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

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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 |lnM(r;f)|⩽Aln2r. This solves the problem of constructing such entire functions from their values at [aqn+qn/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.

Keywords

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

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Research partially supported by NSF grants DMS 99-70865, DMS 02-03282.