# A Comprehensive Treatment of q-CalculusPre q-Analysis

A Comprehensive Treatment of q-Calculus: Pre q-Analysis [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Comprehensive Treatment of q-CalculusPre q-Analysis

33 pages

/lp/springer-journals/a-comprehensive-treatment-of-q-calculus-pre-q-analysis-srt7hlZrPu
Publisher
Springer Basel
ISBN
978-3-0348-0430-1
Pages
63 –95
DOI
10.1007/978-3-0348-0431-8_3
Publisher site
See Chapter on Publisher Site

### 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.]

Published: Jun 18, 2012

Keywords: Hypergeometric Function; Elliptic Function; Theta Function; Hermite Polynomial; Jacobi Polynomial