Some properties and difference equations for the (q,h)-Frobenius-Genocchi polynomials

Volume 37, Issue 4, pp 406--422 https://dx.doi.org/10.22436/jmcs.037.04.05

Publication Date: November 02, 2024 Submission Date: March 24, 2024 Revision Date: May 23, 2024 Accteptance Date: September 14, 2024

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Authors

A. Alshejari - Department of Mathematical Science, ‎Princess Nourah Bint Abdulrahman University, ‎Riyadh 11671, ‎Saudi Arabia. W. A. Khan - Department of Electrical Engineering, ‎Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia. U. Duran - Department of Basic Sciences of Engineering‎, ‎Faculty of Engineering and Natural Sciences, ‎Iskenderun Technical University, ‎Hatay 31200, ‎Turkiye. Ch.-S. Ryoo - Department of Mathematics‎, Hannam University‎, Daejeon 34430, South Korea.

Abstract

‎In recent years‎, ‎utilizing the generalized quantum exponential function (or say \((q,h)\)-exponential function) that unifies‎, ‎and extends \(q\)‎- ‎and \(h\)-exponential functions in an efficient and convenient form‎, ‎\((q,h)\)-generalizations of the several numbers and polynomials‎, ‎such as Euler and tangent numbers and polynomials‎, ‎have been considered and studied‎. ‎Inspired by the mentioned studies‎, ‎in this work‎, ‎we consider \((q,h)\)-extension of Frobenius-Genocchi polynomials‎, ‎and then we analyzed and derived some of their formulas and relations using their implicit summation formulas and symmetric identities‎. ‎Also‎, ‎we show that these polynomials are solutions of some higher-order \((q,h)\)-difference equations‎. ‎In addition‎, ‎we provide some symmetric \((q,h)\)-difference equations equality including \((q,h)\)-Frobenius-Genocchi polynomials‎. ‎Moreover‎, ‎we observe that \((q,h)\)-Frobenius-Genocchi polynomials are solutions of higher-order \((q,h)\)-difference equations combined with the \(q\)-Euler‎, ‎\(q\)-Bernoulli‎, ‎and \(q\)-Genocchi numbers and polynomials‎, ‎respectively‎. ‎Furthermore‎, ‎we utilize a computer program to provide the structures and shapes of the approximate roots of the aforesaid polynomials‎.

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Keywords

• \(q\)-numbers
• \((q,h)\)-derivative operator
• degenerate \(q\)-Frobenius-Genocchi polynomials
• \((q,h)\)-difference equation‎

MSC

•  11B83
•  34A34
•  34A30
•  33B10

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