Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.

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“…In order to prove our theorems, we give several lemmas below. For previous papers using this method, please see [25][26][27][28][29][30][31].…”

Section: Some Applications Of Frobenius-euler-genocchi Polynomials Of...mentioning

confidence: 99%

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

et al. 2023

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On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

2023

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Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (p,q)-Bernoulli–Fibonacci and generalized (p,q)-Bernoulli–Lucas polynomials and numbers by using the (p,q)-Bernoulli numbers, unified (p,q)-Bernoulli polynomials, h(x)-Fibonacci polynomials, and h(x)-Lucas polynomials. We also introduce the generalized bivariate (p,q)-Bernoulli–Fibonacci and generalized bivariate (p,q)-Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (p,q)-Bernoulli–Fibonacci polynomials.

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Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials

Cited by 2 publications

References 51 publications

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

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