An Euler product transform applied to q-series

Abstract

This paper introduces the concept of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The main result in this paper is a transform, based on an Euler product over the primes. Examples given are D-analogues of the q-binomial theorem and the q-Gauss summation.

The q-Dixon sum Dirichlet series analogue

“…The application of the transform in my paper Campbell [3] to this yields the Theorem 2.1. The Dirichlet series analogue of Dixon's theorem is…”

D-analogues of q-shifted factorial and the q-Kummer sum

“…In the present paper, as we did in [23], we derive two separate classes of Danalogues. One of these involves the above ζ(a; γ) n function, and the other involves the Jordan totient function J n (k) extended in a similar way to our extending ζ(a) into ζ(a; γ) n .…”

“…In a recent paper [23] the idea of a D-analogue was introduced. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series.…”

“…A further comparison between the three kinds of analogue is shown by the three Gauss hypergeometric sum formulae in terms of gamma functions, q-shifted gamma functions as given in Askey [11,12,13] for example, and (perhaps) D-shifted gamma functions. These are, for various conditions given in respectively, Bailey [15, pages 2-3], Gasper and Rahman [27, pages 9-11], and the author's paper [23], as…”

“…We shall continue to explore this interpretation for σ −γ (a; k). (1.14) was given in [23] for the conditions: positive integers a, b and c and ℜγ ≥ 0, with ℜcγ, ℜ(c…”