relation and considering the members of set S , one can approximate a third degree polynomial for )(xf as )7123624()31020()16()( 23

32

210 +−−+−−+−+≅ xxxcxxcxccxf , which eventually yields

.)7123624()()7123624)(exp(arctan)1(

)31020()()31020)(exp(arctan)1(9

)16()()16)(exp(arctan)1(85

)()exp(arctan)1(629)()2

sinh288(

232342

2242

42

42

+−−⎟⎠⎞⎜

⎝⎛ +−−+

+−−⎟⎠⎞⎜

⎝⎛ −−+

+−⎟⎠⎞⎜

⎝⎛ −+

+⎟⎠⎞⎜

⎝⎛ +≅

∫

∫

∫

∫

∞

∞−

−

∞

∞−

−

∞

∞−

−

∞

∞−

−

xxxdxxfxxxxx

xxdxxfxxxx

xdxxfxxx

dxxfxxxfπ

This approximation is exact for any arbitrary polynomial function of degree at most 3. Moreover, note that the monic polynomials set S can be obtained by applying the Gram-Schmidt orthogonalization process if the moments of weight function

)1,0,0,1;()1,4( xW exist on ),( ∞−∞ . In this case we have

∫

∫

∫

∫∞

∞−−

−

∞

∞−−−

−

∞

∞−−

−

∞

∞−−

−

−−

+

+=

+

+=

−==−−=

dxxPxx

dxxPxPxxxC

dxxPxx

dxxPxxxB

BxxPxPxPCxPBxxP

n

nn

n

n

n

n

nnnnn

22

42

2142

21

42

21

42

11021

))()(exp(arctan)1(

)()()exp(arctan)1(;

))()(exp(arctan)1(

))()(exp(arctan)1(

and)(;1)(s.t.)()()()(

After calculating the coefficients nn CB , for 3≤n , the monic orthogonal set S will be derived as

}247

21

23,

203

21,

61,1{ 232 +−−−−−= xxxxxxS .

One of the other advantages of defined polynomials is to estimate a type of definite integrals using Gauss integration theory. By employing the zeros of polynomials

),,,;(),( dcbaxJ qpn as the interpolator points in the Lagrange interpolation, one can

approximate the integrals ∫∞

∞−

−

++

+++ dxxfdcxbaxqdcxbax p )()arctanexp())()(( 22 with

the precision degree 12 −= nN . This subject is clarified by the following example.


56

Example 2. By replacing the zeros of 31020)1,0,0,1;( 2)1,4(2 −−= xxxJ in the Lagrange

interpolation, a two-points approximation will be derived in the form

))20

855()8551()20

855()8551((2

sinh10693

48)()1(

)exp(arctan42

+−+

−+≅

+∫∞

∞−

ffdxxfx

x π

This approximation is precise for 32 ,,,1)( xxxxf = and any linear combination of these elements. Anyway, it should be noted that the maximum precision degree of the

numerical approximation ∫∞

∞−

−

++

+++ dxxfdcxbaxqdcxbax p )()arctanexp())()(( 22 is:

1}21|{max2 −−<= pnnN . For example if ]9,7(∈p then we have 5=N . See also

[6] and [7] in this regard. [6] is applied for the weight function of polynomials )()( xI pn

and [7] for the weight function of )()( xN pn .

4.6. A connection between infinite and finite classical orthogonal polynomials The Rodrigues representation of the finite classical orthogonal polynomials are useful tool to find some limit relations between ),,,;(),( dcbaxJ qp

n and Hermite polynomials and also between )(),( xM qp

n and Laguerre polynomials respectively. For this purpose, the following limits should first be considered

.1)1(lim

,)arctanexp()1(lim)1,0,0,1;(lim

2

2),( 2

=+

=+=

∞→

−−

∞→∞→

n

p

xp

p

qp

p

px

ep

xqp

xp

xW (4.56)

Hence we have (4.57)

).()()1(

))1,0,0,1;()1((

)1,0,0,;()1(lim)1,0,0,1;(lim

22

),(2

2/1),(),(

xHdxede

dxp

xWp

xd

pxWpxJ

nn

xnxn

n

qpnn

qp

n

p

qpnp

=−=

+

×−

=

−+

−∞→∞→

Similarly, this technique can be applied to derive a limit relation between )(),( xM qpn

and Laguerre polynomials. In this sense we have

n

qpnqnnqpqnqp

n dtpttd

ptt

ptM ))/1(()1()1()(),(

−−++− +

+−= . (4.58)

Now, taking limit as ∞→p yields

)(!)1()()1()(lim )(),( tLndt

etdetptM q

nn

n

tqnntqnqp

np−=−=

−+−

∞→. (4.59)


57

At the end of this chapter we can summarize that generally there exist six sequences of classical orthogonal polynomials that are generated by the main differential equation (4.1). Also, the parameters EDCBA and,,, corresponding to each six equations of mentioned sequences determine their characteristics. The following table shows this matter. Table(1) Kind Notation A B C D E 1.Infinite )(),( xPn

βα -1 0 1 2−−− βα βα +− 2.Infinite )()( wxLn

α 0 1 0 w− 1+α 3.Infinite )2/( wvwxH n + 0 0 1 22w− v− 4. Finite ))/((),( xvwM qp

n w v 0 wp )2( +− vq )1( + 5. Finite )()( wxN p

n w 0 0 wp )2( +− 1 6. Finite ),,,;(),( dcbaxJ qp

n *a *b *c *d *e

where .))(1(2)(,))(1(2

,,)(2,*22*

22**22*

cdabpbcadqecapddbccdabbcaa

+−+−=+−=

+=+=+=

Moreover, the following table shows the general properties of these six classes such as shifted orthogonal polynomials and their weight function, kind of polynomials (Finite or Infinite), orthogonality interval, the distribution corresponding to the weight function and finally the conditions of parameters: Table(2) Shifted polynomials

Weight function Kind, Interval, Distribution

Parameters conditions

1. )2

(wvwxH n + ))(exp( 2 xwvx +− Infinite, ),( ∞−∞

Normal

R∈≠∀ vwn ,0, 2*. )()( vwxI p

n + )21(2 ))(1(

−−++

pvwx

Finite, ),( ∞−∞ T Sampling

1max −< pn ,,0 R∈≠ vw

2. ),,,;(),( dcbaxJ qpn ),,,;(),( dcbaxW qp Finite, ),( ∞−∞

Generalized T 2/)1(max −< pn RR ∈∈≠ qvw ,,0

3. )()( wxLnα xwex −α Infinite, ),0[ ∞

Gamma

1,0, −>>∀ αwn

4. )(),( xvwM qp

n )()( qpq vwxx +−+ Finite, ),0[ ∞

F sampling 2/)1(max −< pn

0,0,1 >>−> vwq5. )()( wxN p

n xwpex1−

− Finite, ),0[ ∞ Inverse gamma

2/)1(max −< pn ,0>w

6. )(),( xPnβα βα )1()1( xx +− Infinite, ]1,1[−

Beta 1,1, −>−>∀ βαn


58

Chapter 5 A Generalization of Student’s t-distribution from the Viewpoint of Special Functions 5.1. Introduction Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this chapter, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the Fisher F-distribution has a close relationship with the t-distribution, we also introduce a generalization of the F-distribution and prove that it converges to the chi-square distribution as the number of samples tends to infinity. Hence, we start our discussion again with the Pearson differential equation with a simpler form in comparison with (3.15.1), Sec. 3.3:

)(2 xWcbxax

edxdx

dW++

+= , (5.1)

which is directly connected with classical orthogonal polynomials and defines their weight functions ( )W x [56]. The solution of equation (5.1) can be indicated as

)exp()( 2∫ +++

=⎟⎟⎠

⎞⎜⎜⎝

⎛= dx

cbxaxedxx

cbaed

WxW , (5.2)

where edcba ,,,, are all real parameters. There are several special sub-cases of (5.2). One of them is the Beta distribution, which is usually represented by the integral ∫

C

ba dttLtL ))(())(( 21 , (5.3)

where 1( )L t and 2 ( )L t are linear functions, ba , are complex numbers and C is an appropriate contour [61]. The Euler and Cauchy integrals [18] are two important sub-classes of Beta type integrals, which are often used in applied mathematics. The Euler integral is given by (5.4)

1 1 1( ) ( )( ) ( ) ( ) (Re 0 , Re 0 , 0 , 0) ,( )

bc d c d

a

c dt a t b dt a b c d a bc d

− − + −Γ Γ− − = + > > > >

Γ +∫

A generalization of Student’s t-distribution

60

while the Cauchy integral is represented by the formula

)(1)()()()1(

)()(21 dc

dc badc

dcitbita

dt +−∞

∞−

+ΓΓ−+Γ

=−+∫π

, (5.5)

in which 1 , Re( ) 1 , Re 0i c d a= − + > > and Re 0b > . Note that in both relations

(5.4) and (5.5) ∫∞

−−=Γ0

1)( dxexa xa denotes the Gamma function. The relation (5.5) is a

suitable tool to compute some different looking definite integrals. For this purpose, we use the relation

exp(2 arctan ) ( , , )iqa ib bq a b q

a ib a−⎛ ⎞ = ∈⎜ ⎟+⎝ ⎠

R , (5.6)

which rewrites the complex left hand side in terms of the real right hand side. Consequently we have

2 2( ) ( ) ( ) exp(2 arctan )p iq p iq p tb it b it b t qb

+ −− + = + . (5.7)

Now if (5.7) is substituted, then the integral (5.5) changes towards

2 2 2 11 ( 2 1)( ) exp(2 arctan ) (2 )2 ( ) ( )

p pt pb t q dt bb p iq p iqπ

∞+

−∞

Γ − −+ =

Γ − + Γ − −∫ . (5.8)

The above integral plays a key role to introduce a generalization of the t-distribution. 5.2. A generalization of the t-distribution [4] The Student t-distribution [69, 73] having the probability density function (pdf)

12 ( )

2(( 1) / 2)( , ) (1 ) ( , )( / 2)

mm tT t m t mmm mπ

+−Γ +

= + −∞ < < ∞ ∈Γ

N (5.9)

is perhaps one of the most important distributions in the sampling problems of normal populations. According to a theorem in mathematical statistics, if X and 2S are respectively the mean value and variance of a stochastic sample with the size m of a normal population having the expected value μ and variance 2σ , then the random

variable /

XTS m

μ−= has the probability density function (5.9) with )1( −m degrees of

freedom [73]. This theorem is used in the test of hypotheses and interval estimation theory when the size of the sample is small, for instance less than 30.


61

Now, by using (5.8) one can extend the pdf of the t-distribution. To do this task, we

substitute 1, 1 ,

2t mt b pm

+→ = = − and

2qq → in (5.8) to get

12 1( )

2 2 ( )(1 ) exp( arctan ) 1 1( ) ( )2 2

m mt t m mq dt m iq m iqm mπ∞ + −−

−∞

Γ+ =

+ + + −Γ Γ

∫ . (5.10)

Since the right hand side of (5.10) is an even function with respect to the variable q , we can take a linear combination and get accordingly (5.11)

1 12 ( )1 22

1 2( ) 2 ( )(1 ) exp( arctan ) exp( arctan ) 1 1( ) ( )

2 2

m mm mt t tq q dt m iq m iqm m mλ λ πλ λ

∞ + −−

−∞

+ Γ⎛ ⎞+ + − =⎜ ⎟ + + + −⎝ ⎠ Γ Γ∫ .

The above integral can be used to generalize (5.9) as (5.12)

12 ( )2

1 2 1 211 2

1 1( ) ( )2 2( , , , , ) (1 ) exp( arctan ) exp( arctan )

( ) 2 ( )

m

m

m iq m iqt t tT t m q q qmm m m m

λ λ λ λλ λ π

+−

−

+ + + −Γ Γ ⎛ ⎞= + + −⎜ ⎟+ Γ ⎝ ⎠ where , ,t m−∞ < < ∞ ∈N q is a complex number and 1 2, 0λ λ ≥ . Note that 0, 21 ≥λλ is a necessary condition for (5.12), because the probability density function must always be positive. Also note that the normalizing constant

11 2

1 1( ) ( ) /(( ) 2 ( ) )2 2

mm iq m iq m mλ λ π−+ + + −Γ Γ + Γ

of (5.12) is real, because the corresponding integrand is a real function on ),( ∞−∞ . It is clear that for 0=q in (5.12) the usual t-distribution is derived. Moreover, for 0=q the normalizing constant of distribution (5.12) is equal to the normalizing constant of the t-distribution. This fact can be proved by applying the Legendre duplication formula [18]

)(2

)2

1()2

( 1 zzzz Γ=

+ΓΓ

−

π . (5.13)

But, according to one of the basic theorems in sampling theory, ),( mtT converges to the pdf of the standard normal distribution )1,0,(tN as ∞→m [61, 73], that is lim ( , ) ( ,0,1) .

mT t m N t

→∞= (5.14)


62

Here we intend to show that this matter is also valid for the generalized distribution ),,,,( 21 λλqmtT . To prove this claim, we use the dominated convergence theorem

(DCT) [31] to the real sequence of functions

12 ( )(1) 2

1 2 1 2( , , , ) (1 ) exp( arctan ) exp( arctan )m

mt t tS t q q qm m m

λ λ λ λ+

− ⎛ ⎞= + + −⎜ ⎟⎝ ⎠

. (5.15)

For every m ∈N it is not difficult to see that

(1)1 2 1 2| ( , , , ) | ( ) exp( ) ( )

2mS t q q tπλ λ λ λ≤ + ∈R . (5.16)

On the other hand, we have (5.17)

12 2( )2

1 2 1 2lim (1 ) ( exp( arctan ) exp( arctan )) ( )exp( )2

m

m

t t t tq qm m m

λ λ λ λ+

−

→∞+ + − = + − .

