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

Mourad E.H. Ismail, Dennis Stanton

Research output: Contribution to journalArticlepeer-review

42 Scopus citations


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 languageEnglish (US)
Pages (from-to)125-146
Number of pages22
JournalJournal of Approximation Theory
Issue number1
StatePublished - Jul 1 2003

Bibliographical note

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

Funding Information:
Part of this work was done while the first author was visiting the Liu Bie Ju Center of Mathematical Sciences of City University of Hong Kong and the Hong Kong University of Science and Technology. He gratefully acknowledges the financial support and the hospitality of both universities. Both authors thank Yik-Man Chiang and Mizan Rahman for many discussions and for their interest in this work. Last but not least we thank two very kind referees for many useful comments and remarks on the first version of this paper.

Copyright 2017 Elsevier B.V., All rights reserved.


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

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