Identities related to the Stirling numbers and modified Apostol-type numbers on Umbral Calculus

Komatsu, Takao; Şimşek, Yılmaz

doi:10.18514/mmn.2017.1458

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…Substituting λ = k = 1 into (22), we arrive at the following corollary, which was proved by Roman (p. 103, Equation (4.2.11) [13]), see also (cf. [32]).…”

Section: Identities Including the Numbers Y 1 (N K; λ) Combinatoriamentioning

confidence: 99%

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Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

Şimşek

2018

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In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 (n, k; λ). Finally, we make some remarks and observations regarding these identities and relations.

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“…Substituting λ = k = 1 into (22), we arrive at the following corollary, which was proved by Roman (p. 103, Equation (4.2.11) [13]), see also (cf. [32]).…”

Section: Identities Including the Numbers Y 1 (N K; λ) Combinatoriamentioning

confidence: 99%

“…Another proof of the Equation (27) is given by Dere et al [6] and see also (cf. [32]). The following theorem was proved in (cf.…”

Section: Corollary 2 Bmentioning

confidence: 99%

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

Şimşek

2018

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“…Then it is obviously easier to compute the coefficients C n,k by viewing β n,λ (x) and (x) n,λ as λ-Sheffer sequences and using (20) than by viewing them as Sheffer sequences and using (22).…”

Section: Introductionmentioning

confidence: 99%

Representations of degenerate poly-Bernoulli polynomials

et al. 2021

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As is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed ‘λ-umbral calculus’. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials.

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Recurrence relations, associated formulas, and combinatorial sums for some parametrically generalized polynomials arising from an analysis of the Laplace transform and generating functions

Kilar

Şimşek

2022

Ramanujan J

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Identities related to the Stirling numbers and modified Apostol-type numbers on Umbral Calculus

Cited by 5 publications

References 14 publications

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

Representations of degenerate poly-Bernoulli polynomials

Recurrence relations, associated formulas, and combinatorial sums for some parametrically generalized polynomials arising from an analysis of the Laplace transform and generating functions

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