Since the dominated convergence theorem states that if for a continuous and integrable function )(xg we have | ( ) | ( )mf x g x≤ , then

lim ( ) lim ( )b b

m mm ma a

f x dx f x dx→∞ →∞

=∫ ∫ , (5.18)

by considering the limit relation (5.17) we obtain (5.19)

( )( )

1( )2 21 2

1 2 1( )2 21 2

21 2

1

lim(1 / ) exp( arctan( / )) exp( arctan( / ))lim ( , , , , )

lim(1 / ) exp( arctan( / )) exp( arctan( / ))

( )exp( / 2) (

m

mmm

m

t m q t m q t mT t m q

t m q t m q t m dt

t

λ λλ λ

λ λ

λ λ

λ

+−

→∞∞ +→∞ −

→∞−∞

+ + −=

+ + −

+ −=

∫2

22

1 1exp( ) ( ,0,1) .22)exp( / 2)

t N tt dt πλ

∞

−∞

= − =+ −∫

Remark 1. Taking the limit on both sides of (5.11) as ∞→m , the following asymptotic relation is obtained for the Gamma function

(2 1)

( ) ( ) 2lim2 2 1 (2 1) 2xx

x iy x iyx x π− −→∞

Γ + Γ −=

− Γ −. (5.20)

To compute the expected value of the distribution given by the pdf (5.12) it is sufficient to consider the definite integral


63

12 1( )2 2 ( ) (1 ) exp( arctan ) ( )1 1 1( ) ( )

2 2

m mt t m m q mt q dt m iq m iqm mmπ∞ + −−

−∞

Γ+ =

+ + + − −Γ Γ∫ , (5.21)

which gives the expected value of (5.12) as

1 2

1 2

[ ] ( )1

q mE Tm

λ λλ λ

−=

+ − . (5.22)

On the other hand, since 2 [1 / ]E T m+ can easily be computed, after some calculations, we get for the variance measure of (5.12)

2

2 21 22 2

21 2

( 1) [ ] [ ] [ ] ( )( 2)( 1) ( 1)m q m mqVar T E T E Tm m m

λ λλ λ

⎛ ⎞−+ −= − = − ⎜ ⎟⎜ ⎟− − + −⎝ ⎠

. (5.23)

It is valuable to point out that as expected 0=q in (5.22) and (5.23) gives the expected value and variance of the usual t-distribution, respectively. But it is known that the t-distribution has a close relationship with the Fisher F-distribution [56], defined by its pdf (5.24)

/ 2 1 ( )2 2(( ) / 2)( / )( , , ) (1 ) ( , , 0 )

( / 2) ( / 2)

k m kkm k k m kF x m k x x m k xk m m

+− −Γ +

= + ∈ < < ∞Γ Γ

N ,

where 2x t= and 1k = in (5.24). In other words we have )1,,(),( 2 mtFmtT = . (5.25) By referring to the above relation and the fact that the t-distribution was generalized by relation (5.12), it is now natural to generalize the pdf of the F-distribution (5.24) as follows (5.26)

1 ( )2 2

1 2 1 2( , , , , , ) (1 ) ( exp( arctan ) exp( arctan )),k m kk k kF x m k q Bx x q x q x

m m mλ λ λ λ

+− −

= + + −

where

1 ( )2 2

1 20

1 (1 ) ( exp( arctan ) exp( arctan ))k m kk k kx x q x q x dx

B m m mλ λ

∞ +− −

= + + −∫ . (5.26.1)

For 0=q , (5.26) is the usual F-distribution defined in (5.24). According to the following theorem, the generalized function (5.26) converges to a special case of the Gamma distribution [73], defined by

1( , , ) exp( ) ( , 0 , 0 )( )

xG x x xα

αβα β α βα β

−− −

= > < < ∞Γ

. (5.27)


64

Theorem 1. If the Gamma distribution is given by (5.27), then we have

21 2lim ( , , , , , ) ( , , 2)

2 km

kF x m k q G xλ λ α β χ→∞

= = = =

where 2

kχ denotes the pdf of the chi-square distribution. Proof. Let us define the sequence

1 ( )(2) 2 21 2 1 2( , , , , ) (1 ) ( exp( arctan ) exp( arctan )).

k m k

mk k kS x k q x x q x q xm m m

λ λ λ λ+

− −= + + −

It is easy to show that

1(2) 2

1 2 1 2| ( , , , , ) | ( ) exp( ) ( [0, ) , )2

k

mS x k q x q x kπλ λ λ λ−

≤ + ∈ ∞ ∈N , (5.28)

and

1(2) 2

1 2 1 2lim ( , , , , ) ( ) exp( / 2)k

mmS x k q x xλ λ λ λ

−

→∞= + − . (5.29)

Therefore, according to the DCT we have (5.30)

(2) ( / 2) 11 2

1 2(2) ( /2) 1

1 20 0

lim ( , , , , ) exp( / 2)lim ( , , , , , ) ( , ,2)2

lim ( , , , , ) exp( / 2)

kmm

mk

mm

S x k q x x kF x m k q G xS x k q dx x x dx

λ λλ λ

λ λ

−→∞

∞ ∞→∞−

→∞

−= = =

−∫ ∫.

Moreover, it is not difficult to show that ),,,,(),,,1,,( 2121

2 λλλλ qmtTqmtF = . (5.31) 5.3. Some particular sub-cases of the generalized t (and F) distribution In this section, we intend to study some symmetric and asymmetric sub-cases of the generalized distributions (5.12) and (5.26). 5.3.1. A symmetric generalization of the t-distribution, the case q ib= and

1 2 1/ 2λ λ= = If the special case 1 2and 1/ 2q ib λ λ= = = is considered in (5.12), then (5.32)

12 ( )2

1

1 1( ) ( )1 1 2 2( , , , , ) ( , , ) (1 ) cos( arctan )2 2 2 ( )

m

S m

m b m bt tT t m ib T t m b bmm m mπ

+−

−

+ + + −Γ Γ

= = +Γ

is a symmetric generalization of the ordinary t-distribution in which 11 ≤≤− b .


65

The usual pdf of the t-distribution is obviously derived by 0=b in (5.32). Note that according to the Legendre duplication formula we will reach the normalizing constant of the t-distribution if 0=b is considered in (5.32). In other words, we have

)2/()2/)1((

)(2)2/)1((0

1

2

mmm

mmmb

m Γ+Γ

=Γ

+Γ⇒=

− ππ. (5.33)

Also note that the parameter b in the generalized distribution (5.32) must belong to [-1,1], because the probability density function must always be positive and therefore

we ought to have cos( arctan( / )) 0b t m ≥ . On the other hand, since for 22πθπ

≤≤−

we have cos 0θ ≥ , therefore to prove cos( arctan( / )) 0b t m ≥ it is sufficient to show that

1 1 arctan [ , ] ( , )2 2

tb b t mm

π π− ≤ ≤ ⇔ ⊆ − ∈ ∈R N . (5.34)

For this purpose, let us define the sequence ( ) arctanmtU tm

= to get (5.35)

21( ) ( ) /(1 ) 0 [min ( ), max ( )] [ ( ), ( )] [ , ]2 2m m m m m

tU t U t U t U Umm

π π′ = + > ⇒ = −∞ ∞ = − .

Now if we demand the sequence ( ) arctanmtbU t bm

= to belong to ]2

,2

[ ππ− , it is

clear that we must have 1|| ≤b , which proves (5.34). The following figures clarify this matter for ]1,1[−∈b and ]1,1[−∉b in the interval (-10,10).

Figure 1: 4,2/1 == mb Figure 2: 4,3 == mb Fig. 1 shows the pdf )2/1,4,(tTS with normalizing constant 128/235 and Fig. 2 shows the non-positive function ( , 4,3)ST t = 2 5/ 2(4 / )(1 / 4) cos(3arctan( / 2))t tπ −+ in the interval ( 10,10).− As the above figures show, the generalized distribution (5.32) is symmetric, i.e. ( , , ) ( , , ) ( ) .S ST t m b T t m b t− = ∈R (5.36)


66

Moreover, according to (5.22) and (5.23) the expected value and variance of distribution (5.32) take the forms

)2)(1()1(][,0][

2

−−−−

==mm

bmmtVartE . (5.37)

Clearly 0=b in these relations gives the expected value and variance of the t-distribution. Theorem 2. ),,( qmtTS converges to )1,0,(tN as ∞→m .

Proof. If the sequence 12 ( )(3) 2( , ) cos( arctan )(1 )

m

mt tS t b b

mm

+−

= + is considered, then one

can show that

12 ( )(3) 2| ( , ) | | cos( arctan ) || (1 ) | 1 ( )

m

mt tS t b b t

mm

+−

= + ≤ ∈R . (5.38)

Consequently we have

1( )2 2

1( )2 2

1( )2 2 2

1( )2 22

cos( arctan( / ))(1 / )lim ( , , ) limcos( arctan( / ))(1 / )

lim cos( arctan( / ))(1 / ) exp( / 2)

lim cos( arctan( / ))(1 / ) exp( / 2)

m

S mm m

m

mm

m

b t m t mT t m bb t m t m dt

b t m t m t

b t m t m dt t dt

+−

∞ +→∞ →∞ −

−∞

+−

→∞∞ +

−

→∞−∞ −∞

+=

+

+ −= =

+ −

∫

∫( ,0,1) .N t∞ =

∫

(5.39)

By referring to (5.26), we can now define the generalized F-distribution corresponding to the first given sub-case as follows (5.40)

1 ( )2 2

11 1( , , , , , ) ( , , , ) (1 ) cos( arctan ) ( 1 1)2 2

k m kk kF x m k ib F x m k b B x x b x bm m

+− −

= = + − ≤ ≤

where

1 ( )2 2

0

/ 2/ 2 ( 1) ( 1)

0

1 (1 ) cos( arctan )

2( ) sin cos cos( ) .

k m k

k k m

k kx x b x dxB m m

m b dk

π

θ θ θ θ

∞ +− −

− −

= +

=

∫

∫ (5.40.1)

Theorem 3. ),,,(F1 bkmx converges to the chi-square distribution as ∞→m .

Proof. We define the sequence 1 ( )(4) 2 2( , , ) (1 ) cos( arctan )

k m k

mk kS x k q x x b xm m

+− −

= + to

get

1(4) 2| ( , , ) | ( [0, ) , , | | 1)

k

mS x k b x x k b−

≤ ∈ ∞ ∈ <N . (5.41)


67

Hence, according to DCT we find out that

(4) ( / 2) 1

1(4) ( / 2) 1

0 0

( , , ) exp( / 2)lim ( , , , ) lim ( , , 2)2

( , , ) exp( / 2)

km

m mk

m

S x k b x x kF x m k b G xS x k b dx x x dx

−

∞ ∞→∞ →∞−

−= = =

−∫ ∫. (5.42)

It is not difficult to verify that the generalized distributions ),,( bmtTS and ),,,(1 bkmxF are related to each other as follows

),,(),1,,( 2

1 bmtTbmtF S= . (5.43) Remark 2. Here is a good position to return to previous chapter and remember that the weight function of orthogonal polynomials ( , ) ( )p q

nM x corresponds to the ordinary F-distribution such that if 0=n is considered in (4.11), then an integral is derived that corresponds to the F distribution. 5.3.2. An asymmetric generalization of the t-distribution, the case 02 =λ Again let us come back to the chapter 4 and consider the weight function of finite orthogonal polynomials ( , ) ( ; , , , )p q

nJ x a b c d , i.e. (5.44)

( , ) 2 2( ; , , , ) (( ) ( ) ) exp( arctan ) ( )p q pn

ax bW x a b c d ax b cx d q xcx d

− += + + + −∞ < < ∞

+,

where a, b, c, d, p, q are all real parameters. This function is a sub-case of the Pearson distribution (5.2), because the logarithmic derivative of (5.44) is a rational function. For

convenience, if 1 , 0, 0, 1a b c dm

= = = = and 1 ( )2

mp m+= − ∈N is selected

in (5.44) then

1 12( , ) ( )2 21( ; ,0,0,1) (1 ) exp( arctan ) ( , )

m mq t tW t q m qmm m

+ +− −

= + ∈ ∈N R . (5.45)

On the other hand since

/ 212 ( ) ( 1)2

/ 2

(1 ) exp( arctan ) cos m

q mt tq dt m e dm m

πθ

π

θ θ∞ +

− −

−∞ −

+ =∫ ∫ , (5.46)

we have

2 2/ 2

( 1)( 1) / 2

2 2/ 2

0

1 ( 1)( 1)!( ( )( 1)) ( ( 1) )2cos

( ( 2 1) )

q qmm

q mm

k

m q q e ee d

q m k

π π

πθ

π

θ θ

−

−−⎢ ⎥⎣ ⎦

−

=

+ −− − − + −

=+ − −

∫∏

. (5.47)

Therefore, an asymmetric generalization of the t-distribution may be defined as


68

12 ( )2( , , ) (1 ) exp( arctan ) ( , , )

m

At tT t m q K q t m qm m

+−

= + −∞ < < ∞ ∈ ∈N R (5.48)

where

( 1) / 22 2

0

2 2

( ( 2 1) )

1 ( 1)( 1)!( ( )( 1)) ( ( 1) )2

m

kq qm

m

q m kK

m m q q e eπ π

−⎢ ⎥⎣ ⎦

=−

+ − −=

+ −− − − + −

∏. (5.48.1)

The distribution (5.48) with normalizing constant given by (5.48.1) was defined in [2] based on this particular approach. But we can still modify and simplify it. To do this task, we set 02 =λ in (5.12) to get

12 ( )

21

1 1( ) ( )2 2( , , ) (1 ) exp( arctan ) ,

2 ( )

m

A m

m iq m iqt tT t m q qmm m mπ

+−

−

+ + + −Γ Γ

= +Γ

(5.49)

which is an explicit representation for the asymmetric distribution (5.48). For the latter distribution, we clearly have ( , , ) ( , , ) .A AT t m q T t m q− = − (5.49.1) The asymmetry of distribution (5.48) (or (5.49)) is shown by Fig. 3 and 4 for specific values of q and m .

Figure 3: 4,1 == mq Figure 4: 3,1 == mq According to (5.48) and (5.48.1), the explicit definitions of the two mentioned figures have respectively the forms

Fig. 3: 52 arctan

2 25( , 4,1) (1 )6 cosh( / 2) 4

t

AtT t e

π

−

= +

Fig. 4: 2 arctan

2 35 3( ,3,1) (1 )12 sinh( / 2) 3

t

AtT t e

π−= +

Now, the following statements (A1 to A5) collect the properties of the asymmetric distribution (5.48) (or (5.49)).


69

A1) The expected value and variance of (5.49) are respectively represented by

2

22

)1)(2())1((][,

1][

−−−+

=−

=mmmqmtVar

mmqtE . (5.50)

0=q in these relations gives the expected value and variance of the t-distribution.

A2) ),,( qmtTA converges to )1,0,(tN as ∞→m . The proof is similar to the first case if one chooses 02 =λ and 11 =λ in the defined sequence ),,,( 21

)1( λλqtSm . A3) By the definition (5.26) and considering the case 02 =λ we can define

1 ( )2 2

1 2( , , , , ,0) ( , , , ) (1 ) exp( arctan )

( , , , 0 ) ,

k m kk kF x m k q F x m k q D x x q xm m

q m k x

λ+

− −= = +

∈ ∈ < < ∞R N (5.51)

where (5.51.1)

/ 21 ( ) / 2 ( 1) ( 1)2 2

0 0

1 (1 ) exp( arctan ) 2( ) sin cosk m k

k k m qk k mx x q x dx e dD m m k

πθθ θ θ

∞ +− − − −= + =∫ ∫ .

A4) ),,,(2 qkmxF converges to the chi-square distribution as ∞→m . The proof is similar to the proof of Theorem 1 when 02 =λ and 11 =λ . A5) The distributions ),,,(2 qkmxF and ),,( qmtTA are related to each other by ),,(),1,,( 2

2 qmtTqmtF A= . (5.52) Remark 3. There is another symmetric generalization of the t-distribution when we set

21 λλ = in (5.12). Its pdf is given as (5.53)

12 ( )2

1 1 1 1 1

1 1( ) ( )2 2( , , , , ) ( , , , , ) (1 ) cosh( arctan ) .

