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q-functions of several variables

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A Comprehensive Treatment of q-Calculus

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Abstract

We begin with the vector notation for the most important functions and q-Taylor formulas for power series and functions of inverse q-shifted factorials. We continue with a historical introduction to the rest of this long and interesting chapter and to the next chapter as well. We will also define q-Appell functions together with the normal form. Then follows the two definitions of q-Kampé de Fériet functions due to Karlsson and Srivastava. The q-analogue of Appell and Kampé de Fériet’s transformation formulas require the Watson q-shifted factorial in the definition. We continue with Carlitz’ Saalschützian formulas, Andrews’s formal transformations and Carlson’s transformations.

We show that the Jacksonian formula for the q-integral of the q-Appell function Φ1 is equivalent to the first of Andrews’s formal transformations. We give several examples of multiple reduction formulas with general terms. These are used to find many q-analogues of reduction formulas for Appell and Lauricella functions and other similar functions. A relation for Γ q functions with negative integer argument from chapter eight as well as the Bayley-Daum formula will be used in the proofs. Many summation formulas appear as doublets, which is a legacy of the two q-Vandermonde summation formulas. We introduce the inverse pair of symbolic operators ▽ q (h) and △ q (h) due to Jackson. Then we derive expansions for q-Appell and q-Kampé de Fériet functions. Each of these expansions is equivalent to a combinatorial identity, which resembles a well-known q-summation formula.

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  1. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Thomas Ernst

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Ernst, T. (2012). q-functions of several variables. In: A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0431-8_10

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