SOME IDENTITIES OF A NEW TYPE OF DEGENERATE POLY-FROBENIUS-EULER POLYNOMIALS AND NUMBERS
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Many mathematicians have been studying various degenerate versions of special poly- nomials and numbers not only in some arithmetic and combinatorial aspects but also in applications to differential equations, identities of symmetry and probability theory. Moreover, umbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim-Kim (J. Math. Anal. Appl.(2020) 23, 124521) introduced the l -Sheffer sequence and the degenerate Sheffer sequence by replacing the l - linear functional and l - differential operator, respectively, instead of the linear functionalal and the differential operator used by Rota. This paper is divided in two parts. In the first part, we introduce a new type of degenerate poly-Frobenius-Euler polynomials and numbers, and derive several combinatorial identities related to those polynomials and numbers without by using degenerate Sheffer sequences. In the second part, we also derive various combinatorial identities related to a new type of degenerate poly-Frobenius-Euler polynomials by using degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bell polynomials, the degenerate Bernoulli polynomials, the degenerate Euler polynomials, the degenerate Daehee polynomials, the Stirling number of the first and second kind, etc.
Abstract
Many mathematicians have been studying various degenerate versions of special poly- nomials and numbers not only in some arithmetic and combinatorial aspects but also in applications to differential equations, identities of symmetry and probability theory. Moreover, umbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim-Kim (J. Math. Anal. Appl.(2020) 23, 124521) introduced the l -Sheffer sequence and the degenerate Sheffer sequence by replacing the l - linear functional and l - differential operator, respectively, instead of the linear functionalal and the differential operator used by Rota. This paper is divided in two parts. In the first part, we introduce a new type of degenerate poly-Frobenius-Euler polynomials and numbers, and derive several combinatorial identities related to those polynomials and numbers without by using degenerate Sheffer sequences. In the second part, we also derive various combinatorial identities related to a new type of degenerate poly-Frobenius-Euler polynomials by using degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bell polynomials, the degenerate Bernoulli polynomials, the degenerate Euler polynomials, the degenerate Daehee polynomials, the Stirling number of the first and second kind, etc.
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