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- Publisher
- Springer Basel
- Copyright
- © Springer Basel 2012
- 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