 Pre q-AnalysisThomas Ernst
Chapter Chapter 3 in A Comprehensive Treatment of q-Calculus, 2012, pp 63-95 from Springer Abstract: 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. Keywords: Hypergeometric Function; Elliptic Function; Theta Function; Hermite Polynomial; Jacobi Polynomial (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0431-8_3 Ordering information: This item can be ordered from DOI: 10.1007/978-3-0348-0431-8_3 Access Statistics for this chapter More chapters in Springer Books from Springer |
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