A KEONECKER KLTA.. CU) JOHNS HOPKINS UMY LAUREL NDFPPIED PHYSICS LS It A FARRELL ET AL. SEP 85

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DEMONSTRATION OF AJACOBI POLYNOMIAL REPRESENTATIONOF A KRONECKER DELTA FUNCTION

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ROBERT W. HART

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S,'.- P SUB GROUP i Kronecker delta function

Jacobi polynomials. .. . . .... . fluid shells -

S- 1-- ' -cPssa,, and dertfy by block number,

\ ' n , ! .' Nholc ha' brci gencraltied to obtain an expansion of the Kronecker delta function as an infinite series invols.ing the products

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ERNEST P. GRAY

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- . THE JOHNS HOPKINS UNIVERSITY APPLIED PHYSICS LABORATORYJohns Hopkins Road, Laurel, Maryland 20707

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ABSTrRACT

N\ formula dens ed bN Noble has been generalized to obtain an espansiollof the K roitecker delta f'unction as an inl'inite series insolsi ni the products~ ofmo5 la,:obi polx'noinlials.

T HE JOH~NS HOPKINS UNIVE 14SITYi

APPLIED PHYSICS LABORATORYAUALL MARYLAND

INTRODUCTION

In our ins estigations of the inviscid, incompressible flow of rotating fluidshells confined between concentric, spherical, rigid, co-rotating boundaries.

m~k sC cnCOUntercd the need for a proof of a Kronecker delta function representa-tion in terms of a series of products of two Jacobi polynomials. In particular,%c required a proof that

2"

kkhcrc

F, (A) - E - --- ---- _ __-

An(k+,A+l+ )(k-+p,+ I

X x~j '(A 1'1' (A) ,( 0/ 1 . (2)

For (%Aj )> -- 1, the P,'3 '(A) are the Jacobi polynomials which are orthogo-nal over the interval (-1, + 1) with the integration weight factor w M)(I -- x) (l + x)". They are normalized by the relation Pn " )(I) = i(a, +n + )

*[F'(n i- i)F(c± + )], w ith F(z) the gamma (factorial) function. For (a,,(3) !s - 1,the Jacobi polynomials' P,('' 3)(A) are defined in terms of the polynomials for((x,0) > -I through repeated applications of the contiguous relations2

(2n + o t 1) P,, " (A) = (n + o(+ 3) P,,"' (A) - (n + 3) P,('-', (A) (3a)

K and

(2n - (k+ 3) 1),," ' (A) = (n + u + i) P,4'H + (it + i)P'(A ,) (3b)

with P4'-" (Al I and P-1'((A) =0.

S Certain sums involving products of two Jacobi polynomials have been previ-ously evaluated (see, for example, Refs. 3 - 6). Our method for evaluating thesumn in Eq. 2 employs mathematical procedures analogous to those whichNoble' used to show that 6

*~ ~ ". M1ag[Ws, F. Oberhettinger. and R. P. Soni. Fojrmulas anfd rheoretsOfor the Special Fun(-,tions of Aathernatical Physics. Springer-Verlag. New York (1966).1. S. (iradshtevn and 1. M. Ry/hik, Tables of Integrals, Series, and Products, Alan Jeffrey,led , Academnic Press, Ness York (1965).11. [. %ianocha and B. I . Sharma. "Some Formulae for Jacobi PQlynornials," in Proc. cami,.Phil. So(., pp. 459-462 (1966).1I, 1 - Manocha and B. I . Sharnma. Geinerating Functions of Jacobi Polynomials," in Proc,('utnh. P'hil. Soc., pp. 431-433 (1966).

* II1. Noble, "'Some Dual Series Equations Ins ols'ing Jacobi Poly nornials.'' in Proc. ('arth. Phil.So(.. pp. 363-3*7I (1963).V.. R. Ian~en, A1 Table of Series and Products, F'rent ice- Hall. Ne%\ York I 1975).

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2''1(1 V) a(l +x) (x--y) +h -- IA(xy)

(a+ b-c) kPF(k+c+l)(2k+c+I) p( X ) P-,,. a (y) (4)-- ) r(k+a+l)r(k+b+l)

where

= I if -1 !; y < X ! 1

0 if I < x _< Y I

PROOF

The proof is done in two stages: first the I = 0 case is treated, and thenthe I e 0 case is analyzed. Both stages use the expansion of a function of twovariables in an infinite series of Jacobi polynomials of one variable and coeffi-cients that depend on the other variable. In particular, the coefficients in theexpansion are obtained by respresenting the Jacobi polynomial as a finite se-ries, evaluating the resulting elementary integrals, and re-summing the resul-tant series.

