Explicit relations for the modified degenerate Apostol-type polynomials

Abstract

Recently, the degenerate Bernoulli numbers and polynomials and the degenerate Euler numbers and polynomials have been studied by several authors. In this paper, we consider the modified Apostol-Bernoulli polynomials and the modified Apostol-Euler polynomials. We give explicit relation for the modified degenerate Bernoulli polynomials and the modified degenerate Euler polynomials. Also, we prove some identities between the modified Apostol-Bernoulli polynomials and the degenerate Stirling numbers of two kinds.

Keywords

• Bernoulli polynomials and numbers,Euler polynomials and numbers,Modified Bernoulli numbers and polynomials,Modified Euler numbers and polynomials,Degenerate Stirling numbers of the second kind,Degenerate μ-multiple sums,Degenerate μ-multiple alternating sums

Modifiye dejenere Apostol-tipi polinomlar için kesin bağıntılar

Abstract

Son yıllar da dejenere Bernoulli sayıları, polinomları ve dejenere Euler sayıları, polinomlarını birçok yazarlar tarafından çalışılıyor. Bu makale de modifiye Apostol-Bernoulli polinomları ve modifiye Apostol-Euler polinomlarını tanımladık. Modifiye dejenere Bernoulli polinomları ve modifiye dejenere Euler polinomları için kesin bağıntı verdik. Ayrıca, ikinci çeşit dejenere Stirling sayıları ve modifiye Apostol-Bernoulli polinomları arasında bazı özellikler ispatlandı.

Keywords

• Bernoulli polinomları ve sayıları,Euler polinomları ve sayıları,Modifiye Bernoulli polinomları ve sayıları,Modifiye Euler polinomları ve sayıları,İkinci Çeşit Dejenere Stirling sayıları,Dejenere μ-katlı toplamlar

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