Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials

Srivástava, H. M.; Kucukoglu, Irem; Şimşek, Yılmaz

doi:10.1016/j.jnt.2017.05.008

Cited by 35 publications

(40 citation statements)

References 19 publications

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“…in the previous equation with the necessary calculations yields the assertion of the theorem. 2 Remark 3.7 Substituting k = 1 into (3.6), we get Theorem 10 in [34].…”

Section: )mentioning

confidence: 90%

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

Kucukoglu¹,

Şimşek²,

Şimşek³

2019

Turk J Math

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The aim of this paper is to not only provide a definition of a new family of special numbers and polynomials of higher-order with their generating functions, but also to investigate their fundamental properties in the spirit of probabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities and formulas involving the Chu-Vandermonde-type convolution formulas, combinatorial sums, Bernstein basis functions, and the other well-known special numbers and polynomials. Moreover, we provide a computational algorithm which returns special values of these numbers and polynomials. In addition, we show that our new identities and formulas are connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present some theoretical and applied details on probabilistic distributions arising from the aforementioned Chu-Vandermonde-type convolution formulas.

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“…in the previous equation with the necessary calculations yields the assertion of the theorem. 2 Remark 3.7 Substituting k = 1 into (3.6), we get Theorem 10 in [34].…”

Section: )mentioning

confidence: 90%

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

Kucukoglu¹,

Şimşek²,

Şimşek³

2019

Turk J Math

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“…(cf. [16], [24]). Moreover, in the special case when q goes to 1, the numbers Y n,χ (λ, q) are reduced to the numbers Y n,χ (λ), given by the following generating functions…”

Section: Introductionmentioning

confidence: 99%

“…Simsek [16]. Recently, Srivastava et al [24] gave various novel identities and relations including the numbers Y n (λ), the Apostol-Bernoulli numbers and polynomials, the Apostol-Euler numbers and polynomials, the Stirling numbers of the first kind. In [7], Choi modified the numbers Y n (λ) as follows:…”

Section: Introductionmentioning

confidence: 99%

On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers

Kucukoglu

Şimşek

2019

Appl Anal Discrete Math

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In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for these numbers and polynomials. By using these algorithms, we provide several values of these numbers and polynomials. Furthermore, some new identities, formulas and combinatorial sums are obtained by using relations derived from the functional equations of these generating functions. These identities and formulas include the Apostol-type numbers and polynomials, and also the Stirling numbers. Finally, we give further remarks and observations on the generating function including λ-Apostol-Daehee numbers, special numbers, and finite sums.

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“…Similarly when λ = 1, the above equation reduces to the following Euler numbers: (1) n (1) (cf. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references cited therein).…”

Section: Introductionmentioning

confidence: 99%

“…[6,13]). We now give some special values of this polynomials as follows: When x = 0, we have the Peters numbers [15,20]:…”

Section: Introductionmentioning

confidence: 99%

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

Şimşek

2019

J Inequal Appl

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The main purpose and motivation of this work is to investigate and provide some new identities, inequalities and relations for combinatorial numbers and polynomials, and for Peters type polynomials with the help of their generating functions. The results of this paper involve some special numbers and polynomials such as Stirling numbers, the Apostol-Euler numbers and polynomials, Peters polynomials, Boole polynomials, Changhee numbers and the other well-known combinatorial numbers and polynomials. Finally, in the light of Boole's inequality (Bonferroni's inequalities) and bounds of the Stirling numbers of the second kind, some inequalities for a combinatorial finite sum are derived. We mention an open problem including bounds for our numbers. Some remarks and observations are presented.

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Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials

Cited by 35 publications

References 19 publications

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

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