Reduction of certain multiple hypergeometric functions

Proceedings A 85 (3), September 20, 1982

MATHEMATICS

Reduction

of certain

multiple

hypergeometric

functions

by Per W. Karlsson

of Denmark, DK-2800 Lyngby, Denmark

The Technical University

Communicated by Prof. C.J. Bouwkamp at the meeting of September 28, 1981

SUMMARY It is shown that a generalized KampC de Ftriet function whose variables are all equal and whose parameters satisfy certain relations reduces to a single hypergeometric function.

It is a well-known fact that a generalizedKampCde FCriet function $j:::i~ whosevariablesare all equalreducesto a singlehypergeometricfunction p+ rFg; relations between parameters are not required. Some applications of this reduction formula have been discussedrecently by Srivastava[6]. In the presentnote we establisha reduction formula for a generalizedKampe de Feriet function with two more parameters per variable; the variables are again equal, whereasthe parametershave to satisfy certain relations. The reduction formula reads,

9:2;...;2

al,,..,ap:

q:l;...;l

I cl ,..., cq:

(1) =p+2&

it holds for all z if

1

p

bI,&+bl;...;

1+2br;...;

bn,++b,;z 1+2b,;

al, . . . . ap, bl + .. . +b,,++bl+...+b,; 1+2b,+...+2b,;’ [ Cl, .*.,cq,

‘*--”

1 ;

< 4, and for z in the cut plane if p = q. 285

Equation (1) is a particular caseof the multiple seriesidentity

where (C) is an arbitrary sequenceand (h,,@= T(h +&/T(h) denotesthe Pochhammer symbol; absolute convergenceis of course assumed. Indeed, (1) follows from (2) if we take

and, if p = q, apply the principle of analytical continuation. To prove (2), introduce for brevity (3)

B=bl+...+b,,

M=m,+...+m,,

and considerthe product P(Z)= fI zFl(bi, 3 + bi; 1+ 2bi; Z) i=l

Now, by virtue of the well-known reduction formula 2Fl(b,++b; 1+2b;z)=(+++1/(1 -z))-~~ (cf. e.g. [2, Q2.81) we have

P(z)=##,++B;

1+2B;z)=

OD c (4 k)(+t-4 k)zk;

k-o (1 + 2B,k)k!

and the uniquenessof the Taylor expansionthus yields (4)

n (bi, mi)(3 + 4, mi) = (4 k)(+ + 4 k) M?, i!i (1+2b iy m)ml (I+ 28, k)k! ’ i i+

Clearly, (2) now follows. The reduction formula (1) leadsto further results when known theoremsare applied to its right-hand side. As an illustration we shall consider someapplicationsof(l)forp=2=qandz=l. 286

The summation theorem

follows from (1) by application of Bailey’s summation formula [l] (6)

4F3

a+N, -N,+g,f-++g; +a,+++a,l +g;

NE (0,1,2, ...I.

Equation (5) was derived in a different way by Srivastava [5], and for double series it was given by Sharma [3]. The restriction upon N cannot, as claimed in [3], by replaced by the condition that the real parts of a - g, a + N, a-g + N be positive: indeed, for a = 2g + 2, N= -g- 1, Re g> - 1, the two sides of (6) become

which are different in general. Further summation theorems involving n + 2 free parameters are obtained by means of Gauss’s theorem for Cl =b1+ . ..+b.,

c$=++q,

and from Watson’s theorem for q=b*+...+b,,

cz=+(l +a, +az).

We finally remark that two summation theorems for double hypergeometric series given by Sharma and Abiodun [4, eqs. (5), (6)] are readily, by virtue of (I), generalized to an arbitrary number of variables.

REFERENCES 1. Bailey, W.N. - Some identities involving generalized hypergeometric series, Proc. London Math. Sot. (2) 29, 503-516 (1929). 2. Erdelyi, A., W. Magnus, F. Oberhettinger and F.G. Tricomi - Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, Toronto and London 1953. 3. Sharma, B.L. - Sum of a double series, Proc. Amer. Math. Sot. 52, 136-138 (1975). 4. Sharma, B.L. and F.A. Abiodun - Some new summation formulae for hypergeometric series of two variables, Rend. 1st. Mat. Univ. Trieste 8, 94-100 (1976). 5. Srivastava, H.M. - The sum of a multiple hypergeometric series, Nederl. Akad. Wetensch. Proc. Ser. A 80 = Indag. Math. 39, 448-452 (1977). 6. Srivastava, H.M. - A note on certain summation theorems for multiple hypergeometric series, Simon Stevin 52, 97-109 (1978).

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