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.
Bibliographical noteFunding Information:
Research partially supported by NSF grants DMS 99-70865, DMS 02-03282.
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.
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- Askey-Wilson operators
- Carlson's theorem
- Integral representations
- Polynomial bases
- q-Taylor series
- q-exponential function