q-Taylor theorems, polynomial expansions, and interpolation of entire functions

Mourad E.H. Ismail; Dennis Stanton

doi:10.1016/S0021-9045(03)00076-5

q-Taylor theorems, polynomial expansions, and interpolation of entire functions

Mourad E.H. Ismail, Dennis Stanton

School of Mathematics

Research output: Contribution to journal › Article › peer-review

51 Scopus citations

Abstract

We 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 1n M(r;f) ≤A 1n² r. This solves the problem of constructing such entire functions from their values at [aqⁿ + q^-n /a]/2, for 0<q<1. Our technique is constructive and gives an explicit representation of the sought entire function. Applications to q-series identities are given.

Original language	English (US)
Pages (from-to)	125-146
Number of pages	22
Journal	Journal of Approximation Theory
Volume	123
Issue number	1
DOIs	https://doi.org/10.1016/S0021-9045(03)00076-5
State	Published - Jul 1 2003

Keywords

Askey-Wilson operators
Carlson's theorem
Integral representations
Polynomial bases
q-Taylor series
q-exponential function

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10.1016/S0021-9045(03)00076-5

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AB - We 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 1n M(r;f) ≤A 1n2 r. This solves the problem of constructing such entire functions from their values at [aqn + q-n /a]/2, for 0

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KW - Carlson's theorem

KW - Integral representations

KW - Polynomial bases

KW - q-Taylor series

KW - q-exponential function

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q-Taylor theorems, polynomial expansions, and interpolation of entire functions

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