Elementary derivations of identities for bilateral basic hypergeometric series

Abstract

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For motivation, we review our previous simple proof (Proc. Amer. Math. Soc. 130 (2002), 1103-1111) of Bailey's very-wellpoised 6 ψ 6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8 ψ 8 transformation, a very-well-poised 10 ψ 10 transformation, by induction, Slater's general transformation for very-well-poised 2r ψ 2r series, and Slater's transformation for general r ψr series. Finally, we derive some new transformations for bilateral basic hypergeometric series of a specific type. Mathematics Subject Classification (2000). Primary 33D15.

Multilateral Inversion of A r , C r , and D r Basic Hypergeometric Series

“…This way of establishing (2.5) works so far only in the classical one-dimensional case, as no multiple series extension of (2.6) is yet known. Our multivariate extensions of Proposition 2.1 in Section 4 of this paper, see Theorems 4.1, 4.3, and 4.5 (obtained by suitable extensions of the analysis applied in [23]), which we find attractive by themselves, can be understood as a first step in the quest of finding multivariate extensions of the very-well-poised 8 ψ 8 transformation formula (2.6), or of even more general transformations.…”

“…In [23], several applications of (1.1) to bilateral basic hypergeometric series were given. One of them included a new very-well-poised 8 ψ 8 summation formula, see Proposition 2.1 in this paper.…”

“…Here we apply part of the analysis of [23] to multiple sums. In fact, by appropriately specializing Gustafson's A r and C r 6 ψ 6 summations [11,12], and an A r 6 ψ 6 summation by the author [25], we derive three multidimensional extensions of the bilateral matrix inverse (1.1) and deduce three multidimensional 8 ψ 8 summations, associated with the respective root systems of types A r and C r , as applications.…”

Positivity preserving transformations for 𝑞-binomial coefficients

“…Here the second equation follows from the first and the fact that f L,r (q) is nonzero if and only if 0 ≤ r ≤ L. Inverse relations such as (4.1) have been much studied in the theory of basic hypergeometric series. Most importantly, they are related to the Bailey transform [5,9,13,17,18,54], the problem of q-Lagrange inversion [31,32] and summations and transformations of q-hypergeometric series [1,22,23,24,27,38,44].…”

On the Sears–Slater basic hypergeometric transformations

“…3, they can be explained as special cases of one formula in spite of their types. Moreover, though the condition 'very-well-poised' is usually explained as a special case of 'well-poised,' considering the Sears or Slater cases, we can conversely deduce the well-poisedness from the very-well-poisedness as a special case, i.e., these two conditions are equivalent at least on the Sears-Slater transformations, as Schlosser pointed out in [13].…”