TRANSFORMATION FORMULAS FOR TERMINATING SAALSCHUTZIAN HYPERGEOMETRIC SERIES OF UNIT ARGUMENT

Transformation formulas for terminating Saalschfitzian hypergeometric series of unit argument p + 1Fp(1) are presented. They generalize the Saalschfitzian summation formula for 3F2(1). Formulas for p 3,4, 5 are obtained explicitly, and a recurrence relation is proved by means of which the corresponding formulas can also be derived for larger p. The Gaussian summation formula can be derived from the Saalschiitzian formula by a limiting process, and the same is true for the corresponding generalized formulas. By comparison with generalized Gaussian summation formulas obtained earlier in a different way, two identities for finite sums involving terminating aF2(1) series are found. They depend on four or six independent parameters, respectively.

When b 2 a 3 or b 1 a3, (1.6) reduces to (1.4).The limit mc of (1.6) yields (1.5), provided that Re(a3) > 0, which is the condition for convergence of the resulting 3F2(1) series on the right- hand side, and provided that Re(s)> 0, which ensures that the left-hand side remains bounded.Formula (1.6) is essentially a known formula ([1] or (3.13.56) of [5]) if the finite series on the right is turned around.
Ig is the main purpose of the present work to obtain such pairs of formulas as (1.5) and (1.6) for hypergeometric series with even more parameters.A motivation was that they are useful for the analytical continuation near z-1 of the hypergeometric functions p+ 1Fp(z).The case of the 3F2(z), in fact, could be solved [2], [4] by means of (1.5), although some authors [3], [7], [8]   do not need such formulas.

A Recurrence Relation for Terminating Saalschfitzian Hypergeometric Series of Unit Argument
In order to generalize (1.6) let us first consider the product of two hypergeometric functions [z (2.1) L: 2F1 afl z pFp_ 1 B1,B2,...,Bp It is convenient to introduce the non-trivial characteristic exponent at z-1 for each of the hyper- geometric functions, or respectively, and to consider in addition zm. (2.5) We now assume that the parameters of the hypergeometric functions in (2.1) are not all indepen- dent of each other but satisfy a + S O, (2.6) so that we have L-R. (2.7) Expanding both sides in powers of z and equating the coefficients of Z m we obtain p Identifying the parameters as A1,A2,...,Ap, -k p+lFp B1,B2,...,Bp_l,l_ eo_kll where s is determined by (2.6) and is explicitly given below in (2.10), we obtain the following re- currence relation.
Theorem 1, for p 4 and the right-hand side rewritten by means of Corollary 1, yields the fol- lowing new formula.
Corollary 2: For rn O, 1, 2,..., and s b 1 -4-b 2 -+b 3 + b 4 a 1 a 2 a 3 a 4-a 5 not a negative integer or zero, it holds true that 1 6F I al' a2' a3' a4' aS' rn We might proceed further this way, but at each step, one more sum appears, and so, the formulas become more and more complicated and lengthy as the number of parameters increases.

The Limit m oo
It is straightforward to let m---<x in the above corollaries, but it seems to be difficult to establish the conditions for convergence of the resulting series on the right-hand sides.If Re(s) > 0, we get from (2.12): a a a a 4,/)2 + ba aa a4 1 When a 4 or a a are equal to any of bl,b,ba, then the new formula (a.1) reduces to (1.).The condition for convergence of aF(1) on the right-hand side then is Re(aa) > 0 or Re(a4) > 0, res- pectively.Based on this information, we conjecture that Re(aa) > 0 A Re(a4) > 0 is the condition for the convergence of the series on the right-hand side of (.1).The occurrence of the gamma functions r(aa)r(a4) in the denominator in front of the series supports this hypothesis, for if a a or a 4 are equal to zero (or a negative integer), the right-hand side of (a.1) can be different from zero only if the sum of the series is infinite.
Generalizations of the Gaussian summation formula like (3.1) or (3.2) were also obtained in a different way [3].This includes the proof that the conditions for convergence of the series (3.1) and (3.2) conjectured above are really true.While that way of proof again yields (1.5) as expected, it leads to formulas which look significantly different from (3.1) or (3.2).