2 ( )

m

m

m iq m iqt tT t m q T t m q qmm m m

λ λ λ λπ

+−

−

+ + + −Γ Γ

− = = +Γ

Therefore, at the end of this chapter, we in fact considered the three following particular sub-cases of the general distribution (5.12): a) 1 2and 1/ 2 ; symmetric caseq ib λ λ= = = b) 2 0 ; asymmetric caseλ = c) 1 2 ; symmetric caseλ λ=


70

Chapter 6 A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it 6.1. Introduction In this chapter, we will present a generic formula for the polynomial solutions of the well-known differential equation of hypergeometric type 0)())1(()()()()( 2 =−+−′++′′++ xyandnxyedxxycbxax nnn , and show that all the three infinite classical orthogonal polynomial families as well as three finite orthogonal polynomial families, investigated in chapter 4, can be identified as special cases of this derived polynomial sequence [3]. We will also present some general properties of the mentioned sequence. For this purpose, we should reconsider the differential equation 0)()()()()( =−′+′′ xyxyxxyx nnnn λτσ , (6.1) in which, as before, cbxaxx ++= 2)(σ is a polynomial of degree at most 2,

edxx +=)(τ is a polynomial of degree 1 and ndannn +−= )1(λ is the eigenvalue parameter depending on ,...2,1,0=n , and suppose that the polynomial solution of (6.1) is denoted by

⎟⎟⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn.

So far extensive research has been done on equation (6.1) and its polynomial solutions. In 1929 Bochner [21] classified the polynomial solutions of (6.1) and showed that the only polynomial systems up to a linear change of variable arising as eigenfunctions of the differential equation (6.1) are (see also [14]) Jacobi polynomials ( ),...}2,1{1,,,)}({ 0

),( −−∉++∞= βαβαβα

nn xP Laguerre polynomials ( ),...}2,1{,)}({ 0

)( −−∉∞= αα

nn xL Hermite polynomials ∞

=0)}({ nn xH Bessel polynomials ( ). 0and,...}2,1,0{,)}({ 0

),( ≠−−∉∞= βαβα

nn xB

Then, in 1988, Nikiforov and Uvarov [56] gave some general properties of ⎟⎟⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn,

such as a generating function for the polynomials, a Cauchy integral representation and

A generic polynomial solution for the hypergeometric differential equation

72

so on in terms of the given )(xσ and )(xτ . Of course their approach is based on the Rodrigues representation of the polynomials and is not expressed in an explicit form, the task that we will do in this chapter. Some other approaches in this regard are [14, 44, 72]. But before deriving a generic solution for (6.1), we should introduce an algebraic identity, which is easy to prove but important. 6.1.1. An algebraic identity If ba, and ),...,1,0( nkCk = are real numbers then

.)()()(0 00 00∑ ∑∑ ∑∑=

−

=−

−

=

−

=−

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+

n

k

knk

iin

ikn

k

kkn

iin

inn

k

kk axCb

ikin

baxCb

kin

baxC (6.2)

For instance, if we set !)...()()()...()()()(

121

32

kqqqpppn

Ckmkk

kmkkkk

−

−= such that ∏

−

=

+=1

0

)(k

ik irr then

we get (6.3)

kn

k m

mmm

nmnn

nmnn

kn

k m

mmm

nmnn

nmnn

m

mmm

axb

bnpnpnpnqnqk

Fkn

qqqppax

bax

bnpnpnpnqnqnk

Fkn

qqqppb

baxqqqppn

F

)(1|1,....,1,1

1,....,1,)...()()()...()()(

)(1|1,....,1,1

1,....,1,)...()()()...()()(

|,....,,,....,,

0 32

111

121

2

0 32

111

121

2

121

21

∑

∑

=

−−

−

=

−−

−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−−−−−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−−−−−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

where !)...()()(

)...()()(|

,....,,,....,,

0 21

21

21

21

kx

bbbaaa

xbbbaaa

Fk

k kqkk

kpkk

q

pqp ∑

∞

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛ denotes the generalized

hypergeometric function of order (p, q) (see e.g. [43], Chapter 2). Note that to compute the relations (6.3) we have generally used the two identities

( ) ( )( )

( 1) ( )( ) ; ; , , .( )( 1)( ) (1 )(1 )

k ink

n ii

k

r k r rrr r k n irr k r n

r

+−

Γ + = Γ⎧−⎪ ⇒ = ∈ ∈Γ −⎨Γ − = − −⎪ −⎩

R Z (6.4)

An interesting case takes place for (6.3) when 1and2 === bam . In this case we have

k

n

kn

nn

kn

kn

nn

xnp

nqkF

kn

qpx

xnp

nqnkF

kn

qp

xq

pnF

−

=

=

∑

∑

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

0 2

112

1

2

0 2

112

1

2

1

212

1|1

1)(

)()(

1|1

1)(

)()1(1|

(6.5)

which according to Gauss’ identity (introduced in chapter 3, formula (3.38))


73

)()()()(112 bcac

baccc

baF

−Γ−Γ−−ΓΓ

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ,

simplifies to

. 1|1)(

)()(

|1)(

)(1|

2

2112

1

2

21

212

1

21

1

212

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

xnppqn

Fq

px

xnpq

pnF

qpq

xq

pnF

n

nn

n

n

(6.6)

The first identity of (6.6) can be found in [13], (15.3.6) and the second identity in (6.6) is a special case of [13], (15.3.7) for integer upper parameter. Now it is a good position to propound the main theorem of finding a generic polynomial solution for equation (6.1). 6.2. The main Theorem. The monic polynomial solution of the differential equation 2( ) ( ) ( ) ( ) (( 1) ) ( ) 0 ;n n nax bx c y x dx e y x n n a d y x n +′′ ′+ + + + − − + = ∈Z (6.7) is given by the formula

kn

k

nknn xedcbaG

kn

xcbaed

Pxy ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

0

)( ),,,,()( , (6.8)

where

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−+

−

−−

−−+−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

−

acbbacb

nad

na

d

acba

bdaenkF

acbbaG

nk

nk

442

2/22

142

2

42

2

22122

)( .

For 0=a the equality can be adapted by limit considerations and gives (6.8) in the form

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−+−

−== −

→ cdbn

bbecdnkF

cbedcbaGedcbG nkn

ka

nk

22

02)(

0

)( 1,)(),,,,(lim),,,,0( ,

which is valid for , 0, ≠dc leading to

.)()(0 2

2112 ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−−−

−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛bcdx

bd

bcdeb

nF

bcdeb

dbx

cbed

P nn

n

Finally for 0== ba and 0≠d (6.8) is transformed to

.)(

22

1,2)(lim

00 20200 ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+−

−−−+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

→→ edx

cdnnF

dexx

cbaed

Pxced

P nn

ban


74

Proof. Consider the differential equation (6.7) and suppose that qptx += . Hence it will change to

. 0))1(()()2( 2

2

2

22 =−+−

+++

+++

++ yand

an

dtdy

padqet

ad

dtyd

apcbqaqt

apbaqt (6.9)

If 1/)2(and02 −=+=++ apbaqcbqaq are assumed in (6.9), then

. 2

4and4 22

aacbbq

aacbp −±−

=−

= ∓ (6.10)

Therefore (6.9) is simplified as

. 0))1(()42

42()1(2

2

2

2

=−+−−

−±−++− yand

an

dtdy

acbaacbdbdaet

ad

dtydtt

∓ (6.11)

On the other hand, equation (6.11) is a special case of the Gauss hypergeometric differential equation (see e.g. [43], p. 26):

0))1(()1( 2

2

=+−+++− ydtdyt

dtydtt αβγβα (6.12)

for acba

acbdbdaeadnn42

42and/1,2

2

−

−±−=+−=−=

∓γβα respectively. So, by

considering ⎟⎟⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn as a pre-assigned solution of (6.7) and comparing the relations

(6.11) and (6.12), we must have (6.13)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−±

−±−+−−

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −±−+

− t

acba

acbdbdaeadnn

FKa

acbbta

acbcbaed

Pn

42

42/1

2

44

2

212

22

∓ .

(6.13) can also be written in terms of the variable x so that we have (6.14)

. )424

4(

4242

/1

2

2

22

212

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+

−−±

−±−+−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

acbacbb

acbax

acbaacbdbdae

adnnFKx

cbaed

Pn∓∓

From (6.14) two following sub-cases are concluded (6.15)

, 424

44242

/1(i)

2

2

22

212

*

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−−−

−

−

−

−+−+−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

acbacbb

acbax

acbaacbdbdae

adnnFKx

cbaed

Pn


75

. 424

4424)2(

/1(ii)

2

2

22

212

**

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−++

−−

−+−−+−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

acbacbb

acbax

acbaacbdbdae

adnnFKx

cbaed

Pn

Note that both above relations only differ by a minus sign in the argument of the second formula (ii), which does not affect on the differential equation (6.7). In other words, if in (i) we consider the case ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−

−−x

cbaed

Pn, we will reach the second formula of

(6.15). Therefore, only the formula (ii) must be considered as the main solution. To compute **K , it is sufficient to obtain the leading coefficient of ( )......12 F corresponding to formula (ii), which is given by

. )/1()(

))42/()42(()4(! 222**

nnn

nn

adnnaacbaacbdaebdacbn

K+−−

−−+−−= (6.16)

But according to identity (6.3)

. )(1|1

1)(

)()1(|

01212

kn

k

kn

n

nn

rxsnp

nqnkFs

kn

qp

srxq

pnF ∑

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

− (6.17)

So, by considering the main solution (6.15) and assuming

424,

4

4242,1

2

2

2

2

2

⎪⎪

⎩

⎪⎪

⎨

⎧

−

−+=

−=

−

−+−=+−=

acbacbbs

acbar

acbaacbdaebdq

adnp

(6.18)

(6.17) is changed to (6.19)

. 4

42|22/42

4))22((2

)4

()42

4(

)42

42(

)1()1(

42

4

442

42/1

2

2

2

2

12

0 22

2

2

22

2

22

212

k

n

k

kkn

n

nn

xacbb

acb

nadacba

acbandbdaenkF

acb

a

acb

acbbkn

acba

acbdaebd

nad

acb

acbb

acb

ax

acba

acbdaebdadnn

F

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−+

−

+−−−

−−+−−−

×−−

−+⎟⎟⎠

⎞⎜⎜⎝

⎛×

−

−+−

−+−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−++

−−

−+−+−−

∑=

−

Simplifying this relation and substituting **K by (6.16) finally gives the monic


76

polynomial solution of equation (6.7) in the form (6.8). Hence the first part of the theorem is proved. To deduce the limiting case when 0→a , one should use the limit

relation rr

r

adn

ada )()22(lim

0−=−+−

→and the following identity

,)(1|1

)1()1(|

00211

kn

kx

sr

snpnk

Fkn

pnpsrx

pn

F ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−Γ−−Γ

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

− (6.20)

which is a special case of identity (6.2) for !)(

)(kp

nC

k

kk

−= .

Here we would like to point out that the general formula ),,,,()( edcbaG nk is a suitable

tool to compute the coefficients of kx for any fixed degree k and arbitrary a. If for example the coefficient of 1−nx is needed in the generic polynomial

⎟⎟⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn then it

is enough to calculate the term

andbne

anda

baandbdae

ab

ba

anda

andbdae

Fb

aedcbaG nn

)22()1()

)22(2

2))22((21)(

2(

2|))22((

2))22((21

)2(),,,,( 121)(

1

−+−+

=−+

×Δ+Δ

×Δ

Δ−+−−+

Δ+=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Δ+Δ

−+−

ΔΔ−+−−

−

Δ+= −

−

(6.21)

in which acb 42 −=Δ . Note that in the above simplified relation, all parameters

edcba and,,, are free and can adopt any value including zero since it is easy to find out that neither both values a and d nor both values b and e in (6.8) can vanish together. After simplifying ),,,,()( edcbaG n

k for ...,2,1 −−= nnk we eventually get (6.22)

. 4

42|))22((

42

4))22((2

)2

4(...

))32()()22(())22(())2()()1((

2)22()1(

1

2

22

2

12

2

21

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+

−

−+−

−

−−+−−−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−+−+−++−+−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−

acbb

acb

aandacba

acbandbdaenF

aacbb

nn

xandand

andcbnebnenx

andbnen

xxcba

edP

n

nnnn

The above relation implies that (6.22.1)

. 4

42|))22((

42

4))22((2

)2

4(02

22

2

12

2

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+

−

−+−

−

−−+−−−−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛

acbb

acb

aandacba

acbandbdaenF

aacbb

cbaed

P nn

Moreover, (6.22) shows, for example, that if n = 0,1,2,3 then (6.22.2)


77

.)2)(3)(4(

)2)(()4()2)(3(2)3)(4(

)2)(()4(3423

,))(2(

)()2(2

2

,

,1

233

22

1

0

adadadbebeeadcebeadc

xadad

bebeadcxadbexx

cbaed

P

adadbeeadcx

adbexx

cbaed

P

dexx

cbaed

P

xcbaed

P

++++++++++

+

++++++

+++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

+++++

+++

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛

6.3. A special case of the generic polynomials (6.8)

In the sequel, let us apply the Gauss identity again and assume that 14

422

2

=−+

−

acbbacb

in (6.8). This assumption implies that 0=ac . If 0=c , the following special case for the generic polynomial (6.8) is derived (6.23)

. /

/1)/1()/2(

)/1()/22()/1()/2(

)/1()/22()(

12/2

21

22

)(0

12

0

012

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−−×

−−Γ−−Γ−Γ−−Γ

=−−Γ−−−Γ−−Γ−−Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

−−+−

−⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

∑

∑

=

−

=

−

xba

beadnn

F

benadnabeadnbx

benadknbekadn

ba

kn

xnad

na

dab

bdaenkFba

kn

xba

edP

n

nk

n

k

nk

kn

k

nkn

Furthermore, if one comes back to the Nikiforov and Uvarov approach and consideres the differential equation (6.7) as a self-adjoint form, then according to (3.23), chapter 3, the Rodrigues representation of ⎟⎟

⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn is given by (6.24)

,))((

))2((

12

1

n

nn

n

k

n dx

xcbaed

cbxaxd

xcbaed

akndx

cbaed

P⎟⎟⎠

⎞⎜⎜⎝

⎛++

×

⎟⎟⎠

⎞⎜⎜⎝

⎛−++

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∏=

ρ

ρ

where ))()2(exp( 2∫ ++−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛dx

cbxaxbexadx

cbaed

ρ .

