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JOURNAL OF COMPUTATIONAL AND APPLIEDMATHEMATICS

Journal of Computational and Applied Mathematics 72 (1996) 293-300

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Generalizations of Whipple's theorem on the sum of a 3F2 J.L. Lavoie a'*, F. Grondin b, A.K. R a t h i e c aD~partement of Mathbmatiques et de Statistique, Universitb Laval, Qua., Canada G1K 7P4 bForintek Canada Corp., Facult~ de Foresterie et de G~omatique, Universitb Laval, Qub., Canada G1K 7P4 CDepartment of Mathematics, Dungar Autonomous College, Bikaner 334001, Rajasthan, India

Received 11 August 1995; revised 29 October 1995

Abstract

Thirty-eight summation closely related to Whipple's theorem, in the theory of the generalized hypergeometric series, are obtained. Some limiting cases are also considered. A M S classification: Primary 33C20

1. Introduction This is the last of a set of three notes dealing with generalization of the classical theorems of Watson, Dixon and Whipple, on the sum of certain 3F2, in the theory of the generalized hypergeometric series. The extensions of the results of Watson [6] and Dixon [-7] were obtained, in the first two notes, from a methodic exploitation of the simplest relations existing between contiguous series. These are particular cases of the general contiguous function relations given by Rainville [-8, 9, p. 80] and more specific details will be found in [4, 5]. Clearly, the same procedure could have been utilized to extend Whipple's result. But, instead, it was decided to deduce these generalizations from the different extensions of Dixon's theorem, contained in the second note. There, the convergent series

aF2Ia,

b, l +i+a-b,

c l +i+j+a-c of i,j -- - 3, - 2,

1]

(1)

is evaluated for a subset - 1, 0, 1, 2, 3. More specifically, 39 distinct formulas are given, including Dixon's result which appears when i = j = 0.

* Corresponding author. 0377-0427/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 0 4 2 7 ( 9 5 ) 0 0 2 7 9 - 0

294

J.L. Lavoie et al./Journal of Computational and Applied Mathematics 72 (1996) 293-300

We start from the first fundamental relation between 3Fz given by Bailey [1, p. 14, Eq. (3.2.1)]. We reapply that formula on itself to find F ( e ) -F (ae)+Tf ~- a --+?--b-c) b 3F2 I a, b, f 1I = ~-~ e,

c) 3F2 [ a , f - b ,

f,

f-c e+f-b-c

11 .

Hence, with b = 1 + i + j - a and f = 1 + i + 2c - e, we obtain 3F2Ia, l - a =

+ i + j, c e, l+2c-e+i

F(e)F(c-j)3Fz[a+2c-e-j, F ( e - - a)F(a + e - - j )

1] l+c-e-i, a + c -j

a [1]. 1 + 2c - e + i

(2)

Dixon's theorem can be used on the right of (2), when i = j = 0, and Whipple's theorem is obtained. With the help of the duplication formula for the gamma function, this result can be written in the form I-a, 1 - a, 3F2 / e, 1 + 2 c - - e

C lll

nF(e)F(1 + 2c -- e) = 2 zc- 1F(a/2 + e/2)F(1/2 - a/2 + e/2)F(1/2 + a/2 + c -- e/2)F(1 -- a/2 + c - e/2)'

(3)

where the condition for convergence is R(c) > O. When the remaining 38 known evaluations of (1) are used on the right of (2), then generalizations of Whipple's theorem are obtained.

2. Main result

In [7], the 38 formulas generalizing Dixon's theorem were obtained separately and then incorporated into a single artificial formula, in order to control the length of the paper. The same procedure is followed here. Our results generalizing Whipple's theorem are contained in the general formula

r ( e ) r ( f ) r ( c - k(j + [J I)) F(e - c - ½(i + l il))F(a - ½(i + j + l i + J I)) 22"-~-Jr(e - a ) F ( f - a)F(e - c)r(a)r(c) r ( e / 2 - a/2 + ¼(1 - ( - 1)'))F(f/2 - a/2) x A~,j F(e/2 + a/2 - i/2 + [ - j / z ] ) r ( f / 2 + a/2 - i/2 + ( ( - 1)J/4)(( - 1)~ - 1) + [ - j / 2 ] ) F(e/2 - a/2 + ¼(1 + (-- 1)i)) + Bi, j F(e/2 + a/2 -- 1/2 -- i/2 + [ - j / 2 + 1/2])

r(f/2 - a/2 + 1/2) × r ( f / 2 + a/2 - 1/2 - i/2 + ( ( - 1)i/4)(1 Z ( - _ 1)i) + [ - j / 2

