Partial derivative formulas and identities involving $$\mathbf {2}$$2-variable Simsek polynomials

Khan, Subuhi; Nahid, Tabinda; Riyasat, Mumtaz

doi:10.1007/s40590-019-00236-4

Cited by 15 publications

(17 citation statements)

References 2 publications

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“…Recently, many applications of these numbers and polynomials have been studied and investigated by different authors (cf. [11,13,[28][29][30]34]). Recently, Khan et al [11] constructed 2-variable of the polynomials Y n (x; λ) .…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…[11,13,[28][29][30]34]). Recently, Khan et al [11] constructed 2-variable of the polynomials Y n (x; λ) . They gave quasimonomial properties of these polynomials on the Weyl group structure.…”

Section: Introductionmentioning

confidence: 99%

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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“…By using these functions, many computation formulas and relations including these sums and various kinds of special numbers and polynomials have been given (cf. References [1][2][3][4][5][6][7][8][9]).…”

Section: Introductionmentioning

confidence: 99%

“…If k > n, then we have S 2 (n, k) = 0 (6) (cf. References [1][2][3][4][5][6][7][8][9]11,). The Bernoulli polynomials of the second kind are defined by…”

Section: Introductionmentioning

confidence: 99%

“…where x, t ∈ C (cf. References [2,[16][17][18][19][20][21][22][23]). The Peters polynomials are including some well-known families of special polynomials and numbers such as the Boole polynomials and numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the Changhee polynomials and numbers and other combinatorial polynomials and numbers.…”

Section: Introductionmentioning

confidence: 99%

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Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Kim

Şimşek

2019

Symmetry

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The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

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Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

Kucukoglu

2022

Math Methods in App Sciences

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The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism.

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Partial derivative formulas and identities involving $$\mathbf {2}$$2-variable Simsek polynomials

Cited by 15 publications

References 2 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

Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

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