Unified Apostol–Bernoulli, Euler and Genocchi polynomials

Özarslan, Mehmet Ali

doi:10.1016/j.camwa.2011.07.031

Cited by 60 publications

(42 citation statements)

References 22 publications

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“…The case when f n (x) = B n (x) in (7) was first studied by Agoh and Dilcher [14] who proved an existence theorem and also derived some explicit expressions for k = 3 involving the Bernoulli polynomials. We now state the following higher-order convolution for the general Apostol-type polynomials Y n,β (x; κ, a, b) defined by (5).…”

Section: Resultsmentioning

confidence: 99%

“…This paper is organized as follows. In Section 2, we first give the higher-order convolution for the polynomials defined by (5) Y n,β (x; κ, a, b) and then present the corresponding higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Moreover, several corollaries and consequences of our main theorems are also deduced.…”

Section: Introductionmentioning

confidence: 99%

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Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

Aracı

Srivástava

et al. 2018

Mathematics

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In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

Aracı

Srivástava

et al. 2018

Mathematics

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show abstract

“…If we set x = 0 in (4.18), we obtain [31], [21], [45], [26], [32], [34], [35], [47], [49], [46], [48]). If we set α = 1 in (4.19) and (4.18), we have…”

Section: From This Equation We Getmentioning

confidence: 99%

Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

Şimşek

2013

Fixed Point Theory Appl

100

132

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The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers. We derive various functional equations and differential equations using these generating functions. The second aim is to provide a novel approach to derive identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations.Furthermore, we derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.

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“…Let us recall the definition of the unified family of the Apostol–Bernoulli, Euler and Genocchi polynomials introduced recently by Özarslan . Definition Let

a MathClass-punc, b MathClass-rel\in double-struckR MathClass-bin∖ {0}

, α and β be two arbitrary complex numbers,

r MathClass-rel\in {double-struckN}_{0}

x MathClass-rel\in double-struckR

, and

P_{n MathClass-punc, β}^{(α)} (x MathClass-punc; r MathClass-punc, a MathClass-punc, b)

be the unified Apostol polynomials which are defined by the following generating function:

f_{a MathClass-punc, b}^{(α)} (x MathClass-punc; t MathClass-punc; r MathClass-punc, β) MathClass-punc: MathClass-rel= {((), \frac{2^{1 MathClass-bin- r} t^{r}}{β^{b} e^{t} MathClass-bin- a^{b}})}^{α} e^{xt} MathClass-rel= {MathClass-op\sum}_{n MathClass-rel= 0}^{MathClass-rel\infty} P_{n MathClass-punc, β}^{(α)} (x MathClass-punc; r MathClass-punc, a MathClass-punc, b) \frac{t^{n}}{n MathClass-punc!}

where, for the convergence of the series involved in , one needs the following: If a b > 0 and

r MathClass-rel\in double-struckN

, then

(||, t MathClass-bin+ b normallog ((), \frac{β}{a})) MathClass-rel< 2 π MathClass-punc; 1^{α} MathClass-punc: MathClass-re...

…”

Section: Unified 2d‐apostol Polynomialsmentioning

confidence: 99%

Srivastava-Pintér theorems for 2D-Appell polynomials and their applications

Özarslan¹,

Gaboury

2013

Math. Meth. Appl. Sci.

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Unified Apostol–Bernoulli, Euler and Genocchi polynomials

Cited by 60 publications

References 22 publications

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

Srivastava-Pintér theorems for 2D-Appell polynomials and their applications

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