] + 1/2]j;'

(4)

J.L. Lavoie et al./Journal o f Computational and Applied Mathematics 72 (1996) 293-300

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J.L. Lavoie et al./Journal o f Computational and Applied Mathematics 72 (1996) 293-300

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J.L. Lavoie et al./Journal of Computational and Applied Mathematics 72 (1996) 293-300

297

where a + b = 1 + i + j, e + f = 2c + 1 + i, Ix] is the greatest integer less than or equal to x and its modulus is denoted by [xl. Also, i, j takes values in a subset of 0, _ 1, + 2, __+3 and the coefficients At, j, B~,i are given in Tables 1 & 2. The 3F2(1) will converge for R(c) >j,j = O, + 1, ___2, + 3. More to the point, if h~,j is the left-hand side of (4), then the available formulas appear in the array ha, - 3

h3,

2

ha, - 1

ha, 0

--

--

h2, - 3

h2, - 2

h2, - 1

h2, 0

h2, 1

h2, 2

hl,-3

hl,-2

hi,-1

hi,0

h1,1

hl,2

ho. - 3

ho, 2

ho, 1

ho, o

ho, 1

ho,

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--

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2

--

h-l,-2

h-1,-1

h-l,o

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--

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h-2,0

h-2,1

h-2,2

h-2,3

--

--

--

h-3,o

h-3,1

h-3,2

h-3,3

(5)

The dashes represents results that were judged inappropriate for publication. Note that ho, - 1 and ho, 1 have appeared in print before [4, p. 270, Eqs. (3) and (4)], while h0. o is Whipple's result.

3.

Some

limiting

cases

If, as in the preceding section, h~,i is the left-hand side of (4), then limc-.oohi, j = 2F1 [a, 1 + i + j - a; e; ½] and the sum of this series can be found from any element, in a given diagonal of the array, for i + j = 0, _ 1, _ 2, _ 3, + 4. In fact, after a simple change of variables, the following formula can be constructed:

2Flla'l+n--a~]b

r(½)r(b)r(1 - a) - 2b-"- 1 - - - - ~ a + + r(b/2

-

f

~ n 7+ Inl)) ]F(b/2

a/2)r(b/2 + a/2

-

1/2 -

Ctn -- a/2 + 1/2)F(b/2 + a/2 [n/2])

[1/2 + n/23)

"

(6)

The en and fin are given in Table 3 and the values have been extended to include n = __ 5. Note that n = 0 is Bailey's formula [1, p. 11, Eq. 2.4.3)] while the cases corresponding to n = - 1 and n = 1 have been given before in [4, p. 273, Eqs. (10) and (11)]. F o r m u l a (6) can be useful for other related series since

2Fl[a' l + n - - a

1]=2a2Fl[a' a + b - nb- l l

--1 b

"

These are immediately obtained from Euler's transformation [9, p. 60, Ths. 20 and 21].

298

J.L. Lavoie et a l . / J o u r n a l o f C o m p u t a t i o n a l a n d A p p l i e d M a t h e m a t i c s 72 (1996) 2 9 3 - 3 0 0 Table 3 • . a n d ft.

n

fin

an - (4b 2 - 2ab - a 2 - 22b + 13a + 20)

5 4 3 2 1

2(b - 2)(b - 4) - (a - 1)(a - 4) a-2b+3 b-2 -1

0 -1

1 1

-2 -3 - 4 - 5

b 2b - a 2b(b + 2) - a(a + 3) 4b z - 2 a b - a 2 + 8b - 7a

4b 2 + 2 a b -

a 2 - 34b - a + 62

- 4 ( b - 3) a+2b-7 -2 1 0 1 2 a+2b+2 4(b + 1) 4b 2 d- 2ab - a 2 -Jr 16b - a + 12

4. A g e n e r a l f o r m u l a

The results derived up to now were obtained from a systematic exploitation of the simplest relations existing between contiguous functions. Clearly, these multiple relationships seem to be special cases of a more general formula which would be interesting to obtain. Let us restrict our attention to t h e special case given in preceding section. The simple nature of F [a, 1 + n - a; b; ½] invites a search for a general representation which will be obtained for n a nonnegative integer. Consider the transformation

(fl)"~ (-- l)k(n~(fl -~)kp+2F,+ F(ap),m, 1 + ~ - = (a).k = 0

\kJ

fl wl

' L(bq), 1 + a -- fl -- k I

J'

