The main propose of this article is to investigate and modify Hermite type polynomials, Milne-Thomson type polynomials and Poisson–Charlier type polynomials by using generating functions and their functional equations. By using functional equations of the generating functions for these polynomials, we not only derive some identities and relations including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Poisson–Charlier polynomials, the Milne-Thomson polynomials and the Hermite polynomials, but also study some fundamental properties of these functions and polynomials. Moreover, we survey orthogonality properties of these polynomials. Finally, by applying another method which is related to p-adic integrals, we derive some formulas and combinatorial sums associated with some well-known numbers and polynomials.
Generating function Functional equation Orthogonal polynomials Bernoulli numbers and polynomials Euler numbers and polynomials Stirling numbers Milne-Thomson polynomials Poisson–Charlier polynomials Hermite polynomials Special functions Special numbers and polynomials p-adic integral
Mathematics Subject Classification
05A15 11B83 11B68 11S80 33C45
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The present paper was supported by Scientific Research Project Administration of Akdeniz University.
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