A note on Henrici’s triple product theorem

Abstract

Making use of certain known transformations in the theory of hypergeometric functions, the authors prove a general triple series identity which readily yields Henrici's recent result expressing the product of three hypergeometric OF, functions in terms of a hypergeometric 2F7 function. Recently, Henrici [2] derived the elegant formula: (1) OF0F O [6O; I [21 6c; L-6c; j -6c; 27L6c 2c, 2c + 1, 2c + 2, 4c 4, 4c, 4c + 9J 3' 3' 3 where (2) co = exp (3 1 ), by consideration of the differential equations satisfied by the functions on either side of (1). In the present note we give a shorter proof of (1), utilizing certain known transformations of hypergeometric functions. Indeed, we shall first prove the following general triple series identity: oo n+2p m?n+p (3) mn A m+n+p (b) (b)n (b)p m! n! p! oo A3r (12b -4 )r ( 2 b + 4)r (9X) E (b)r ( 3 b) r ( 3 b + 3 ) r ( 3 b + 23 ) r ( 3 b-3 ) r ( 3 b) r ( 3 b + 3 ) r ! where {/A0}n=O is a bounded sequence of complex numbers, and w is defined by (2). For b 6c and A0 =1 (n > 0), the identity (3) evidently yields Henrici's formula (1). Received by the editors March 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 33A30. KeY words and phrases. Hypergeometric functions, Chu-Vandermonde theorem, Appell functions, triple series identity, quadratic and cubic transformations, Pochhammer symbol. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page