(7)

where n = 0, 1, 2, ..., and (7p) stands for the set of parameters 71, 72, ... ,7p. We assume that the parameters are such that the right-hand side is not singular and that when p + 1 ~< q, w is finite and when p = q, Iw[ < 1. That finite s u m m a t i o n is substantially close to several results given by Joshi and M c D o n a l d [3] and Gottschalk and Maslen [2, p. 1997]. Here is a short proof where, besides the usual manipulations involving the factorial function, K u m m e r ' s first formula for the confluent hypergeometric function, 1Fl[a; b; z] = e ' l F l [ b -- a, b, - z], with e" = Z~=o(zn/n!) [9, p. 125, Eq. (2)] will be used in the analysis. We have ~?(~)"

1Fq[(aP)i~;nw]Zn

J.L. Lavoie et al./Journal of Computational and Applied Mathematics 72 (1996) 293-300 = e " ~ [(ap)], (e), w, [fl-e-r ,=o r! [(bq)], 1F1 fl

299

1 -- Z

=eZ~(fl--Ct)k ~ (ap), ct, 1 + ct-- fl ] ,=o k!(fl)k ( - z)kp+2Fq+~ l(bO, 1 + ~ - fl - k w ,,k (fl -- O:)kp+zFq+x F (ap), ~, 1 + ~ -- fl = .=ok=o ~ ~ ( - x' fci~(~-~ k(bq), 1 + ~ - [ l - k l

lz,+ k wJ

~ l)k(n'~ (fl Z~)kp+2Fq+ F (av)'~' 1 + c t - - fl =,=Ok=O ( \k] (fl)k l[(bq),l + c t - f l - k

]z" w n~..

(8)

Equating the coefficients of z" in the first and last m e m b e r of (8), we arrive at (7). Now, if p = q = 1, w=½, e=l-a and f l = 2 - a - b , the infinite series on the right of (7) reduces to 2F1 [a, 1 - a; b - k; ½] which can be evaluated [1, p. 11, Eq. (2.4.3)]. Thus, after separating the even from the odd order terms, we find

r ( 1 / 2 ) r ( b ) ( 2 - a - b), = 2b- l(a + b -- 2)(1 -- a),

{

1

x r ( ( b + a - 2 ) / 2 ) r ( ( b - a + 1)/2) - r ( ( b + a -- 1)/2)r((b - a)/2)3F2

1/2, 3/2,

3/2 - a/2 - b/2 1 2 - a/2

b/2 1

1

,

for n = 0, 1, 2 , . . . . This is the desired general formula. It can be used to check and extend Table 3 for ~, and ft,, for n a nonnegative integer.

Acknowledgements The authors want to express their thanks to Prof. Jet Wimp, principal editor, for a very valuable remark.

References [-1] W.N. Bailey, Generalized Hypergeometric Series, Cambridge Math. Tract No. 32 (Cambridge Univ. Press, Cambridge, 1935; reprinted by Stechert-Hafner, New York, 1964). [2] J.E. Gottschalk and E.N. Maslen, Reduction formulae for the generalized hypergeometric functions of one variable, J. Phys. A: Math. Gen. 21 (1988) 1983-1998. [3] C.M. Joshi and J.B. McDonald, Some finite summation theorems and an asymptotic expansion for the generalized hypergeometric series, J. Math. Anal. Appl. 40 (1972) 278-285.

300

J.L. Lavoie et al./Journal of Computational and Applied Mathematics 72 (1996) 293-300

[4] J.L. Lavoie, Some summation formulas for the series 3F2, Math. Comp. 49 (1987) 269-274. [5] J.L. Lavoie, Notes on a paper by J.B. Miller, J. Austral. Math. Soc. Ser. B 29 (1987) 216-220. [-6] J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Watson's theorem on the sum of a 3F2, Indian J. Math. 34 (1992) 23-32. I-7] J.L. Lavoie, F. Grondin, A.K. Rathie and K. Arora, Generalizations of Dixon's theorem on the sum of a 3F2, Math. Comp. 62, 267-276. [-8] E.D. Rainville, The contiguous function relations for pFq with applications to Bateman's j~,v and Rice's H,((, p, v), Bull. Amer. Math. Soc. 51 (1945) 714-723. [-9] E.D. Rainville, Special Functions (Macmillan, New York, 1960; printed by Chelsea, Bronx, New York, 1971).