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Pre q-Analysis

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A Comprehensive Treatment of q-Calculus

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Abstract

We begin with the duality between analytic number theory, combinatorial identities and q-series, to indicate the historical development of the allied disciplines. It is irrelevant what notation we use for the Γ-function, the essential part is that we keep this notation. Section 3.7 is devoted to this important function and the hypergeometric function. We use a vector notation for the Γ-function and introduce the concepts well-poised and balanced series. The binomial coefficients also play an important part since a finite hypergeometric series can always be expressed in two equivalent ways. The three Kummerian summation formulae (and their multiple q-analogues) will follow us in future chapters. We summarise the different schools for Theta functions and show that the elliptic function snu can be written as a balanced quotient of infinite q-shifted factorials. We conclude this chapter with definitions of the most important orthogonal polynomials; we keep Jacobi’s definition for the Jacobi polynomials.

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

    Thomas Ernst

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Ernst, T. (2012). Pre q-Analysis. In: A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0431-8_3

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