Lidstone expansions of entire functions

Abstract: Lidstone expansions express an entire function $f(z)$ in terms of the values of the derivatives of even orders at $0,1$. The polynomials in the expansion are called Lidstone polynomials. They are Bernoulli polynomials; many authors introduced necessary and (or) sufficient conditions for the absolute convergence of the series in the expansion. The classical exponential function plays an essential role in deriving the Lidstone series. In the $q$ theory, we have three $q$-difference operators, the Jackson $q$-difference operator, the symmetric $q$-difference operator, and the Askey-Wilson $q$-difference operator. Each operator is associated with a $q$-analog of the exponential function. In this talk, we introduce $q$-extensions to the Lidstone expansion associated with these operators. New three $q$-analogs of Bernoulli polynomials with nice properties are coming out.

Joint work with Professor Mourad Ismail, University of Central Florida, USA.

complex variables

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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