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Ismail, Mourad E. H.; Stanton, Dennis

\(q\)-Taylor theorems, polynomial expansions, and interpolation of entire functions. (English) Zbl 1035.30025

J. Approximation Theory 123, No. 1, 125-146 (2003).
The authors establish \(q\)-analogues of Taylor series expansions in special polynomial bases for functions analytic in bounded domains and for entire functions whose maximum modulus \(M(r;f)\) satisfies \(| \ln M(r;f)| \leq A \ln^2 r\). This solves the problem of constructing such entire functions from their values at \([aq^n +q^{-n}/a]/2\), for \(0<q<1\). Their technique is constructive and gives an explicit representation of the sought entire function. Applications to \(q\)-series identities are given.
Reviewer: Wolfram Koepf (Kassel)

Cited in 4 Reviews
Cited in 37 Documents

MSC:

30E10 Approximation in the complex plane
41A05 Interpolation in approximation theory
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

\(q\)-Taylor series; polynomial bases; Askey-Wilson operators; integral representations; \(q\)-exponential function; Carlson’s theorem
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Full Text: DOI

References:

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