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The computation of 3 F 2(1)Jet Wimp aa Drexel University , Philadelphia, PA, 19104Published online: 20 Mar 2007.

To cite this article: Jet Wimp (1981) The computation of 3 F 2(1), International Journal of Computer Mathematics, 10:1,55-62

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Intern J Con~plcrer Murh 1981, Vol 10, pp 55-62 0320-7160/81 1001-0055 $06 50 0 L' Gordon and Breach Sclence Publishers Inc , 1981

Ipr~nted In Great Brltdln

The Computation of ,F,(I ) JET WIMP

Drexel University, Philadelphia, PA 19 704

(Recewed July 1980, in finalform March 1981)

We present an algorithm for computing the generalized hypergeometric function with unit argument, ,F,( l) . The algorithm, based on a 3-term recursion relation, is effective for computing the analytic continuation of the function in any of its parameters. As an application, we give a new algorithm for computing the Beta function.

KEY WORDS: Hypergeometric function, recursion relations, Miller's algorithm, Beta function

C R. CATEGORIES: 5 12, 5.13

1. INTRODUCTION

The function ,F2(1) is one of the fundamental special functions of applied mathematics. It occurs in a surprising diversity of contexts in many different disciplines.

In mathematical physics the function sometimes arises in the evaluation of a beta integral of a solution of the important hypergeometric differential equation. In fact, it was in working on the problem of computing

1

1 (1 - t)"(l + t)bPf,P)(t) dt, - 1

(1)

Pf,P'(t) a generalized Jacobi function, that I was led to develop the present algorithm. I thank Y. L. Luke for bringing this problem to my attention.

It has been said that if ,F2(l) could be determined as a closed-form expression, e.g., as a ratio of finite products of Gamma functions (as can the function ,F, (I)), much of applied mathematics could be substantially

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5 6 J . WIMP

simplified. For special values of its parameters, ,F,(l) can be expressed in closed form, see [I , Section 4.41 and, more recently [2. v.l. Chapter 3). However, attempts to extend these tables in any dramatic way have met with failure.

It is surprising that so little attention has been devoted to the computation of the ,F, ( l ) . In fact its computation, except for special values of the parameters, is a difficult unresolved problem in numerical analysis.

In what follows let r , , 'r,, r,. /I,, / i 2 be complex numbers, P i + O , - 1, - 2, . , .,

I use the notation M(x , ) to denote a product of values ,tf(x, )M(r,)!LI(r,) , e.g.

Whenever the quantity s appears. this convention is being invoked The series defintiion

converges iff

In almost any given problem the series (4) is an unsatisfactory way of computing F. Note. however. that if any r j is zero or a negative integer the series terminates and the computation of I; is trivial. Throughout I will assume this does not happen,

xi*O,-1,-2 ,..., i=1 ,2>3 . (6)

Now, for large k the general term of the series behaves like

On the other hand the series (4) with Pl replaced by Dl + N , N an

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COMPUTATION OF ,F, (1) 5 7

"Integer, has a general term which behaves like

N in this series can always be chosen large enough to produce any desired accuracy. Thus if there were a formula to relate ,F, with a denominator parameter p, + N for one or more values of N to F itself, a computational algorithm could be constructed from such a formula.

The purpose of this paper is to present and analyze the application of such a formula, which turns out to be a three-term recursion relation. The algorithm serves to compute ,F,(1) for all values of a,, P, for which the $unction is analytic, not merely those which guarantee the convergence of the series (4). Further the algorithm does not appear to be very sensitive to 'the values of a,, p,.

2. THE RECURSION RELATION

The recursion relation on which the algorithm is based is a special limiting case of a recursion relation for a generalized hypergeometric function. I gave this relation in a 1967 paper, 131. A generalized version is given in the reference [2, pp. 135-1371 and I will refer to this treatment. In formulas (a), (16) of that reference replace z by z / i and let i+ x. (This is called a confluent limit). After making an obvious identification of parameters (s = 3, r =2, etc.) and performing some algebra one finds that the recursion relation, originally of fourth order, dramatically reduces to one of only second order:

where

Ao(n)=(n+p,)(n+P1 +l ) (n+/ j -x ) ,

I

A solution of (9), considered as a second order difference equation, is

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58 J. WIMP

The method of derivation of the general formula from which (9) was derived makes clear that 4,, satisfies the recursion relation provided n is suficieiztly large and the parameters xi, pj fulfil certain conditions (c.f. the theorem to follow). Then the series for 4, converges uniformly and thus defines an analytic function of its parameters.

