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Abstract
In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.
Highlights
For i ≥ 0, the degenerate Stirling numbers of the first kind are defined by means of the following generating function: 1 i!
For i ≥ 0, the degenerate Stirling numbers of the second kind are defined by means of the following generating function: 1 i!
We note that limλ⟶0S2,λðj, kÞ = S1ðj, kÞ are the Stirling numbers of the second kind given by the following: 1 i!
Summary
Many mathematicians, namely, Carlitz [1, 2], Kim and Kim [3,4,5], Kim et al [6,7,8,9], Muhiuddin et al [10,11,12], and Sharma et al [13,14,15] have introduced and studied various degenerate versions of special polynomials and numbers like degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Fubini polynomials, and degenerate Stirling numbers of the first and second kinds. The classical Frobenius-Euler polynomials HðnαÞðx ; uÞ (u ∈ C with u ≠ 1) of order α are defined by means of the following generating function (see [16, 17]):. For i ≥ 0, the degenerate Stirling numbers of the first kind are defined by means of the following generating function (see [4]):. For i ≥ 0, the degenerate Stirling numbers of the second kind are defined by means of the following generating function (see [18]): ðeλ ðzÞ. We note that limλ⟶0S2,λðj, kÞ = S1ðj, kÞ are the Stirling numbers of the second kind given by the following (see [3,4,5,6,7, 18]): ðez j=i.
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