Abstract
Transformation formulas for terminating Saalschützian hypergeometric series of unit argument p+1Fp(1) are presented. They generalize the Saalschützian 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 Saalschützian 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 3F2(1) series are found. They depend on four or six independent parameters, respectively.
Highlights
This paper is concerned with hypergeometric series [1], [5], [9]
We are looking for such pairs of formulas for hypergeometric series with more parameters
Theorem 1, for p 4 and the right-hand side rewritten by means of Corollary 1, yields the following new formula
Summary
Which are presented in terms of Pochhammer symbols (x), x(x + 1)...(x + n- 1) r(x + n)/F(x). We are looking for such pairs of formulas for hypergeometric series with more parameters. 2. A Recurrence Relation for Terminating Saalschfitzian Hypergeometric Series of Unit Argument. We assume that the parameters of the hypergeometric functions in (2.1) are not all independent of each other but satisfy a + S O,. Expanding both sides in powers of z and equating the coefficients of Zm we obtain. With p 3, 4F3(1) on the right-hand side of (2.9) rewritten by means of (1.6) yields the following new formula which generalizes (1.6). Theorem 1, for p 4 and the right-hand side rewritten by means of Corollary 1, yields the following new formula. 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
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