The recursion relation will be used in the backward direction. For a suitably large value of N, 4,v+ ,, 4, will be computed. Then the recursion relation is used for n = N - 1, N - 2, . . ., 0, to obtain 4, = F. In fact, (9) may be used to deduce the domain of analyticity of F, a result which does not seem to be available in the standard references.

def THEOREM Let p = ( r l , a2, u,, bl, p2) he a rector in C5 , and let U be the subset oj C 5 satisfying

Then F is analytic for p~ CT.

Proof The proof is quite simple. n may be chosen, by virtue of (8), so the series definitions of $,+, and 4,,+, converge uniformly, and hence are analytic, for p on any compact subset K of U . Since A,(iz) cannot vanish, 4,, 4 , , 1 , . . ., 4o = F can be computed by (9). For r,,b, fixed this can be considered a functional equation in the variable Bl = z. Since the equation holds for 8 , sufficiently large and the coefficients are analytic when p is restricted by (13) the permanence principle of functional equations (see, for example 14, p. 1071) states the equation is satisfied by the analytic continuation in f i , = z of F, and, in fact, serves to analytically continue F. Obviously, this analytic continuation is also analytic in r,,b,.

Perhaps a more convenient form of the coefficients for computations is

where N=D, + 1.

3. THE ALGORITHM

The algorithm proceeds in 4 steps. I assume cc,, 0, are such that F exists and is nontrivial, i.e., (6) and (13) are satisfied. I also assume cn,,P, are real.

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COMPUTATION OF ,F, (1) 59

This assumption isn't necessary for the convergence of the algorithm, but it simplifies the presentation somewhat.

1 ) Pick R l , R 2 to be the smallest positive ( 2 0 ) integers such that

- R 1 < a , , b 1 5 R 2

and set

2 ) Pick N ( > 1 ) an integer such that

where E is the desired accuracy and

(Note a ballpark estimate for N can be obtalned by approximating the LHS of (17) by K N " + ~ - ~ I ~ (4127 )*.

3) Compute, as approximations to 4, +.+,, j = 0,1,

4) Compute @, i z=N+P-1, N + P - 2 ,..., 0,

from

Then

It can be shown, using the standard asymptotic techniques of difference equations, see [ 5 ] , that when N is determined via (17), then I4* h + P + I

- 4N+P+JI < E and, further, ij all computations are subsequently performed

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60 J. WIMP

in exact arithnzetic,

In the accompanying table 4; is computed for increasing vnlues of N for values of r i , / l j for which F can be evaluated in closed form (see [I, p. 189, formula (5)l). The computations were done in APL which has 16 figure precision. Increasing N increases the accuracy of 4; up to a point. After that the algebraic accumulation of roundoff error becomes significant, and to further increase N produces a deterioration in accuracy in accordance with formula (22).

4. COMPUTING THE BETA FUNCTION

The same considerations lead to an interesting algorithm for computing B ( s , y ) for

The computation is based on the evaluation of a ,!, with u n ~ t argument and denominator parameter c = 11 + 1. The appropr~ate recursion formula can be found from [ I , p. 104, formula (4611.

1) Pick R , , R , to be the smallest positive ( 2 0 ) integers such that

and set

2) Pick N ( > 1 ) an integer such that

K N ~

( N - 1 )

where c is the desired accuracy and

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COMPUTATION OF ,F,(1)

cc's 0.85, 0.36, 0.2 0.71, -0.36, 2.315 0.35, 0.73, 1.56

a-/1-1 -2.73 -. . . - .- -- - -

N 5 1.03618 7 1.03618

10 1.03618 12 1.03618 15 1.03618 20 1.03618 25 1.03618

TRUE 1.03618

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62 J . W I M P

3) Compute, as an approximation to d),+,+ ,.

4) Compute $:, n = N + P , N+P-1, ..., O

Then

It can be shown that error accumulates like

where 4;+, = $ ,+ , + ci, ci being the initial error. Since T(l -x), T( l -y) are not, strictly speaking, known, it is necessar!

to obtain lower bounds for them to use (27) . However, this is easily done For instance for a > 0,

5 . CONCLUSIONS

The methods of t h ~ s paper can be used to construct algorithms for thc evaluation of ,F,(z), / z / = 1. and for ,+,F, (I), p > 2 . Of course, thr recursion formulas involved are of higher order.

References

A. Erdelyi et al.. Higher Trunscerzdei~tal Function.,. \ . 1. McCraw-Hill, N.Y. (1953).

Y. L. Luke, The Special Functio~is and Their Approsi~nations. 2v., Academic Press, N.Y (1969). Jet Wimp, Recursion Formulae for Hypergeometric Functions. Math. Comp. 21 (1967). 363-373. Z. Nehari, Cor~fornlul Mapping, McGraw-Hill, N.Y. (1952) . Jet Wimp. Sequt~~zce Trumfbrrncrtions ulul Their Appiiccrrions. Academic Press, N.Y. (1981).

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