Now if we suppose 0=c in this relation and refer to (6.23), we get (6.24.1)


78

22 2

1

0

( 2 ) ( ) ( 2 ) ( )exp( ) (( ) exp( ))

( ( 2) )

(2 2 / ) (1 / )( ) .(1 / ) (2 / )

n n

n n

k

nk n k

k

d a x e b d a x e bdx d ax bx dxax bx ax bx

dxd n k a

nn d a a k e b xkn e b b n k d a

=

−

=

− + − − + −− +

+ +×+ + −

⎛ ⎞Γ − − Γ − −= ⎜ ⎟Γ − − Γ − − −⎝ ⎠

∫ ∫

∏

∑

6.4. Some further properties of the main polynomials (6.8) 6.4.1. A linear change of variables Using the representation (6.24) one can derive a linear change of variables for the monic polynomials ⎟⎟

⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn explicitly. Assuming vwtx += , the mentioned

relation changes to (6.25)

.))2((

)))()2(((

1

222

nn

k

nnn

n

dtvwtcba

edaknd

vwtcbaed

cbvavtbwawvtawdwvwt

cbaed

P

⎟⎟⎠

⎞⎜⎜⎝

⎛+−++

⎟⎟⎠

⎞⎜⎜⎝

⎛++++++

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

∏=

ρ

ρ

On the other hand ⎟⎟

⎠

⎞⎜⎜⎝

⎛+ vwt

cbaed

ρ can be simplified as

. ,)2(,

)(,

))2(

)())(2(exp(

22

2

222

⎟⎟⎠

⎞⎜⎜⎝

⎛

++++

=

+++++−++−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+ ∫

tcbvavwbavaw

wedvdw

dtwcbvavwtbavtaw

bevwtadvwtcba

ed

ρ

ρ

(6.26)

Therefore (6.25) is transformed to

⎟⎟⎠

⎞⎜⎜⎝

⎛

++++

=⎟⎟⎠

⎞⎜⎜⎝

⎛+ t

cbvavwbavawwedvdw

Pwvwtcbaed

P nn

n 22

2

,)2(,)(, , (6.27)

which shows the effect of a linear change of variables on the polynomials (6.8). For instance, if 0and1 =−= vw in (6.27), then we have

. ,,

,)1( ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛− t

cbaed

Ptcbaed

P nn

n (6.28)

6.4.2. A generic three-term recurrence equation The second formula of (6.15) is a suitable relation to compute the recurrence equation of the generic polynomials (6.8). In other words, it can be applied along with various


79

identities of the Gauss hypergeometric function for generating a recurrence relation. For example, the following identity holds for the function );,,(12 trqpF [18], (6.29)

( ). 0);,1,1()1)(();,1,1()1)((

);,,()1(2)1)(1()(

1212

12

=−+−−−++−+−−+−−+++−−−−

trqpFqpqrptrqpFqpprqtrqpFqprpqtqpqpqp

By using the formula (ii) in (6.15) and its coefficient in (6.16) if one assumes in (6.29) that (6.29.1)

, 42

4

4and

42

42,1,2

2

22

2

acb

acbbxacb

atacba

acbdaebdrnadqnp

−

−++

−=

−

−+−=−+=−=

then after some computations, one finally gets (6.30)

( ) )())12(())22()()32((

))()(())2(())22(())2((

)())22()(2(

)2)(2()1(2)(

12

22

1

xPandandand

bdbeabeandnbandcandn

xPandnad

nbeadabnnxxP

n

nn

−

+

−+−+−+−+−+−+−−+−+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−++

+−+++=

in which )(xPn denotes the monic polynomials of (6.8) and the initial values

)(and1)( 10 dexxPxP +== are given. For other approaches to extract (6.30), see

[52] and [45]. 6.4.3. A generic formula for the norm square value of the polynomials Let ],[ UL be a predetermined orthogonality interval which (besides for finite families) of course consists of the zeros of cbxaxx ++= 2)(σ or ∞± . By using the Rodrigues representation of the polynomials (6.8) we have

. )/))(((

)2(

1

2

1

22

∫

∏∫

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛×

−++=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

U

L

nnnn

n

k

U

Lnn

dxdxxcbaed

cbxaxdxcbaed

P

aknddxx

cbaed

xcbaed

PP

ρ

ρ

(6.31)

Consequently integrating by parts from right hand side of (6.31) yields

. ))()2((exp)()2(

)1(!2

2

1

2

∫ ∫∏ ++

−+−++

−++

−=

=

U

L

nn

k

n

n dxdxcbxax

bexadcbxaxaknd

nP (6.32)


80

6.5. Six special cases of the generic polynomials (6.8) as classical orthogonal polynomials As we mentioned in chapter 4, from the main equation (6.1) one can extract six special sequences of orthogonal polynomials on the real line. Jacobi, Laguerre and Hermite polynomials are three of them, which are infinitely orthogonal and three other ones are finitely orthogonal for some restricted values of n. In this section we intend to use the previous generic formulas to redetect the properties of each of these six sequences. 6.5.1. Jacobi orthogonal polynomials If βαβα +−=−−−===−= edcba and2,1,0,1 are selected in (6.8) then

,22

,)1(

!2)1(

1,0,1,2

!2)1(

)(

120

),(

kn

k

nkn

n

nnn

n

xn

nnkF

kn

nn

xPn

nxP

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+++=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−−−−+++=

∑=

−

βααβα

βαβαβαβα

(6.33)

are the Jacobi orthogonal polynomials with weight function

βααββαβαβαρ )1()1()

1)(exp(

1,0,1,2

2 xxdxx

xx +−=+−

−++−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+−−−−∫ , (6.34)

and orthogonality relation (6.35)

mnmn nnn

nndxxPxPxx ,

11

1

),(),(

)1()1()1(

)12(!2)()()1()1( δ

βαβα

βα

βαβαβαβα

+++Γ++Γ++Γ

+++=+−

++

−∫ .

These polynomials can also be represented as

1

22

,)1()1(

1,0,1,2

12 ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−−−−−

−−=⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−−−−xn

nnFxxP nn

n βααβαβα , (6.36)

which is one of the known hypergeometric representations for the Jacobi polynomials (see e.g. [13], [70]). Since the Gegenbauer (ultraspherical), Legendre and Chebyshev polynomials of the first and second kind are all special sub-cases of the Jacobi polynomials, the following representations are straightforwardly concluded for them. Gegenbauer polynomials: (6.37)

, 1

2221

2/1,)1()1(

!)(2

101012

!)(2

)( 12)(

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

−−−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−=

xnnn

Fxn

xPn

xC nnnn

nn

n

n λλλλλλ

Legendre polynomials:


81

, 1

22,

)1()1(2)!()!2(

10102

2)!()!2()( 1222 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

−−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=xn

nnFx

nnxP

nnxP nn

nnnn (6.38)

Chebyshev polynomials of first kind:

, 1

2212/1,

)1()1(2101

012)( 12

11⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

= −−

xnnn

FxxPxT nnnn

nn (6.39)

Chebyshev polynomials of second kind:

. 1

2212/1,

)1()1(2101

032)( 12 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−

−−−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=xn

nnFxxPxU nnn

nn

n (6.40)

6.5.2. Laguerre orthogonal polynomials If one puts 1,1,0,1,0 +=−==== αedcba in (6.8) then

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−= x

nF

nxP

nxL n

n

n

n 1!)1(

01011

!)1()( 11

)(

αααα , (6.41)

are the Laguerre orthogonal polynomials with weight function

xexdxx

xx −=+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +−∫ ααα

ρ )exp(0,1,0

1,1, (6.42)

and orthogonality relation

. !

)!()()( ,0

)()(mnmn

x

nndxxLxLex δαααα +

=∫∞

− (6.43)

6.5.3. Hermite orthogonal polynomials If 0,2,1,0,0 =−==== edcba are selected in (6.8) then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

−

−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −= 202

12

1,2)2(

10002

2)(x

nnFxxPxH n

nn

n , (6.44)

are the Hermite orthogonal polynomials with weight function

)exp())2(exp(1,0,00,2 2xdxxx −=−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−∫ρ , (6.45)

and orthogonality relation

. 2!)()()exp( ,2

mnn

mn ndxxHxHx δπ=−∫∞

∞−

(6.46)


82

6.5.4. Finite classical orthogonal polynomials with weight function )(

1 )1(),,( qpq xxqpxW +−+= on ),0[ ∞ According to the approach explained in chapter 3, section 3.11.1, if one computes the

logarithmic derivative of the given weight as xxqpx

xWxW

++−

=′

21

1

)()( , then by referring to

the Pearson’s differential equation one respectively gets . 1,2,0,1,1 +=+−==== qepdcba (6.47) In [53] the family of this type of polynomials are called Romanovski-Jacobi polynomials, see also [65, 54]. In [10] the related polynomials are denoted by

)(),( xM qpn , for which we get (6.48)

. 11,

)1()1(011

12)1()1()( 12

),(⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−+−

+−=⎟⎟⎠

⎞⎜⎜⎝

⎛ ++−−+−= x

qpnn

Fqxqp

PpnxM nn

nnnqp

n

Also, as it was shown in chapter 4, section 4.2, the finite set Nn

nqp

n xM ==0

),( )}({ is orthogonal with respect to the weight function ),,(1 qpxW on ),0[ ∞ if and only if

12and1 +>−> Npq . Let us add that to compute the norm square value of the polynomials one can also use the general relation (6.32). According to the foresaid relation we have

. )1(

)()(

)1(!011

12)1( 0

2

1

0

2 ∫∏

∫∞

+

=

∞

+ ++

++−

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++−

+dx

xxxx

knp

ndxx

qpP

xx

qp

qn

n

k

n

nqp

q

(6.49)

Thus, noting (6.48) yields

.

)!1()12()!()!1(!))((

)1(0

2),(

−−+−−+−−

=+∫

∞

+ nqpnpnqnpndxxM

xx qp

nqp

q

(6.50)

6.5.5. Finite classical orthogonal polynomials with weight function xpexpxW /1

2 ),( −−= on ),0[ ∞

If the fraction 22

2 1)()(

xpx

xWxW +−

=′

is generated then the main parameters are derived as

. 1,2,0,0,1 =+−==== epdcba (6.51) In [53], the polynomials of this type are called Romanovski-Bessel polynomials. In [10] these polynomials are denoted by )()( xN p

n , for which we have


83

. 1,

)1( 001

12)1()1()( 02

)(⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−+−= x

npnFx

pPpnxN n

nnnp

n (6.52)

Moreover, in chapter 4, section 4.3, it was shown that the finite set Nn

np

n xN ==0

)( )}({ is orthogonal with respect to the weight function ),(2 pxW on ),0[ ∞ if and only if

12 +> Np . Nevertheless, using the general relation (6.32) helps us obtain the norm square value of these polynomials as follows

.

)(

)1(!001

12

0

12

1

0

21

∫∏

∫∞ −−

=

∞ −−

++−

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−dxexx

knp

ndxxp

Pex xpnn

k

n

nxp (6.53)

Consequently, the complete orthogonality relation takes the form (6.54)

. 2

1,...,2,1,0,for )12())!1((!)()( ,

0

)()(1 −

<=⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−

=∫∞ −

− pNnmnpnpndxxNxNex mn

pm

pn

xp δ

6.5.6. Finite classical orthogonal polynomials with weight function

)arctan exp())()((),,,;( 22),(3 DCx

BAxqDCxBAxDCBAxW pqp

++

+++= − on ),( ∞−∞

Similar to the two previous cases, by computing )()(

3

3

xWxW ′

we get the parameters

. ))(1(2)(,))(1(2

,,)(2,22

2222

CDABpBCADqeCApdDBcCDABbCAa

+−+−=+−=

+=+=+= (6.55)

In [53], the polynomials of this type are called Romanovski-Pseudo-Jacobi polynomials. In [9] the related polynomials are denoted by ),,,;(),( DCBAxJ qp

n , for which we have

( )

. ))()((

)(2

222

,F

)()()21()1(

),(2,))(1(2)(),)(1(2

)()21()1(),,,;(

12

22

2222

22

22),(

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+++−−

−

−+−−×

++−++−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

++++−+−+−

×

+−+−=

CiAxDiBCiABCADi

np

iqpnn

CAxiBCADCDABpn

xDBCDABCA

CDABpBCADqCApP

CApnDCBAxJ

nn

n

n

nn

nqpn

(6.56)

6.6. How to find the parameters if a special case of the main weight function is given?


84

Similar to the chapter 3, section 3.11.1, it is easy to find out that the best way for finding edcba ,,,, is to compute the logarithmic derivative )(/)( xWxW ′ and match

the pattern with cbxax

bexadxx

++−+−

=′

2

)()2()()(

ρρ . Let us clarify the subject by some

examples given below. Example 1. Consider the weight function 21;)23()( 102 <<−+−= xxxxW . If the logarithmic derivative of this weight function is computed, then we get

. 33,22,2,3,1)()2(23

3020)()(

22 =−=−==−=⇒++−+−

=−+−

+−=

′edcba

cbxaxbexad

xxx

xWxW

Consequently the related monic orthogonal polynomials are ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−xPn 2,3,1

33,22 . Note

that these polynomials are orthogonal with respect to the given weight function on [1,2] for every value n. Therefore it is not necessary to know that these polynomials are the shifted Jacobi polynomials on the interval [1,2] since they can explicitly be expressed by the generic polynomials (6.8). Example 2. The weight function ∞<<∞−++= − xxxxW ;)122()( 102 is given. Thus

, )18,38,1,2,2(),,,,(122

2040)()(

2 −−=⇒++

−−=

′edcba

xxx

xWxW

and the related monic orthogonal polynomials are ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−xPn 1,2,2

18,38. These

polynomials are finitely orthogonal for 9≤n , because according to (4.44) in chapter 4 we must have )2/1(10}max{ −<= nN . Hence, the finite set 9

0)};18,38,1,2,2({ ==−− n

nn xP is orthogonal with respect to the weight function 102 )122( −++ xx on ),( ∞−∞ . Example 3. Consider the weight function ( ) exp( ( )) ; ,W x x x xθ θ= − −∞ < < ∞ ∈R . Then we have

, ),2,1,0,0(),,,,(2)()( θθ −=⇒+−=

′edcbax

xWxW

which gives the related monic orthogonal polynomials as ⎟⎟⎠

⎞⎜⎜⎝

⎛ −xPn 1,0,0

,2 θ for every

value n.

Example 4. For the weight function ∞<<+

= xx

xxW 0;)2(

)( 8 , which is a special

case of the second kind of the Beta distribution, we have

)6,11,0,4,2(),,,,(42

215)()(

2 −=⇒++−

=′

edcbaxx

xxWxW .


85

Therefore, the monic orthogonal polynomials are ⎟⎟⎠

⎞⎜⎜⎝

⎛ −xPn 0,4,2

6,11 for 3≤n ,

because according to (4.6) in chapter 4, )2/1(812 −<+n . This means that the finite set 3

0});6,11,0,4,2({ ==− n

nn xP is orthogonal with respect to the weight function 8)2/( +xx on ),0[ ∞ .

6.7. A generic formula for the values at the boundary points of monic classical orthogonal polynomials [2] In the previous sections of this chapter, we found a generic formula for the polynomial solution families of the well-known differential equation (6.1). Now, in the section (6.8) we intend to obtain another such formula, which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition. For this goal, we should again use the general form of the Rodrigues representation of the polynomials in (6.24) and recall its corresponding weight function, i.e.

2

( 2 ) ( )exp( ) .d e d a x e bx dx

a b c ax bx cρ⎛ ⎞ − + −

=⎜ ⎟ + +⎝ ⎠∫

Without loss of generality, let us suppose that ))(( 21

2 θθ ++=++ xxacbxax in which

2 2

1 24 4and .

