Hypergeometric Functions with Integral Parameter Differences

Abstract

For a generalized hypergeometric function pFq(z) with positive integral differences between certain numerator and denominator parameters, a formula expressing the pFq(z) as a finite sum of lower-order functions is proved. From this formula, Minton's two summation theorems for p = q + 1, z = 1 are deduced, one of these under less restrictive conditions than assumed by Minton.

Some Generalizations of Pochhammer’s Symbol and their Associated Families of Hypergeometric Functions and Hypergeometric Polynomials

“…For κ = α, Gould's identity (18) reduces at once to the expansion formula (16). Moreover, in its limit case when |κ| → ∞, (18) corresponds (at least formally) to the expansion formula (15).…”

“…, the expansion formula (15) can easily be shown to imply the expansion formula (16). In fact, it is not difficult to show that the expansion formulas (15) and (16) are equivalent (see, for details, [42, pp.…”

“…Moreover, in its limit case when |κ| → ∞, (18) corresponds (at least formally) to the expansion formula (15). Thus, for bounded κ (κ = α), Gould's identity (18) may naturally be looked upon as being independent of the equivalent expansion formulas (15) and (16).…”

“…In fact, it is not difficult to show that the expansion formulas (15) and (16) are equivalent (see, for details, [42, pp. 354-356]).…”

“…The following interesting generalization (and unification) of the equivalent expansion formulas (15) and (16) was given by Gould [14, p. 196, Equation (6.1)]:…”