×

Parameter augmentation for basic hypergeometric series. I. (English) Zbl 0901.33008

Sagan, Bruce E. (ed.) et al., Mathematical essays in honor of Gian-Carlo Rota’s 65th birthday. Boston, MA: Birkhäuser. Prog. Math. 161, 111-129 (1998).
Let \(E(z)= (-z; q)_\infty\) be the \(q\)-exponential function; the authors define the operator \(E(b\theta)\), where \(\theta f(a)= [f(aq^{-1})- f(a)]/(aq^{-1})\); they prove that \(E(b\theta)[(at; q)_\infty]= (at, bt; q)_\infty\), and that \(E(b\theta)[(as, at; q)_\infty]= (as, at, bs, bt; q)_\infty/(abst/q; q)_\infty\). They call these properties “parameter augmentation”. They use parameter augmentation to prove formulaes on \(q\)-hypergeometric functions and \(q\)-integrals. For example, they prove Rogers identity, which contains three parameters, from a special case containing only one parameter. Many other examples are given. For a general method for proving identities on hypergeometric and \(q\)-hypergeometric functions, see H. S. Wilf and D. Zeilberger [Invent. Math. 108, 575–634 (1992; Zbl 0739.05007)].
For the entire collection see [Zbl 0890.00032].
Reviewer: D.Duverney (Lille)

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
05A40 Umbral calculus
05A19 Combinatorial identities, bijective combinatorics
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals

Citations:

Zbl 0739.05007
PDFBibTeX XMLCite