Series with Hermite polynomials and applications (original) (raw)
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arXiv: Mathematical Physics, 2018
For a sequence P=(pn(x))n=0inftyP=(p_n(x))_{n=0}^{\infty}P=(pn(x))n=0infty of polynomials pn(x)p_n(x)pn(x), we study the KKK-tuple and LLL-shifted exponential lacunary generating functions mathcalGK,L(lambda;x):=sumn=0inftyfraclambdann!pncdotK+L(x)\mathcal{G}_{K,L}(\lambda;x):=\sum_{n=0}^{\infty}\frac{\lambda^n}{n!} p_{n\cdot K+L}(x)mathcalGK,L(lambda;x):=sumn=0inftyfraclambdann!pncdotK+L(x), for K=1,2dotscK=1,2\dotscK=1,2dotsc and L=0,1,2dotscL=0,1,2\dotscL=0,1,2dotsc. We establish an algorithm for efficiently computing mathcalGK,L(lambda;x)\mathcal{G}_{K,L}(\lambda;x)mathcalGK,L(lambda;x) for generic polynomial sequences PPP. This procedure is exemplified by application to the study of Hermite polynomials, whereby we obtain closed-form expressions for mathcalGK,L(lambda;x)\mathcal{G}_{K,L}(\lambda;x)mathcalGK,L(lambda;x) for arbitrary KKK and LLL, in the form of infinite series involving generalized hypergeometric functions. The basis of our method is provided by certain resummation techniques, supplemented by operational formulae. Our approach also reproduces all the results previously known in the literature.
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The Electronic Journal of Combinatorics, 2004
Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's combinatorial interpretation of the Hermite polynomials as counting matchings of a set to obtain a triple lacunary generating function for the Hermite polynomials. We also give an umbral proof of this generating function.
A Note on the (p, q)-Hermite Polynomials
In this paper, we introduce a new generalization of the Hermite polynomials via (p, q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also define a (p, q)-analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p, q)-hyperbolic representations of the (p, q)-Bernstein polynomials. In addition, we obtain a correlation between (p, q)-Hermite polynomials and (p, q)-Bernstein polynomials.