Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert-Hermite polynomials via generalized Fibonacci polynomials. (English) Zbl 1535.11032
The purpose of this paper is to study the generalized Humbert-Hermite polynomials, which has been defined by a generating function. An explicit formula and a recursion relation are obtained by using the Faà di Bruno formula and Leibniz’ formula. Furthermore a determinantal representation of generalized Humbert-Hermite polynomials is proved. Then a recursion relation for the generalized Humbert-Hermite polynomials is proved by using the Hessenberg determinant. Finally, multilinear and multilateral generating functions for the generalized Humbert-Hermite polynomials is proved by using the generalized Apostol-Bernoulli polynomials of order \(\alpha\) and a parametric kind of Fubini-type polynomials.
The function \(p(x)\) in theorem 3.2. is not defined.
The function \(p(x)\) in theorem 3.2. is not defined.
Reviewer: Thomas Ernst (Uppsala)
MSC:
| 11B83 | Special sequences and polynomials |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B73 | Bell and Stirling numbers |
| 33C47 | Other special orthogonal polynomials and functions |
Keywords:
Fibonacci polynomials; generalized Fibonacci polynomials; generating function; Hessenberg determinant; Humbert-Hermite polynomials; recursive relationReferences:
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