2 2b b ac b b ac

a aθ θ− − + −= = (6.57)

In the general case, 1θ− and 2θ− in (6.57) are the boundary points of the underlying interval for the corresponding classical orthogonal polynomials. This means that if 1θ and 2θ are finite and equal, the polynomials are of the Bessel type and if both 1θ and

2θ are finite but different from each other, the polynomials are of the Jacobi type, whereas if one of these values tends to ±∞ , the polynomials are of the Laguerre type, and finally if both values are ±∞ , then the polynomials are of the Hermite type. The relation (6.57) implies that the main weight function is simplified as

BA xxRxcba

ed)()( 21 θθρ ++=⎟⎟

⎠

⎞⎜⎜⎝

⎛, (6.58)

where R is a constant and

2 2

2 21 and 12 22 4 2 4d ae bd d ae bdA Ba aa b ac a b ac

− −= − + = − −

− −. (6.59)

Note that the relation (6.58) follows because the logarithmic derivative of the function

BA xxxW )()()( 21* θθ ++= equals the logarithmic derivative of the main weight

function, and since


86

( ) ( ) ( ) ( ) ,( ) ( )

u x v x u x R v xu x v x′ ′

= ⇔ = (6.60)

so (6.58) is valid. On the other hand, according to (6.24) the Rodrigues representation of ⎟⎟

⎠

⎞⎜⎜⎝

⎛x

cbaed

Pn is

. )()2(

1 2

1

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−++

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∏=

=

xcbaed

Wcbxaxdxd

xcbaed

Wakndx

cbaed

P nn

n

nk

k

n

So, if (6.58) is replaced into the above representation, then

1 2 1 2

1 21

1 21 2

( ( ) ( ) ( ) ( ) )

( 2) ( ) ( )

1 (( ) ( ) ) .( 1 / ) ( ) ( )

nn n n A B

n

n k nA B

k

nn A n B

A B nn

d R a x x x xd e dxP xa b c

R d n k a x x

d x xn d a x x dx

θ θ θ θ

θ θ

θ θθ θ

=

=

+ +

+ + + +⎛ ⎞=⎜ ⎟ ⎛ ⎞⎝ ⎠ + + − + +⎜ ⎟

⎝ ⎠

= + +− + + +

∏ (6.61)

But according to the Leibniz rule

∑=

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

nk

k

knkn

n

xgxfkn

dxxgxfd

0

)()( )()())()(( , (6.62)

we have (6.63)

1 21 2

0

(( ) ( ) ) ( 1) ( ) ( ) ( ) ( ) .n n A n B k n

n n A k B kk n kn

k

nd x x n A n B x xkdx

θ θ θ θ+ + =

+ − +−

=

⎛ ⎞+ += − − − − − + +⎜ ⎟

⎝ ⎠∑

Hence, (6.61) is simplified as (6.64)

.)2

4()2

4(

)42

212

()42

212

(

)/22(1

22

022

kkn

nk

kknk

nn

aacbbx

aacbbx

acbabdae

adn

acbabdae

adn

kn

adnx

cbaed

P

−++

−−+×

−

−++−−

−

−−+−−⎟⎟

⎠

⎞⎜⎜⎝

⎛

×−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−

=

=−∑

This is in fact another general representation for the polynomial solution of equation (6.7). Combining (6.8) and (6.64), we get straightforwardly (6.65)


87

0

2 10

2 2( 1 ) ( 1 ) ( ) ( )2 2 2 2 2 2

2 12 2(2 2 / ) .2 22 / 2

k nn k k

k n kk

k nk nk

nk

n d ae bd d ae bd b bn n x xk a a a a a a

ae bd dn k n nan d a F xa ak b bd a n

=−

−=

−=

=

⎛ ⎞ − − − Δ + Δ− − + − − − + + + +⎜ ⎟ Δ Δ⎝ ⎠

−⎛ ⎞− + − −⎛ ⎞ Δ⎛ ⎞ ⎜ ⎟= − − Δ⎜ ⎟⎜ ⎟ ⎜ ⎟+ Δ + Δ⎝ ⎠ ⎜ ⎟⎝ ⎠ − −⎝ ⎠

∑

∑

where again acb 42 −=Δ . Relation (6.64) can also be represented in terms of the hypergeometric form (6.66)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++

Δ−

−

Δ−

−+−−−

−−

+Δ−

++−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

1

212

1

22

2

221

2)/22(

)()2

212

(

θθθ

xx

abdae

ad

abdae

adnn

Fadn

xa

bdaea

dnx

cbaed

Pn

nn

n

where 1 2 and θ θ are defined by (6.57). Of course, this hypergeometric representation can still be simplified. To simplify (6.66), we use the hypergeometric identity

kn

k

knn

n

nn

stsnp

nqkFr

kn

tqp

str

qpn

F )(1|1

1)(

)()1(|

01212 ∑

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−, (6.67)

which was used already in (6.17) with a rather different form. If we choose in particular (6.68)

,2

41

,4,42

22

,42

212

2

2

1222

aacbbxtands

aacbr

acba

bdaea

dqacba

bdaea

dnp

−−+==

−=−=

−

−−=

−

−−+−−= θθ

then by (6.67) relation (6.66) reads as (6.69)

2 1 10

( 1) ( ) ( ) ( ) 1 ( ) .1(2 2 / ) (1 )

n k nn k kn n

nkn n

d e n k n Bn B n AP x F xa b c k An d a B a

θ=

−

=

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − − − − Δ= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟+− − +⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑ On the other hand, using Gauss’s identity (i.e. ...)1;,,(12 =cbaF ) (6.69) can be further simplified as

2

2

2

2 1 2 22

2( 4 ) ( )2 2 4

(2 2 / )

1 /4 ,2

4 2 42 2 4

nn

n nn

d ae bdb acd e a a b acP x

a b c a n d a

n n d aax b b acF d ae bd

b ac b aca a b ac

−− +

⎛ ⎞ −= ×⎜ ⎟ − −⎝ ⎠− − +⎛ ⎞

− − + −⎜ ⎟+−⎜ ⎟+ − −⎜ ⎟−⎝ ⎠

(6.70)


88

which is the same form as the first formula of (6.15). Furthermore, since the identity

0≠∀⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛λ

λλλλλ

xcbaed

Pxcba

edP nn , (6.71)

is also valid for 1−=λ , the relation (6.70) can be brought in the form of the second formula of (6.15) too. 6.7.1. Values of the classical orthogonal polynomials at the boundary points Using the explicit representations for the monic classical orthogonal polynomials, we can now compute the generic value of the polynomials at their boundary points of definition, 1θ− and 2θ− , respectively. If we set in the second formula of (6.15)

2

2 2

04 ,14 2 4

ax b b acb ac b ac

⎧+ −+ = ⎨

− − ⎩ (6.72)

then

2 2

2 14 4and ,

2 2b b ac b b acx x

a aθ θ− − − − + −

= = − = = − (6.73)

respectively. Therefore we get

,)/1()(

))42/()2()2/(()4( 22

2n

nn

n

n adnaacbabdaeadacb

cbaed

P+−−

−−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−θ (6.74)

and

2 2

1( 4 ) (( / 2 ) (2 ) /(2 4 )) .

( 1 / )

nn

n nn

d e b ac d a ae bd a b acPa b c a n d a

θ⎛ ⎞ − + − −

− =⎜ ⎟ − +⎝ ⎠ (6.75)

For example, by noting the section 6.5.1 for the monic Jacobi orthogonal polynomials

)(),( xPnβα we have ),2,1,0,1(),,,,( βαβα +−−−−−=edcba . Consequently, (6.74) and

(6.75) yield

( , ) ( 1) ( 1 ) ( 1 )( 1) 2 2 ,( 1 ) ( 1) (2 1 )

n nnn

n

n nPn n

α β α α α βα β α α β+ Γ + + Γ + + +

+ = =+ + + Γ + Γ + + +

(6.76)

( , ) ( 1) ( 1 ) ( 1 )( 1) ( 2) ( 2) .

( 1 ) ( 1) (2 1 )n nn

nn

n nPn n

α β β β α βα β β α β+ Γ + + Γ + + +

− = − = −+ + + Γ + Γ + + +

(6.77)

Moreover, by noting the section 6.5.2 for the monic Laguerre polynomials ( ) ( )nL xα with ( , , , , ) (0,1,0, 1, 1)a b c d e α= − + we have xcbxax =++2 . Therefore just one root i.e.

021 ==θθ is derived and by computing the corresponding limit one gets


89

( ) (0) ( 1) (1 ) .nn nL α α= − + (6.78)

Furthermore, since the Hermite polynomials can be written in terms of the Laguerre polynomials (see e.g. [13], relation 22.5.40), we can also conclude that

))1(1()!2/(2

!)0( 1n

nn nnH −+= + (6.79)

Similarly, for the Bessel polynomials ( ) ( )nB xα with ( , , , , ) (1,0,0, 2,2)a b c d e α= + we have 22 xcbxax =++ . So, after computing the corresponding limit one can obtain that

( ) 2(0) .( 1 )

n

nn

Bn

α

α=

+ + (6.80)


90

Chapter 7 Application of rational classical orthogonal polynomials for explicit computation of inverse Laplace transforms 7.1. Introduction In the chapters 4 and 6, it was shown that from the hypergeometric differential equation, six finite and infinite classes of orthogonal polynomials could be extracted. In this chapter, we first apply the Mobius transform R∈≠+= − qpqpzx ,0,1 for the mentioned equation to generate the classical orthogonal polynomials with negative powers. Then we show that the generated rational orthogonal polynomials are a very suitable tool to compute the inverse Laplace transform directly, with no additional calculation for finding their roots. In this way, by applying infinite and finite rational classical orthogonal polynomials, we present (for example) three basic expansions of six ones to explicitly obtain the inverse Laplace transform. To do this task, we should again reconsider the hypergeometric differential equation 0)())1(()()()()( 2 =−+−′++′′++ xyandnxyedxxycbxax nnn , (7.1) and suppose that R∈≠+= − qpqpzx ,0,1 .Therefore, the equation (7.1) eventually changes to 0))1(()2()( 4043

2201

22

2 =−+−′+++′′++ yllnnylxlxlxylxlxlx , (7.2) where 01234 ,,,, lllll are real parameters and n is a positive integer. Obviously one of the main solutions of equation (7.2) is a rational (negative power)

polynomial like ∑=

−− =⎟⎟

⎠

⎞⎜⎜⎝

⎛ n

k

kkn xrx

lllll

P0012

34 that depends on the parameters 01234 ,,,, lllll

respectively. According to the Sturm-Liouville theorem, one can find a weight function corresponding to the differential equation (7.2) as

))(

)2()3(2exp(

012

2

04132

2

012

34 dxlxlxlx

llxllxlx

lllll

W ∫ ++−+−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛. (7.3)

Now assume that ],[ UL is a predetermined orthogonality interval. So, we should have

Explicit computation of inverse Laplace transforms using rational classical orthogonal polynomials

92

mnn

U

Lmn Adxx

lllll

Pxlll

llPx

lllll

W ,012

34

012

34

012

34 δ=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∫ −− , (7.4)

where

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= −

U

Lnn dxx

lllll

Pxlll

llWA 2

012

34

012

34 )( , (7.5)

denotes the norm square value. There exist six cases, corresponding to the main equation (7.2), that are orthogonal for some specific values of 01234 ,,,, lllll . As it is seen, the connection between equations (7.1) and (7.2) is a Mobius transform as

qpzx += −1 for different values of qp and . For example, if 1and2 =−= qp are considered then the rational Jacobi polynomials can be defined as

∑=

=

−−−−− ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+−=+−=

nk

k

kknn x

knn

kkn

xPxP0

1),2(),( 22)1()12()(

βααββαβα . (7.6)

For n = 0,1,2 this definition respectively gives

⎪⎪

⎩

⎪⎪

⎨

⎧

−−−+−+−++=

−−+−=

=

−−−

−−

)1)((21))(1()1)(2(

21)(

)1()(

1)(

12),(2

1),(1

),(0

βαβαβαααα

βαα

βα

βα

βα

xxxP

xxP

xP

(7.7)

Since the orthogonality relation of Jacobi polynomials is known, the orthogonality relation of )(),( xP n

βα− will also be known as (7.8)

.1and0)!2(!)12(

)!()!2()()()1( ,1

),(),( −>>⇔−+−++−−+

=−∫∞

−−− βαδ

ααββαβαβαβα

mnmn nnnnndxxPxPxx

Moreover, the differential equation of )(),( xPy n

βα−= is a special case of the main

equation (7.2) for 1,1,0,3,2 01234 −===+−=−= lllll αβα and we have 0)1()2)3(()1(2 =+++′−++−+′′− ynnyxxyxx αααβ . (7.9) Similarly one can obtain the Mobius transforms of other five classes of rational classical orthogonal polynomials. Hence, it is better not to enter in details of them rather just we note that all differential equations of these transforms must however be the special cases of (7.2). For instance, the rational Laguerre orthogonal polynomials

)1()( 1)()( −= −− xLxL nn

αα satisfies


93

0)12()1( 23 =+′−+−′′− ynyxxxyxx α , (7.10) as well as the rational Hermite orthogonal polynomials )()( 1 βα += −

− xHxH nn

satisfies 02)(2 2224 =+′+++′′ ynyxxxyx αααβ . (7.11) 7.2. Evaluation of Inverse Laplace Transform using rational classical orthogonal polynomials [8] It is well known that the Laplace transform provides a powerful method for analyzing the linear systems. However, many physical problems lead to Laplace transforms whose inverses are not readily expressed in terms of tabulated functions. Because of this problem, extensive researches have been done on this matter and its applications up to now. For example, Chandran and Pallath [23] have computed inverse Laplace transforms of a class of non-rational fractional functions. Evans and Chung in [34] have obtained Laplace transform inversions using optimal contours in the complex plane, see also [17]. Iqbal in [40] has stated a classroom note regarding the Fourier method for computation of Laplace transform inversion. The problem of inverse two-sided Laplace transform for probability density functions has been stated by Tagliani in 1998 [71]. Furthermore, the problem of numerical inversion of Laplace transform has been studied by several authors. For example, Cunha and Viloche in [29] have presented an iterative method for the numerical inversion of Laplace transform. In [26], Dong has introduced a regularization method for this purpose. In [30], Crump has used Fourier series approximation (see also [39]) while Miller and Guy in [55] have used Jacobi polynomials and Sidi [68] has applied a window function for Laplace transform inversion. Finally Piessens’ work [60] is a good bibliography in this regard that one can refer to it. But, in cases where the inverse Laplace transform is required for many values of the independent variable, it is convenient to obtain the inverse as a series expansion in terms of a set of linearly independent functions. Procedure based on this idea can be calculated by solving a system of equations, which can be reduced to a triangular system if one chooses to use “orthogonal polynomials”. Such a method, using orthogonal polynomials, gives an approximate evaluation of the inversion integral using “Gauss quadrature” in the complex plane [66, 58, 59, 67]. Of course, the chief disadvantage of this method is the necessity of finding all roots, real and complex, of a polynomial of high degree, and of the calculation of a set of complex Christoffel numbers [74, p. 419]. Hence, we wish here to insist that the orthogonal polynomials with negative powers are in turn suitable tool to compute the inverse Laplace transform without any effort for finding the roots of orthogonal polynomials. To achieve this goal, we should use the orthogonality properties of rational classical orthogonal polynomials introduced in section 7.1. In this way, we present three basic expansions for explicit computation of Laplace transform inversion. 7.2.1. Inverse Laplace transform using rational Jacobi orthogonal polynomials

)(),( xP nβα

− . Let us consider the Laplace transform together with its inverse as (7.12)


94

0;)(2

1)]([)()()]([)( 1

0

>∀==⇔== ∫∫∞+

∞−

−∞

− sdssFei

sFLxfdxxfexfLsFi

i

sxsxλ

λπ.