In proving Eq. 1, it is convenient to use the relationship

-- I +X) . (x) for n > 1 (6)

in order to write Eq. 2 as

*IF,..,._ - p (,-,,:. i (A)(2,u+ l)

(I+A) 2 (2k+,i+I+V)2 A) P(M A',(7)+ _ _ P A( "I / ':'* ( A ) -- k A _+ 1- ' ( A ) , (7 )8 k-I k(k+1)

where the first term is the k=0 term of the sum appearing in Eq. 2.

02

• .

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CASE OF 1 0For I = 0, we expand

(I+x)(I+Y) -(

= . C"(")(y) P, ( ), (8a)k=O

where

h xif y > xy if y < x. (8b)

Multiplying Eq. 8a by (1 +x)(i -x)') P,,' ' (x) and integrating from xI -to I yields

(m + 1)2p 5 /2C,"(

0 (y) 2 512

(m+u+3/2)(2m+p+5/2) -1+y)

x dx~l X) 'A+ : P (11+:' ( ) d ( - ) (p + 3/2)

+[ d(l AMp4~ W() [ t~ -rI-) ~ dl+ v(I -- v| P" 0

+~ ~ ~ ~ -dxx(x)" P,,),+ ' 2 ).)( -)(x) :dI

20(l+y) (+ V- -4 dx(I-x) - g + 'l"(x)

The integrals are readily evaluated by expanding the Jacobi polynomial as'

= -y (m,ao, 3;r)( ~x)r (10a)r =-

3

................................. (-)(') -)+:p...... .

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with

r y(m,a,f3;r) r(af+m+lnr(af+(+m+r+I)(-1/)r/

integrating each term; and re-summing the series. In particular,

(m + 1)2"+5/2 -C,/()=(yV(+ '

(mn+ A+3/2)(2m+,u+5/2)

M -2 r4 -7 /2 2 1 +S/

2+I r+

.y~~p '',lr) (r+Mu+ 3 / 2 ) (r+ 1)

x-Y L+ I IM~ yr~~T

(r+ 1 (k+3/2))r=

x [ 2 r+PA+7/2 2-SVA(+ -Y) r+]

(r+1)(r+,u+3/2)

=-2A+/ (+y) (m + I+3/2)2 P M(++I h(l (y]

___________ _ (11)

where the fact that

* y(m + l,pA+ /,- l;r +1)=

-(m + j± + 3/2)2,y (mM + 11,l;r)/[2(r+)A+ 3/2)(r + 1))

is used to arrive at the next to the last line, and Eq. 6 is used in the final step.Inserting this result in Eq. 8a yields

4

........................................

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2P ~~ (-w) -A")-H(0 ) (xy)=

(+y)(1+X) (+ 1)

(2k ++ 5 /2) P"P+ (V 4A ()(I2a)k=0 (k +1)2

or

2 'A +

\2A l 1-w/ (2 1A+l) 8

(2k +,u+ '2 ."y) (I2b)

k2 Pk,( Y) P_

with w defined by Eq. 8b. Since Po(")~ (A) = 1, comparing Eq. 7 with Io to Eq. 12b with x=y=A yields

_ ( )'+ ) (13)(2)A+1) ]-

which demonstrates that Eq. I holds for 1 0.

CAEFor 1 0, we consider

0for I : y 2) x-2t

H"' (x~v)r~p-+5/2) (I -+x y) - (A +3/2) (x-_y)]

fo f an ~x >y 1 (14)

Application of Leibnitz's rule for differentiating a product yields

I-I Y 1+1

W) (x,y) Es -(x-y)(15

5

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APPLIED PHYSICS LABORATORYLAUREL. MARYLAND

with +1 = 3/2)P(,u+ t+3/2)r (1)) -y)r(A-I+5/2)r(t+2)r(t+ l)rI-t)

and U(x,y) I if 1 > x > y -1, and U(xy) =0 otherwise. The func-tion HUl) (xy) is expanded in terms of Jacobi polynomials

