Abstract
We introduce two sine and cosine types of generating functions in a general case and apply them to the generating functions of classical hypergeometric orthogonal polynomials as well as some widely investigated combinatorial numbers such as Bernoulli, Euler and Genocchi numbers. This approach can also be applied to other celebrated sequences.
Highlights
The generating function of a sequence of polynomials {Pn(x)} is defined by a bivariate function, say G(x, t), whose expansion in powers of t has the form ∞ (1)G(x, t) = Pn(x)tn, n=0 for sufficiently small |t|
The main aim of this paper is to introduce sine and cosine types of generating functions in a general case containing all above-mentioned examples as particular cases
We introduce sine and cosine types of generating functions in a general case and apply them for two main classes, i.e. for the generating functions of classical hypergeometric orthogonal polynomials and for widely-investigated sequences of numbers appeared in number theory
Summary
The generating function of a sequence of polynomials {Pn(x)} is defined by a bivariate function, say G(x, t), whose expansion in powers of t has the form. Generating functions, Classical orthogonal polynomials of Jacobi, Laguerre and Hermite type, Sheffer and Appell polynomials, Bernoulli, Euler and Genocchi numbers. Genocchi numbers Gn := Gn(0) have found various applications in number theory, combinatorics and numerical analysis [8] Another important case of Appell polynomials is the Apostol type of Bernoulli, Euler and Genocchi polynomials which are respectively generated by (4). We introduce sine and cosine types of generating functions in a general case and apply them for two main classes, i.e. for the generating functions of classical hypergeometric orthogonal polynomials and for widely-investigated sequences of numbers appeared in number theory.
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