By referring to the previous sections we can find an explicit solution of the above integral equation provided that )(sF is known and expandable. First by noting (7.8), we clearly have

mnmn nnnnndttPtPtt ,

0

),(),(

)!2(!)12()!()!2()1()1()1( δ

ααββαβαβαβα

−+−++−−+

=+++∫∞

−−− . (7.13)

Now, let )(sF satisfy the Dirikhlet conditions and

∑∑∞

=

∞

=− +

=+=00

),(

)1()1()(

nn

n

nnn s

AsPCsF βα . (7.14)

By applying the property (7.13) in the expansion (7.14), the coefficients nC are found as

∫∞

−− ++

+−−+−+−+

=0

),( )()1()1()!()!2()!2(!)12( dssFsPss

nnnnnC nn

βαβα

ββααα . (7.15)

On the other hand, taking the inverse Laplace transform from the equality (7.14) yields

∑∞

=−

−− +==0

),(11 )]1([)]([)(n

nn sPLCsFLxf βα . (7.16)

Moreover, according to the definition (7.6) we have

,)!1(

)(22)(

22

])1[(22

)1()]1([

1

1

0

1),(1

∑

∑=

=

−−

=

=

−−−

−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−−+

=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+−=+

nk

k

kx

nk

k

kkn

kx

knn

kkn

exn

sLkn

nk

knsPL

βααδ

βαβα

βααβα

(7.17)

in which )(Lim)(0

xfx εεδ

→= such that

⎩⎨⎧

>≤≤

=ε

εεε x

xxf

00/1

)( is the Dirac function.

Consequently the special series (7.18)

∑∑

∫

∫

∞

= =

=

−−

∞

−−

∞−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−−+

×

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+−−+−+−+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−−

=

1

1

1

0

),(

0

)!1()(22

)(2

2

)()1()1()!()!2()!2(!)12(

)()()1(!)!2(

)!1()(

n nk

k

kx

n

kx

knn

kkn

exn

dssFsPssnn

nnn

xdssFssxf

βααδ

βαβα

ββααα

δββα

α

βαβα

βα


95

is an expanded solution for the integral equation (7.12). Note that the foresaid solution is valid if and only if its definite integrals are convergent and )(sF in (7.14) is an expandable function under the Dirikhlet conditions. The function βα ss −+ )1( in the right hand side of (7.18) plays in fact a weighted distribution role for computation of definite integrals (7.18) on ),0[ ∞ . Hence, if this distribution changes, another expanded solution will appear. The next section will specify this subject. 7.2.2. Inverse Laplace transform using rational Laguerre orthogonal polynomials

)()( xL nα− .

First, let us define the sequence

∑=

=

−− ⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−

==nk

k

kk

nn skn

nks

LsL0

)()(

!)1()1()(

ααα , (7.19)

that satisfies the orthogonality relation

mnmns

nndssLsLes ,

0

)()(1

)2(

!)!()()( δαααα +

=∫∞

−−

−+− . (7.20)

If the expansion

∑∑∞

=

∞

=− ==

00

)( )()(n

nn

nnn s

AsLCsF α , (7.21)

is considered, then by applying (7.20) in (7.21), the coefficients nC will be derived as

∫∞

−

−+−

+=

0

)(1

)2( )()()!(

! dssFsLesn

nC ns

nαα

α . (7.22)

Therefore, we similarly have

∑∞

=−

−− ==0

)(11 )]([)]([)(n

nn sLLCsFLxf α . (7.23)

On the other hand, since

∑=

=

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

=nk

kk

k

n sL

knn

ksLL

0

1)(1 ]1[!)1()]([

αα , (7.24)

so, eventually the special series

∑ ∑∫

∫∞

=

=

=

−∞

−

−+−

∞ −+−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛

++

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1 1

1

0

)(1

)2(

0

1)2(

)!1(!)()()()(

)!(!

)()(!

1)(

n

nk

k

k

ns

s

kkx

knn

xn

ndssFsLes

nn

xdssFesxf

αδ

αα

δα

αα

α

(7.25)

is a series solution for the Laplace transform inversion. Again, we mention that (7.25) is valid only if its definite integrals are convergent and the function )(sF is expandable


96

under the Dirikhlet conditions. In relation (7.25), the function ses /1)2( −+− α is in fact a weighted distribution on ),0[ ∞ . But, by noting the section 7.1, finite rational classical orthogonal polynomials can also be applied for approximate computation of inverse Laplace transform, because the function )(sF can be expanded by them finitely, and only some limit conditions are imposed on their parameters. For example, if we define the sequence

∑=

=

−+− ⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−==

nk

k

kknqpn

qpn s

knnq

knp

ns

MsM0

),2(),( 1)1(!)1()1()( , (7.26)

that satisfies the orthogonality relation

mnqp

mqp

nqp

p

nqpnpnqnpndssMsM

ss

,0

),(),(2 )!1)(21(

)!()!1(!)()()1(

δ−++−+

+−+=

+∫∞

−−++ , (7.27)

with },{,12,1 nmMaxNNpq =−>−> , then by considering the approximation

2

1;)()(00

),( +<=≅ ∑∑

==−

pNsA

sMCsFN

nnn

N

n

qpnn , (7.28)

and applying (7.27) on (7.28) we get

∫∞

−++++−+−++−+

=0

),(2 )()(

)1()!()!1(!)!1)(21( dssFsM

ss

nqnpnnqpnpC qp

nqp

p

n . (7.29)

Therefore

∑=

−−− ≅=

N

n

qpnn sMLCsFLxf

0

),(11 )]([)]([)( , (7.30)

in which

∑=

=

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−=

nk

kk

knqpn s

Lknnq

knp

nsML0

1),(1 ]1[1

)1(!)1()]([ . (7.31)

Hence (7.32)

∑∑

∫

∫

= =

=

−

∞

−++

∞

++

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ +×

⎟⎟⎠

⎞⎜⎜⎝

⎛

++−+−++−+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

≅

N

n nk

k

kn

qpnqp

p

qp

p

kx

knnq

knp

nxn

nq

dssFsMss

nqnpnnqpnp

xdssFss

qpqpxf

1

1

1

0

),(2

02

)!1()(1

!)1()(

)()()1()!()!1(!

)!1)(21(

)()()1(!!

)!1()(

δ

δ

is an approximate solution for the integral equation )()]([ sFxfL = . Here we add that the inversion problem can also be propounded for the negative power polynomials


97

),,,;(and)(,)( ),()( dcbaxJxHxN qpnn

pn −−− similarly. It is now a good position to present

two practical examples in this way. 7.3. Special examples of section 7.2 Example 1. Let us consider a special case of rational Jacobi polynomials for

2/1,1 −== βα . This case is defined by

)11arccoscos()

11()1(

)21,1(

+−

=+−

=+−

− xxn

xxTxP nn , (7.33)

and called the rational Chebyshev polynomials of the first kind [70]. By noting the

expansion (7.18) let us also suppose that for instance s

sF+

=1

1)( . This implies that the

integrals of (7.18) are simplified as

∑∑ ∫

∫∫=

=

=

=

∞

+

−

∞−

−

−∞ −

+−+−+−

Γ+Γ

=+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

=+

=+

nk

k

knk

kk

n

kknkkkn

nnds

ss

knn

kkn

dssPs

sdss

s

00 02

2/1

0

)21,1(

2

2/1

02

2/1

)!1()!(!)12()!1()1(

)(2)2/1(

)1(2/11

)()1(

and2)1(

π

π

Therefore the inverse Laplace transform for the given )(sF takes the form

∑ ∑∑∞

=

=

=

−−

=

=

−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−

+Γ+−

+−+−Γ

+Γ

+==+

1 1

1

0

1

))!1(

)(2/11)(

!)2/1()(

)!1()!(!)12()!1()1(

)(2)1((

)(21)

11(

n

nk

k

kx

ni

i

i

x

kx

knn

kkn

exn

niini

iinnnn

xes

L

δπ

π

δ

Clearly the above relation is valid if it is expanded. Example 2. For this example, we use the analytic expansion (7.25) and consider the

equation ss

xfL2

)]([ π= . To derive the solution of this integral equation, first we have

to evaluate the definite integrals corresponding to (7.25). Hence we have

.)1()1(

)2/5()1()2/3(83

11

2/5)1()1(

)2/5()1(!)1(

2

)()(and)25(

2)(

2)(

120 0

1)

27(

0

)(1

)2(

0

1)2(

nnnn

nF

nndses

knn

k

dssFsLesdssFesss

sF

nk

k

skk

nss

−+Γ+Γ+Γ++Γ−−Γ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛++−

+Γ+Γ+Γ++Γ

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

=+Γ=⇒=

∑ ∫

∫∫=

=

∞ −++−

∞

−

−+−

∞ −+−

ααα

αα

αααπαπ

αππ

α

ααα


98

Note that for the latter integral we have used again the well-known Gauss identity. Consequently the expansion (7.25) is transformed to

.

)!1(!)()(

)1()2/5()2/3(

83

)()1(

)2/5(2

)2

(

1 1

1

1

∑ ∑∞

=

=

=

−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛−+Γ

+Γ−−Γ

++Γ+Γ

==

n

nk

k

k

kkx

knn

xn

nn

n

xxss

L

αδ

αα

α

δααππ

The above relation holds if and only if .1and1 −∉−+−> Znαα

Chapter 8 Some further special functions and their applications 8.1. Introduction In this chapter, we will introduce some new classes of special functions and study their applications in the classical equations of applied physics. 8.2. Application of zero eigenvalue for solving the potential, heat and wave equations using a sequence of special functions [5] In the solution of boundary value problems, usually zero eigenvalue is ignored. This case also happens in calculating the eigenvalues of matrices, so that we would often like to find the nonzero solutions of the linear system XXA λ= when 0≠λ . But on the other hand 0=λ implies that 0det =A for 0≠X and then the rank of matrix A is reduced at least one degree. This approach can be stated for the boundary value problems similarly. In other words, if at least one of the eigens of equations related to the main problem is considered zero, then one of the solutions will be specified in advance. By using this note, we can introduce a class of special functions and apply for the potential, heat and wave equations in spherical coordinate. Hence let us first define the following sequences

,)))(ln(sinh(

2))(())(())(;(

,)))(ln(cosh(2

))(())(())(;(

zanzazazazS

zanzazazazC

nn

n

nn

n

=−

=

=+

=

−

−

(8.1)

where )(za can be a complex (or real) function and n is a positive integer number. It is not difficult to verify that both of defined sequences satisfy a unique second order differential equation in the form ( ) 0))(()()())()(()()( 32222 =′−′′′−′+′′′ yzanyzazazazayzaza , (8.2) provided that 0)()(2 ≠′ zaza . Consequently, we deal with only one class of special functions, which is in fact the solution of equation (8.2). The functions ))(;( zazCn and

))(;( zazSn have several important sub-cases that are useful to study. First sub-case is the Chebyshev polynomials if one chooses )arccosexp()( ziza = in (8.1) and uses the well-known Euler identity. In this case, the following sequences will appear

Some further special functions and their applications

100

,)(1)cossin())arccosexp(;(

,)()arccoscos())arccosexp(;(

12 zUziznArcizizS

zTznzizC

nn

nn

−−==

== (8.3)

where )(&)( zUzT nn denote the same as first and second kind of Chebyshev polynomials. Moreover, if the selected )(za is replaced in (8.2), the differential equation of the first kind of Chebyshev polynomials is derived as 0)1( 22 =+′−′′− ynyzyz . (8.4) The second sample is the rational Chebyshev functions that can be generated by

)arccotexp()( ziza = . Thus, for this selected case we have

.)arccotsin())arccotexp(;(,)arccotcos())arccotexp(;(

znizizSznzizC

n

n

==

(8.5)

In this way, replacing the related )(za in (8.2) yields

.)1(

)arccot3exp())((,)1(

)arccot3exp()12()()(

,)1(

)arccot3exp())()((,1

)arccot3exp()()(

323

222

222

22

zziiza

zziizzaza

zzizaza

zziizaza

+=′

+−

=′′

+−

=′+

−=′

(8.6)

Therefore the functions (8.5) eventually satisfy 0)1(2)1( 2222 =+′++′′+ ynyzzyz . (8.7) It should be noted that the explicit forms of the real functions ))arccotexp(;( zizCn and

))arccotexp(;( zizSi n− in (8.5) could be derived by the Moivre’s formula. In other words, let us substitute xarccot=θ in the Moivre formula to get

)arccotsin()arccotcos()1(

)(2

xnixnxix

n

n

+=+

+ . (8.8)

Consequently we have

.)()1/()12

1)1(())arccotexp(;(

,)()1/()2

)1(())arccotexp(;(

*12]2/[

0

21

*2]2/[

0

2

xUxxk

nxixSi

xTxxk

nxixC

nn

n

k

knkn

nn

n

k

knkn

=+⎟⎟⎠

⎞⎜⎜⎝

⎛++

−=−

=+⎟⎟⎠

⎞⎜⎜⎝

⎛−=

+

=

−+

=

−

∑

∑ (8.9)


101

It can be shown that the above rational Chebyshev functions )(and)( ** xUxT nn are

orthogonal with respect to the weight function 211)(x

xW+

= on ),( ∞−∞ and satisfy

the following orthogonality properties respectively

⎩⎨⎧

=≠

=+

⎪⎩

⎪⎨

⎧

===≠

=+

∫

∫

∞

∞−

∞

∞−

.,0

1)()(

.02,,0

1)()(

2

**

2

**

nmifnmif

dxx

xUxU

nmifnmifnmif

dxx

xTxT

nn

nn

π

ππ

(8.10)

Here is worthy to point out that J.P. Boyd in 1987 [22] applied the rational functions

)2/)(( 2/12/1* −− xxTn on the interval ),0[ ∞ in spectral methods, and we should here mention that his functions could be derived only by replacing

)arccot2exp()( ziza = in (8.1). But, so far it has been investigated that the Legendre (or Associated Legendre) differential equation (introduced in chapter 2, equation (2.19))

0)()1

()(2)()1( 22 =

−−+′−′′− xy

xqpxyxxyx , (8.11)

has three solutions in Cartesian coordinate as follows (a) 0,0 ≠≠ qp that generates the associated Legendre functions; (b) 0,0 =≠ qp that generates the Legendre polynomials; (c) 0,0 == qp that is reduced to the simple equation 0)(2)()1( 2 =′−′′− xyxxyx ,

which has the solution 21 11ln)( c

xxcxy +

−+

= .

