Applications of constructed new families of generating-type functions interpolating new and known classes of polynomials and numbers. (English) Zbl 1472.05010
Summary: The aim of this article is to construct some new families of generating-type functions interpolating a certain class of higher order Bernoulli-type, Euler-type, Apostol-type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol-type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A40 | Umbral calculus |
| 05C30 | Enumeration in graph theory |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 26C05 | Real polynomials: analytic properties, etc. |
| 12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 41A05 | Interpolation in approximation theory |
Keywords:
Bernoulli numbers; Bernoulli polynomials; combinatorial numbers and sums; Euler numbers; Euler polynomials; Fubini numbers; generating function; Lah numbers; shift operator; special functions; Stirling numbers; umbral calculus conventionReferences:
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