H~(I (xy) = CA!~ (Y) Pk;' '~ x (16)k=O

with the coefficients determined from

(m + 1) 9 1 5/

= H(")(xy) (x)I (lxPI+ )()dx

1-1 m= S, r,y(m, 14-1+11,1;r) dx( i -xyu-+ 12+r (Xy)1+l 1 (17)i=0 =O

The integral is evaluated by changing the variable from x to z M (x-y)l/(I -y),which leads to

m

J11)2 ~,MI ,1;r)(1 _Y),,/+ V,+r+ 2 r(l + r+ 3/2)r=O

, r(,u-+r+t+7/2) (8

Substituting for S, the inner summation appearing in Eq. 18 reduces to

(,+/)-I+512 _ rs+31 2+t )]

r~,)= (Mu-+/2)=1);-'+1 (-2r)

(1) 1(19)

For r + r2, we note that

6

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LAUREL. MARYLAND

u+/2 a-2- + V Z I- I

(Ma/)(l,+ aI (l) 0 for r 0.ar, (±-I+ 5/2) z'"11(20)

The derivative is zero at Z I because (I - 2 - r) < I1- 1. The fact that thederivative reproduces the series in Eq. 19 can be readily verified by writingthe factor (I - )-1as a polynomial in z, differentiating each term, and thensetting z= 1. For I < r +2, we note that

ds s. I ~S1 . + IS

(l)'-'r(Mu+ 5/2)r(r+ 1)

The multiple integral in Eq. 21 is readily evaluated by recalling that integra-tion by parts leads to

ds1 ... (1 -se (sjf(s 1 )dsl. (22)0 ~(j-l1)!

b The fact that the multiple integral reproduces the series in Eq. 19 can bereadily verified by writing (I - sj) '- as a polynomial in sj and integratingeach term separately. Inserting Eqs. 20 and 21 into Eq. 18 yields

()r ,+5/2) r=1

I(m,1) = r(1-1+r+312)r! for M - -r (MA+52+r) (r+ 1-1)!

0 otherwise. (23)

* .Form 2tI 1, using the definition of y(m,,u-4 I ,l2 ;r) and introducingthe indices r' -r--(I - 1) and m' =m-(I - 1) leads to

7

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III') - __________

- min + A + /) r(4+ 5/2 -1

[(y- 1 )/2y r(in' +.u+l+ +r')

where the definition of -y(m-l+ 1, At+1+ V2, - l;r) was employed to obtainthe last line. Inserting Eqs. 23 and 24 into Eqs. 17 and 16 yields

- GA+ 3 12 'r(/L+5/2) (2m+ji-1+5/2)

~j - r(I+5,2-/) m=I-1 (m+l)(m+±+312)

X P,( J('/2 1-(y) P,,(A-1+'4'I)(X) (25)

Changing the summation index to m' = m - (I - 1), writing the m' =0 termseparately, and using Eq. 6 in both the mn' = 0 and the mn' d0 terms lead to

H(1) (x.y) ( P~+/)[(+.I() + (~)Iy)(l+x)r(,u+5/2 -1) L(2p+ 1) 8

(2XfM y (26)M, ' (in' + 1) rnIpM,+'.)xj

Setting x=y= A in Eq. 26 and comparing with Eq. 7, one finds

H (A1A 2 -)r ( +5/2) )F..f 0, 1 60 , (27)

where the vanishing of W)' (AA) follows directly from the form of Eq. 15.This completes the proof of Eq. 1.

8

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dl REFERENCES

'W. Miagnus, F. Oberhettinger, and R. P. Soni, Formulas and Theoremnsforthe Special Functions of Mathematical Ph sics, Springer-Verlag, New York(1966).

21. S. Gradshteyn and 1. M. Ryzhik, Tables of Integrals, Series, and Products,Alan Jeffrey, ed., Academic Press, New York (1965).'H. L. Manocha and B. L. Sharma, "Some Formulae for Jacobi Polvnomi-als," in Proc. Camb. Phil. Soc., pp. 459-462 (1966).

4 H. L. Manocha and B. L. Sharma, "Generating Functions of Jacobi Poly-nomials," in Proc. Camb. Phil. Soc., pp. 43 1-433 (1966).

5 5B. Noble, "Some Dual Series Equations Involving Jacobi Polynomials, "inProc. Camb. Phil. Soc., pp. 363-371 (1963).

6 E. R. Hansen, A Table of Series and Products, Prentice-Hall, New York(1975).

9

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