Degenerate Miller–Paris Transformations

Abstract

Important new transformations for the generalized hypergeometric functions with integral parameter differences have been discovered some years ago by Miller and Paris and studied in detail in a series of papers by a number of authors. These transformations fail if the free bottom parameter is greater than a free top parameter by a small positive integer. In this paper we fill this gap in the theory of Miller-Paris transformations by computing the limit cases of these transformations in such previously prohibited situations. This leads to a number of new transformation and summation formulas including extensions of Karlsson-Minton theorem.

Alternative Approach to Miller-Paris Transformations and Their Extensions

“…Their derivation hinges on certain simple partial fraction decompositions. The results differ from those in [12]: here the negative parameter difference −p (recall that b − c = −p) may take any (negative) integer value regardless of whether degeneration happens or not in the corresponding general Miller-Paris transformation. We further present two extensions: to several negative parameters differences instead of one and to a pair of additional unrestricted parameters on top and bottom of the generalized hypergeometric function.…”

“…It is instructive to compare identities (29) and (31) with the degenerate Miller-Paris transformations derived in [12, Theorems 1 and 3]. One important difference is that the above theorem holds for any p ∈ N, while in [12] p is restricted to the set {1, . .…”

“…Nevertheless, with a little effort one can make sure that (29) and [12, (16)] are related by a rather simple rearrangement and the polynomial T p−1 is a constant multiple of the polynomial R p from [12]. The same, however, does not hold for (31), and the polynomial T * p−1 is not a constant multiple ofR p from [12]. Proof.…”

“…, m}. In our recent paper [12] we used a careful limit transition to derive the degenerate forms of the transformations (3) and (4) for such values of c. When evaluated at x = 1 these degenerate transformations lead to extensions of Minton-Karlsson summation formulas (1), (2) which we also investigated in [13] using several different techniques. The main purpose of this paper is to present an alternative derivation of both the general transformations (3), (4) and their degenerate forms found in [12].…”

“…In our recent paper [12] we used a careful limit transition to derive the degenerate forms of the transformations (3) and (4) for such values of c. When evaluated at x = 1 these degenerate transformations lead to extensions of Minton-Karlsson summation formulas (1), (2) which we also investigated in [13] using several different techniques. The main purpose of this paper is to present an alternative derivation of both the general transformations (3), (4) and their degenerate forms found in [12]. Our derivation of the general case is presented in Section 2 and is based on the representation of the generalized hypergeometric function by the generalized Stieltjes transform of a particular type of Meijer's G function, namely G p,0 p,p , which for historical reasons we prefer to call the Meijer-Nørlund function.…”