So, a fourth case 0,0 ≠= qp remains, which is different from above mentioned ones

and should be solved. To find the solution of fourth case, we substitute 21

)11()(

zzza

+−

=

in (8.2) to arrive at the differential equation

01

2)1( 2

22 =

−−′−′′− y

znyzyz . (8.12)

Clearly (8.12) is a specific case of (8.11) for 2,0 nqp == . According to (8.1), the solutions of this equation are respectively

.))

11()

11((

21))

11(;(

,))11()

11((

21))

11(;(

2221

2221

nn

n

nn

n

zz

zz

zzzS

zz

zz

zzzC

−

−

+−

−+−

=+−

+−

++−

=+−

(8.13)


102

Here let us claim that the mentioned functions are very applied in the Helmholtz equation in spherical coordinate (see chapter 2, section 2.3.1). In other words, if the equation ∇ =2 2U r k U r( , , ) ( , , )θ θΦ Φ is separated to several ordinary equations, then one of the separate equations takes the form

0)()sin

)1(()(sinsin

12

2

=−++ θθθ

θθθ

ynmmddy

dd , (8.14)

which is equivalent to equation (8.11) for θcos=x . Hence, if θcos=z is considered

in (8.12) or equivalently 2

tan)( zza = in (8.2), the special case of (8.14) for 0=m , i.e.

0sin

)cot( 2

2

=−′+′′ yz

nyzy , (8.15)

has the following solutions

.))

2(tan)

2((tan

21)

2tan;(

,))2

(tan)2

((tan21)

2tan;(

nnn

nnn

zzzzS

zzzzC

−

−

−=

+= (8.16)

These sequences will frequently appear in the given problems of the next section. 8.2.1. Application of defined functions (8.16) in the solution of potential, heat and wave equations in spherical coordinate Usually most of the boundary value problems related to the wave, heat and potential equations in spherical coordinate are reduced to the Helmholtz partial differential equation ∇ =2 2U r k U r( , , ) ( , , )θ θΦ Φ . But, for the special case of k = 0 in this relation we have

0sin1cot12

2

2

2222

2

22

2

=Φ

++++∂∂

θθ∂∂θ

θ∂∂

∂∂

∂∂ U

rU

rU

rrU

rrU , (8.17)

which is known as the potential (Laplace) equation in spherical coordinate. Now, if the related variables in equation (8.17) are separated as )()()(),,( Φ=Φ BArRrU θθ , then the following ordinary differential equations will be derived

.0)sin

()cot(

,0

,02

22

1

2

12

=++′+′′

=−′′

=−′+′′

AAA

BB

RRrRr

θλ

λθ

λ

λ

(8.18)

As we said, the solution of Laplace (potential) equation is generally determined when the boundary conditions are known, nevertheless, if in (8.18) λ 1 0= and λ 2

2 0= − ≠k


103

are assumed, then for the variable r we must have 221

1)( cr

crR +−= ( c1 and c2 are

constant) and for the third equation the same form as equation (8.15). Therefore, the general solution corresponding to the third equation would be

)2

(tan)2

(tan)2

tan;()2

tan;()( 2121θθθθθθθ kk

nn aaSACAA −+=+= , (8.19)

where 1A and 2A & 1a and 2a are all constant values. Accordingly, (8.19) implies to have the general solution of the potential equation 0),,(2 =Φ∇ θrU with the pre-

assigned condition 22

11)( cr

crR +−= as

))2

(tan)2

(tan)(sincos()(),,( 21212

21 θθθ kk aakbkbc

rcrU −+Φ+Φ+

−=Φ . (8.20)

It is interesting to know that the solution (8.20) shows the sensitivity of potential equation with respect to the variable r, so that we have ∞=Φ

→),,(lim

0θrU

r.

As an example, here let us consider the Laplace equation arrU <<=Φ∇ 0;0),,(2 θ (in spherical coordinate) when the variable r takes the pre-assigned form

2

21)( c

rcrR +

−= and the following initial and boundary conditions are given

.),3

,(

,0),2

,(

,0),,2

(

,),,(lim0

Φ=Φ

=Φ

=Φ

∞=Φ→

π

π

θ

θ

aU

rU

aU

rUr

The general solution of this problem, according to the given conditions and assuming 211 cbaAk = , 212 cbaBk = would be finally as

.)sincos())

2(tan)

2()(tan

21(),,(

,),,(),,(

Φ+Φ−−=Φ

Φ=Φ

−

∑

kBkAr

arU

rUrU

kkkk

k

kk

θθθ

θθ

On the other hand, putting the last condition in the above general solution yields

∑ Φ+Φ−=Φ+−

kkk

kkkBkA )sincos()33(2 ,


104

where kA and kB are calculated by

.)1(4sin4)33(

,)1)1((4cos4)33(

0

20

kdkB

kdkA

k

k

kk

k

k

kk

−=ΦΦΦ=−

−−=ΦΦΦ=−

∫

∫+−

+−

π

π

π

ππ

Consequently, the particular solution of potential equation under the given conditions is

.)2

tan2

(tansin)33(

)1(4cos)33(

)1)1((4)2

1(),,(1

2/2/

1

2/2/

2

∑∞

=

−−

−

−

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛Φ

−−

+Φ−−−

−=Φk

kkkk

k

kk

k

kkkkr

arU θθπ

θ

As we see, a special case of the functions (8.1) has appeared in the above solution. Similarly application of zero eigenvalue can be propounded for the heat and wave

equations respectively. For instance, if the classical heat equation ∇ =2UUt

∂∂

is

considered, then separating the variables as U r t S r T t( , , , ) ( , , ) ( )θ θΦ Φ= , where )()()(),,( Φ=Φ BArRrS θθ , yields

⎩⎨⎧

=−′=−∇

⇒′=∇⇒⎪⎩

⎪⎨

⎧

==

∇=∇=∇

00

)()()( 2

2

222

TTSS

StTSTtTS

tTS

tU

STTSU

αα

∂∂

∂∂

∂∂ (8.21)

So, the following ordinary differential equations are derived

.0)sin

()cot(

,0

,0)(2,0

22

1

2

122

=++′+′′

=−′′

=+−′+′′

=−′

AAA

BB

RrRrRrTT

θλ

λθ

λ

λα

α

(8.22)

Again, if λ 1 0= , 0and0 22

2 ≠−=≠−= kn αλ are assumed in (8.22), then the

general solution, when kr

rkJcrkJcrR

)()()( 2/122/11 −+= is pre-assigned, takes the form

,)sincos(

))2

(tan)2

(tan())()((),,,(

21

212/122/11

2

Φ+Φ×

++=Φ −−

−

nana

bbrkJcrkJckr

etrU nntk θθθ

(8.23)


105

in which )(and)( 2/12/1 xJxJ − are two particular cases of the Bessel functions )(xJ p .

For example, let us consider the heat equation t

UtrU∂∂θ =Φ∇ ),,,(2 ; ar <<0 (in

spherical coordinate) when the variable r takes the pre-assigned form

krrkJcrkJc

rR)()(

)( 2/122/11 −+= and the following conditions hold

arbitrary;),()0,,3

,(

,0),0,,(

,0),,2

,(

,),,,(lim0

Φ=Φ

=

=Φ

<Φ→

rfrU

trU

trU

MtrUr

πθ

π

θ

By referring to the general solution (8.23) and using the given conditions we get 012 == ac and 021 =+ bb . So, if 112, cbaA nk = , then

,sin)

2tan

2)(tan(),,,(

,),,,(),,,(

2/1,,

,

2

Φ−=Φ

Φ=Φ

−−

∑∑

nkrJkr

eAtrU

trUtrU

nntk

nknk

k nnk

θθθ

θθ

is the general solution of this specific example. On the other hand, since the orthogonality relation of Bessel functions )(xJ p is denoted by [18]

,)(2

)()( ,),(2

1

2

0),(),( mnmpp

a

nppmpp ZJadxaxZJ

axZJx δ+=∫ (8.24)

where ),( mpZ is thm zero of )(xJ p (i.e. 0)( ),( =mpp ZJ ), it is better for the eigenvalues

k to be considered as 21;),2/1( == p

aZ

k m . Therefore, the general solution of the

problem is simplified as

∑∑ Φ−−

=Φ −

n m

nnm

m

m

mn narZJarZ

ta

Z

AtrU sin)2

tan2

)(tan/(/

))(exp(),,,( ),2/1(

21

),2/1(

2),2/1(

*,

θθθ

in which

aZ

nmn m

AA),2/1(,

*, = . Now, it is sufficient to compute the coefficients *

,mnA . To do

this, substituting the last condition of the problem in above relation yields

∑∑ Φ−

=Φ

−

n mm

m

nn

mn narZJ

ZA

rfar )(sin)(

)33(),( ),2/1(2/1

),2/1(

22*, .


106

Therefore, by applying the orthogonality relation of Bessel functions and using the orthogonality property of the sequence ∞

=Φ 1)}{sin( nn on ],0[ π , *,mnA are found as

)()33(

)(sin)(),(4

),2/1(2

2/323

22

0 0

23

),2/1(2/1),2/1(*,

m

nn

a

mm

mn

ZJa

ddrrnarZJrfZ

A−

ΦΦΦ= −

∫ ∫

π

π

.

Finally, the problem can be stated for the wave equation ∇ =22

2UU

t∂∂

according to

the following stages. First we have

⎩⎨⎧

=−′′=−∇

⇒′′=∇⇒⎪⎩

⎪⎨

⎧

==

∇=∇=∇

00

)()()( 2

2

2

2

2

2

2

2

222

TTSS

StTSTtTS

tTS

tU

STTSU

αα

∂∂

∂∂

∂∂ (8.25)

which results the ordinary equations

.0)sin

()cot(

,0

,0)(2,0

22

1

2

122

=++′+′′

=−′′

=+−′+′′

=−′′

AAA

BB

RrRrRrTT

θλ

λθ

λ

λα

α

(8.26)

Now, if λ 1 0= , 0and0 222 ≠−=≠−= nk αλ are assumed in (8.26), then the general

solution of the classical wave equation when nr

rnJcrnJcrR

)()()( 2/122/11 −+= would be

as follows

.))()(()

2tan

2tan(

).sincos()sincos(),,,(

2/122/1121

2121

nrrnJcrnJcaa

kbkbtndtndtrU

kk −− ++×

Φ+Φ+=Φ

θθθ

(8.27)

Here let us consider a specific problem regarding the wave equation in spherical

coordinate when the variable r has the form nr

rnJcrnJcrR

)()()( 2/122/11 −+= and the

conditions

⎪⎪

⎩

⎪⎪

⎨

⎧

Φ=Φ

=Φ=

=Φ<Φ→

arbitrary;),(),,3

,(.5

0)0,,,(.4,0),0,,(.3

0),,2

,(.2,),,,(lim.10

rgqrU

rUtrU

trUMtrUr

πθθ

πθ


107

are given. To solve the problem, replacing the given conditions in the general solution (8.27) gives 0112 === dbc and 021 =+ aa . If 2121, dcbaB nk = is supposed, then (8.27) becomes

.sin)

2tan

2(tan)()sin(),,,(

,),,,(),,,(

2/1,,

,

Φ−=Φ

Φ=Φ

−

∑∑

knr

nrJntBtrU

trUtrU

kknknk

k nnk

θθθ

θθ

But similar to the previous problem, if a

Zn m),2/1(= is taken, then

∑∑ Φ−=Φ −

k m

kk

m

mmmk k

arZ

arZJatZBtrU sin)

2tan

2(tan

/

)/()sin(),,,(

),2/1(

),2/1(2/1),2/1(

*,

θθθ ,

where

aZ

kmk m

BB),2/1(,

*, = . By substituting the last condition of the problem in the above

relation, i.e.

∑∑ Φ−

=Φ

−

k mm

m

m

kk

mkk

arZJ

ZaqZB

rgar )(sin)(

)sin()33(),( ),2/1(2/1

),2/1(

),2/1(22*

,,

and using the orthogonality relation of Bessel functions )( ),2/1(2/1 arZJ m on [0,a], the

unknown coefficients *,mkB will be derived as

)()33()sin(

)(sin)(),(4

),2/1(2

2/323

22),2/1(

0 0

23

),2/1(2/1),2/1(*,

m

kk

m

a

mm

mk

ZJaaqZ

ddrrkarZJrgZ

B−

ΦΦΦ= −

∫ ∫

π

π

,

which determines the final solution of the given problem straightforwardly. 8.3. Two classes of special functions using Fourier transforms of finite classical orthogonal polynomials [11] Some orthogonal polynomial systems are mapped onto each other by the Fourier transform or by another ones such as the Mellin or Hankel transforms, see [33]. The best-known examples of this type are the Hermite functions, i.e. the Hermite polynomials ( )nH x multiplied by )2/exp( 2x− , which are eigenfunctions of the Fourier transform. More examples of this type are found in [49, 50] and [47]. The latter author showed that the Jacobi and continuous Hahn polynomials can be mapped onto each other in such a way, and the orthogonality relations for the continuous Hahn


108

polynomials then follow from the orthogonality relations of the Jacobi polynomials and the Parseval formula. Now, we intend to introduce two new examples of finite systems of this type in this section and obtain their orthogonality relations. We then estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences. For this purpose, we should come back to the chapter 4 and recall the orthogonality property of the finite classical orthogonal polynomials. Hence, let us here recall the polynomials

( , )

0

( 1)( ) ( 1) ! ( )

np q n k

nk

p n q nM x n x

k n k=

− + +⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

∑ ,

that satisfy the orthogonality relation

mn

qpm

qpnqp

q

nqpnpnqnpndxxMxM

xx

,0

),(),(

))!1(())12(()!())!1((!)()(

)1(δ⎟⎟⎠

⎞⎜⎜⎝

⎛+−++−

++−=

+∫∞

+

for 1and2

1,...,2,1,0, −>−

<= qpNnm . To derive the Fourier transform of the

polynomials )(),( xM qpn we should first refer to the definition of the Fourier transform of

a function, say )(xg , as

( ) ( ( )) ( ) ,isxs g x e g x dx

∞−

−∞

= = ∫F F (8.28)

and consider its inverse transform as

1( ) ( ) .2

isxg x e s dsπ

∞

−∞

= ∫ F (8.29)

In this sense, the Parseval identity of Fourier theory should also be considered as

1( ) ( ) ( ( )) ( ( )) 2

g x h x dx g x h x dsπ

∞ ∞

−∞ −∞

=∫ ∫ F F , (8.30)

for 2, ( )g h L∈ R . Now, to obtain the Fourier transform of )(),( xM qp

n first define the following functions

( , ) ( , )( ) ( ) & ( ) ( ) .(1 ) (1 )

qx axr u x c d x

n mx p q x a b

e eg x M e h x M ee e+ += =

+ + (8.31)

