Abstract
This article aims to introduce a hybrid family of 2-variable Boas–Buck-general polynomials by taking Boas–Buck polynomials as a base with the 2-variable general polynomials. These polynomials are framed within the context of the monomiality principle, and their properties are established. Further, we investigate some members belonging to this family. A general method to express connection coefficients explicitly for the Boas–Buck general polynomial sets is presented. Carlitz theorem for mixed generating functions is also extended to these polynomials. The shapes are shown and zeros are computed for these polynomials using Mathematica software.
Highlights
1 Introduction and preliminaries Special functions of multivariable form have shown remarkable progress in recent years [1,2,3,4,5,6]. These functions arise in diverse areas of mathematics, and they provide a new means of analysis ranging from the solution of large classes of partial differential equations often encountered in physical problems to the abstract group theory
B2(xt) = exp(xt), and C2(t) = t the Boas–Buck polynomials reduce to Euler polynomials
5 Concluding remarks One of the main results of Boas and Buck [15] is that a necessary and sufficient condition for the polynomials Fn(x) to have a generating function of Boas–Buck type is that there exists a sequence of numbers αk and βk such that, for n ≥ 1, the following recursion relation holds: n–1 n–1
Summary
2-variable generalized Laguerre polynomials [3] 2-variable truncated exponential polynomials of order r [4]. Definition 1.1 A general polynomial set is said to be 2-variable general polynomials Pn(x, y) if it has the following generating function [1]: extΦ(y, t) =. The Boas–Buck polynomial set was introduced by Boas and Buck [15] in the year 1956 It includes many important general classes of polynomial sets like Brenke polynomials, Sheffer polynomials, Appell polynomials, etc. The Boas–Buck polynomial set is defined by means of generating function as follows. Definition 1.2 A polynomial set is said to be Boas–Buck polynomial set if it has the following generating function [15]: A(t)B xC(t) = Fn(x) n! The Boas–Buck polynomial set defined by Equation (3) is quasi-monomial under the action of the following multiplicative and derivative operators [16]:.
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