It is clear that the Fourier transform exists for both above functions. However, for example, for the function )(xg defined in (8.31) we have

( , ) 1 ( , )

0

1

0 0

( ( )) ( ) ( )(1 ) (1 )

( 1) ( ) ( 1 )( 1) !( 1) ! (1 )

qx qisx r u x is r u

n nx p q p q

k q is knn k k

p qk k

e tg x e M e dx t M t dte t

u n n n r tn dtn u k t

∞ ∞− − −

+ +−∞

∞ − − +

+=

= =+ +

⎛ ⎞+⎛ ⎞ − − + −= − ⎜ ⎟⎜ ⎟ + +⎝ ⎠ ⎝ ⎠

∫ ∫

∑ ∫

F


109

0

3 2

( 1) ( ) ( 1 )( )! ( ) ( )( 1)! ( 1) ! ( )

1( 1) ( 1) ( ) ( ) 11 1( 1) ( )

knn k k

k k

n

n n ru n q is k p is ku u k p q

n n r q isu n q is p is Fu p isu p q

=

− − + −+ Γ − + Γ + −= −

+ Γ +

− + − −⎛ ⎞− Γ + + Γ − Γ += ⎜ ⎟+ − + −Γ + Γ + ⎝ ⎠

∑ (8.32)

where (...)23 F is a special case of the generalized hypergeometric function defined by

1 2 1 2

01 2 1 2

, , .... , ( ) ( ) ...( )| ,

, , .... , ( ) ( ) ...( ) !

kp k k p k

p qkq k k q k

a a a a a a xF xb b b b b b k

∞

=

⎛ ⎞=⎜ ⎟

⎝ ⎠∑

for ( , ) (3, 2)p q = and again ( ) ( 1)...( 1)kr r r r k= + + − . Note that to derive (8.32) we have used the two identities

( 1) ( )( ) ( )( ) and ( ) .(1 )

k

kk

aa k a a a ka

− ΓΓ + = Γ Γ − =

− (8.33)

Now, by substituting (8.32) in Parseval’s identity we get (8.34)

( ) 1( , ) ( , ) ( , ) ( , )

( )0

3 2

2 ( ) ( ) 2 ( ) ( )(1 ) (1 )

( 1) ( 1) ( 1)( 1) ( ) ( 1) ( )

, 1 ,( ) ( ) ( ) ( )

1, 1

q a x q ar u x c d x r u c d

n m n mx p q a b p q a b

n m

e tM e M e dx M t M t dte t

u n d mu p q d a b

n n r q isq is p is a is b is F

u p

π π∞ ∞+ + −

+ + + + + +−∞

+

=+ +

− Γ + + Γ + += ×Γ + Γ + Γ + Γ +

− + − −Γ − Γ + Γ − Γ +

+ − + −

∫ ∫

3 2

1

, 1 ,1

1, 1

is

m m c a isF ds

d b is

∞

−∞

⎛ ⎞⎜ ⎟⎝ ⎠

− + − −⎛ ⎞× ⎜ ⎟+ − + −⎝ ⎠

∫

On the other hand, if in the left hand side of (8.34) we take 1 and 1 ,u d q a r c p b= = + − = = + + (8.35) then according to the orthogonality relation of polynomials )(),( xM qp

n given above, relation (8.34) reads as

2

,

3 2

3 2

(2 ) !( )!( 1 )! ( ) ( ) ( )( 2 )( 1)! ( 1) ( ) ( )

, ,( ) ( ) ( ) ( ) 1

, 1

, ,1 .

, 1

n mn m

n p b n q a n q a p q a bp b n p q a b n q a n q a m

n n p b q isq is p is a is b is F

q a p is

m m p b a isF ds

q a b is

π δ+

∞

−∞

+ − + − + Γ + Γ + Γ +=

+ − + + + − − − Γ + + Γ + +

− − − −⎛ ⎞Γ − Γ + Γ − Γ + ⎜ ⎟+ − + −⎝ ⎠

− − − −⎛ ⎞× ⎜ ⎟+ − + −⎝ ⎠

∫

(8.36)


110

Theorem 1. The special function

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−+−−−−

+Γ++Γ

= 11,

,,)(

)(),,,;( 23 xcdaxdcbnn

Fda

ndadcbaxAn (8.37)

has a finite orthogonality property as

,

1 ( ) ( ) ( ) ( ) ( ; , , , ) ( ; , , , )2

! ( ) ( 1 ) ( ) ( )( 2 ) ( )

n m

n m

a ix b ix c ix d ix A ix a b c d A ix d c b a dx

n a d n b c n c d a bb c n a b c d n

π

δ

∞

−∞

Γ + Γ − Γ + Γ − −

Γ + + Γ + + − Γ + Γ +=

+ − Γ + + + −

∫ (8.38)

where , 2 , 0 and 0 .a d n b c n a b c d+ > − + > + > + > Remark 1. (i) Replacing n = m = 0 in (8.38) gives the Barnes’ first lemma from 1908, see e.g. Bailey [20] or Whittaker and Watson [76] for the original proof by Barnes. (ii) The weight function of the orthogonality relation (8.38) is positive for

cbda == , or dcba == , . Similarly, the mentioned approach can be applied to the finite orthogonal polynomials

( )

0

( 1)( ) ( 1) ! ( ) ,

np n k

nk

p n nN x k x

k n k=

− +⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

∑

that satisfy the orthogonality relation

1( ) ( )

,0

! ( ( 1))! 1( ) ( ) for , 0,1,2,..., (2 1) 2

p p pxn m n m

n p n px e N x N x dx m n Np n

δ∞ −

− ⎛ ⎞− + −= = <⎜ ⎟− +⎝ ⎠

∫ .

Thus, if we define the sequences

( ) ( )1 1( ) exp( ) ( ) and ( ) exp( ) ( ) ,2 2

x q x x u xn mu x px e N e v x rx e N e− −= − − = − − (8.39)

and take the Fourier transform from ( )u x , then we get

11( ) ( ) 1 ( )22

0

11 2

0 0

( ( )) ( ) ( )

( 1) ( 1) !( 1)!( 1 )! !( )!

( ) ( 1 ) ( 1) 2 ( )(1 )

xpx eisx q x is p qtn n

knn is p k t

k

n p is k k

u x e e N e dx t e N t dt

n q n t e dtq n k k n k

n n qp isp is

−∞ ∞ −− +− − − −

−∞

∞ −− − − +

=

+

= =

⎛ ⎞−= − − − ⎜ ⎟− − − − ⎝ ⎠

− + −= − Γ +

− −

∫ ∫

∑ ∫

F

1

0

2 1

(2 )!

, 1 1 ( 1) 2 ( ) .1 2

kn

k k

n p is

kn n q

p is Fp is

−

=

+ − + −⎛ ⎞= − Γ + ⎜ ⎟− −⎝ ⎠

∑ (8.40)


111

In relation (8.40) the following definite integral has been used

1

1 2

0

2 ( ) .is p k p is ktt e dt p is k∞ −

− − − + + −= Γ + −∫ (8.41)

Now, according to definitions (8.39) apply the Parseval’s identity again to get (8.42)

1( ) ( ) ( ) ( 1) ( ) ( )

0

2 1 2 1

2 ( ) ( ) 2 ( ) ( )

, 1 , 11 1( 1) 2 ( ) ( ) .1 12 2

xp r x e q x u x p r q utn m n m

n m p r

e e N e N e dx t e N t N t dt

n n q m m up is r is F F ds

p is r is

π π−

∞ ∞ −− + − − + +

−∞

∞+ +

−∞

= =

− + − − + −⎛ ⎞ ⎛ ⎞− Γ + Γ + ⎜ ⎟ ⎜ ⎟− − − −⎝ ⎠ ⎝ ⎠

∫ ∫

∫

By assuming 1++== rpuq (8.43) in (8.42) and noting the orthogonality relation of )()( xN p

n given above we can finally deduce the following theorem. Theorem 2. The special function

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−−−

=21

1,

),;( 12 xabann

FbaxBn (8.44)

has a finite orthogonality relation as

mnbamn nbanbandxabixBbaixBixbixa ,2)2(

)1(!),;(),;()()(21 δπ +

∞

∞− −+−++Γ

=−−Γ+Γ∫ (8.45)

if nba 2>+ . Remark 2. (i) If we put n = m = 0 in (8.45) then we obtain

1 ( )( ) ( ) .2 2a b

a ba ix b ix dxπ

∞

+−∞

Γ +Γ + Γ − =∫ (8.46)

(ii) The weight function of (8.46) is positive if ba = . 8.3.1. Evaluating a complicated integral and a conjecture By applying the Ramanujan integral [62] (8.47)

)1()1()1()1()3(

)()()()( −+Γ−+Γ−+Γ−+Γ−+++Γ

=−Γ−Γ+Γ+Γ∫

∞

∞− dbcbdacadcba

xdxcxbxadx


112

we can obtain the explicit value of the following definite integral (8.48)

3 2

, , ( )( ) | 1, 1 (1 )

.(1 ) (1 ) (1 ) (1 )

mn

m

n n b c a x d xI m Fa d c x b x

dxa x d x c x b x

∞

−∞

− − − −⎛ ⎞ −= ⎜ ⎟+ − + − − −⎝ ⎠

×Γ − + Γ − + Γ − − Γ − −

∫

For this purpose, we have (8.49)

0

( ) ( )( )( ) !

( ) ( )(1 ) (1 ) (1 ) (1 ) (1 ) (1 )

nk k

nk k

k m

k m

n n b cI ma d k

a x d x dxc x b x a x d x c x b x

=

∞

−∞

− − −= ×

+

− −×

− − − − Γ − + Γ − + Γ − − Γ − −

∑

∫and since

( ) ( )( 1) ( 1)and(1 ) (1 ) (1 ) (1 )

k mk ma x d x

a x a x k d x d x m− −− −

= =Γ − + Γ − + − Γ − + Γ − + −

(8.50)

according to Ramanujan’s integral defined in (8.47) )(mIn can be simplified towards (8.51)

0

0

( ) ( )( ) ( 1)( ) !

(1 ) (1 ) (1 ) (1 )

( 1) ( ) ( ) (1 ) ( ) ! (1 ) (1 ) (1 ) (1 )

( 1) (1

nk mk k

nk k

k mnk k

k k

m

n n b cI ma d k

dxa x k d x m c x k b x m

n n b c a b c da d k a c b d a b m k c d m k

a

+

=

∞

−∞

+

=

− − −= − ×

+

×Γ − + − Γ − + − Γ − − + Γ − − +

− − − − Γ − − − −=

+ Γ − − Γ − − Γ − − + − Γ − − − +

− Γ −=

∑

∫

∑

3 2

, ,) | 1

, 1(1 ) (1 ) (1 ) (1 )

n n b c a b mb c d F

a d c d ma c b d a b m c d m

− − − + −⎛ ⎞− − − ⎜ ⎟+ − − −⎝ ⎠

Γ − − Γ − − Γ − − + Γ − − −

On the other hand, the Gosper-Saalschütz identity [75] implies that if dcbae −+++= 1 then

2

3 2

, ,| 1

, cos( ) cos( ) cos( ) cos( ) cos( )( ) ( ) .

( ) ( ) ( ) ( ) ( ) ( )

a b cF

d e d e a b cd e

d a d b d c e a e b e c

ππ π π π π

⎛ ⎞= ×⎜ ⎟ +⎝ ⎠

Γ Γ×Γ − Γ − Γ − Γ − Γ − Γ −

(8.52)

Therefore, the final value of the definite integral )(mIn is obtained as


113

(8.53) 2 ( )( )

cos(( ) ) cos(( ) ) cos(( ) ) cos(( ) ) (1 ) (1 )1

( ) ( ) ( ) (1 ) (1 ) (1 )

na dI m

a b b c a d c d a c b d

a d n a b c d n d b m c d m n b d m n a b m

ππ π π π

Γ += ×

+ + − + + Γ − − Γ − −

Γ + + Γ + + + − Γ − + Γ − − − + Γ + − − − Γ − − + But we evaluated the complicated integral (8.48) to be able to claim that the function

),,,;( dcbaxAn defined in Theorem 1 might essentially be orthogonal with respect to the Ramanujan integral. In other words, we conjecture that

3 2 3 2

,

, , , ,| 1 | 1

, 1 , 1

.(1 ) (1 ) (1 ) (1 ) n n m

n n b c d x m m b c a xF F

a d b x a d c xdx K

a x d x c x b xδ

∞

−∞

− − − − − − − −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟+ − + − + − + −⎝ ⎠ ⎝ ⎠

× =Γ − + Γ − + Γ − − Γ − −

∫ (8.54)


114

Bibliography I ) Some parts of the dissertation were published in the following papers: [1]. Mohammad Masjed-Jamei, (2006), A basic class of symmetric orthogonal polynomials using the extended Sturm–Liouville theorem for symmetric functions, Journal of Mathematical Analysis and Applications, In Press, Available online, doi : 10.1016/j.jmaa.2006.02.007 [2]. Wolfram Koepf, Mohammad Masjed-Jamei, (2006), A generic formula for the values at the boundary points of monic classical orthogonal polynomials, Journal of Computational and Applied Mathematics, Volume 191, Issue 1, Pages 98-105 [3]. Wolfram Koepf, Mohammad Masjed-Jamei, (2006), A generic polynomial solution for the differential equation of hypergeometric type and six sequences of classical orthogonal polynomials related to it, Journal of Integral Transforms and Special Functions, Volume 17, Issue 8, Pages 559–576 [4]. Wolfram Koepf, Mohammad Masjed-Jamei, (2006), A Generalization of Student’s t-distribution from the Viewpoint of Special Functions, Journal of Integral Transforms and Special Functions, To appear [5]. Mohammad Masjed-Jamei, Mehdi Dehghan, (2006), Application of zero eigenvalue for solving the Potential, Heat and Wave equations using a sequence of special functions, Journal of Mathematical Problems in Engineering, Volume 2006, Issue 1, Pages 1-9 [6]. E. Babolian, M. Masjed-Jamei, M.R. Eslahchi and Mehdi Dehghan, (2006), On numerical integration methods with the T-distribution weight function, Applied Mathematics and Computation, Volume 174, Issue 2, Pages 1314-1320 [7]. Mehdi Dehghan, M. Masjed-Jamei and M.R. Eslahchi, (2006), Weighted quadrature rules with weight function xpex /1−− on [0, ∞), Applied Mathematics and Computation, Volume 180, Issue 1, pages 1-6 [8]. Mohammad Masjed-Jamei, Mehdi Dehghan, (2005), On Rational Classical Orthogonal Polynomials and their Application for Explicit Computation of Inverse Laplace Transforms, Journal of Mathematical Problems in Engineering, Volume 2005, Issue 2, Pages 215–230 [9]. Mohammad Masjed-Jamei, (2004), Classical orthogonal polynomials with

weight function ),(;)arctanexp())()(( 22 ∞−∞∈++

+++ − xdcxbaxqdcxbax p and a

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