Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials

Khan, Subuhi; Nahid, Tabinda; Riyasat, Mumtaz

doi:10.1007/s10013-020-00472-6

Cited by 6 publications

(9 citation statements)

References 5 publications

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“…That is, when

ω = 1

and ω ≠ 1, the radius of convergence of the series in () is 2 π and

(||, \ln ω)

(cf. previous studies 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 60%

“…Observe that

B_{n} (x) = \lim_{ω \to 1} B_{n} (x; ω) .

Here, B n ( x ) denotes the Bernoulli polynomials (cf. previous studies 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 60%

“…Substituting

x = 0

into (), we have

B_{n} (ω) = B_{n} (0; ω),

which denotes the Apostol–Bernoulli numbers (cf. previous works 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 69%

“…By using (), first few values of the Apostol–Bernoulli numbers and polynomials are computed, respectively, as follows:

\begin{matrix} {scriptB}_{0} false(ω false) = 0, {scriptB}_{1} ((), ω) = \frac{1}{ω - 1}, {scriptB}_{2} false(ω false) = - \frac{2 ω}{{((), ω - 1)}^{2}}, {scriptB}_{3} ((), ω) = \frac{3 ω ((), ω + 1)}{{((), ω - 1)}^{3}}, \end{matrix}

and

\begin{matrix} {scriptB}_{0} ((), x; ω) = 0, \\ {scriptB}_{1} ((), x; ω) = \frac{1}{ω - 1}, \\ {scriptB}_{2} ((), x; ω) = \frac{1}{ω - 1} x - \frac{2 ω}{{((), ω - 1)}^{2}}, \\ {scriptB}_{3} ((), x; ω) = \frac{3}{ω - 1} x^{2} - \frac{6 ω}{{((), ω - 1)}^{2}} x + \frac{3 ω ((), ω + 1)}{{((), ω - 1)}^{3}}, \end{matrix}

and so on (cf. previous studies…”

Section: Introductionmentioning

confidence: 85%

“…Note that there exist a group homomorphism with a period d which is not a Dirichlet character. Thus, the generating functions given in ( 52) and ( 53) are different from the generating functions in ( 24) and (25). By this idea, generating functions given respectively in ( 52) and ( 53) may be investigated in detail by the techniques of both abstract algebra and mathematics.…”

Section: Further Remark and Observation On Decompositions Of The Gene...mentioning

confidence: 99%

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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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“…That is, when

ω = 1

and ω ≠ 1, the radius of convergence of the series in () is 2 π and

(||, \ln ω)

(cf. previous studies 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 60%

“…Observe that

B_{n} (x) = \lim_{ω \to 1} B_{n} (x; ω) .

Here, B n ( x ) denotes the Bernoulli polynomials (cf. previous studies 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 60%

“…Substituting

x = 0

into (), we have

B_{n} (ω) = B_{n} (0; ω),

which denotes the Apostol–Bernoulli numbers (cf. previous works 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 69%

“…By using (), first few values of the Apostol–Bernoulli numbers and polynomials are computed, respectively, as follows:

\begin{matrix} {scriptB}_{0} false(ω false) = 0, {scriptB}_{1} ((), ω) = \frac{1}{ω - 1}, {scriptB}_{2} false(ω false) = - \frac{2 ω}{{((), ω - 1)}^{2}}, {scriptB}_{3} ((), ω) = \frac{3 ω ((), ω + 1)}{{((), ω - 1)}^{3}}, \end{matrix}

and

\begin{matrix} {scriptB}_{0} ((), x; ω) = 0, \\ {scriptB}_{1} ((), x; ω) = \frac{1}{ω - 1}, \\ {scriptB}_{2} ((), x; ω) = \frac{1}{ω - 1} x - \frac{2 ω}{{((), ω - 1)}^{2}}, \\ {scriptB}_{3} ((), x; ω) = \frac{3}{ω - 1} x^{2} - \frac{6 ω}{{((), ω - 1)}^{2}} x + \frac{3 ω ((), ω + 1)}{{((), ω - 1)}^{3}}, \end{matrix}

and so on (cf. previous studies…”

Section: Introductionmentioning

confidence: 85%

Section: Further Remark and Observation On Decompositions Of The Gene...mentioning

confidence: 99%

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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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show abstract

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

Şimşek¹

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Some properties of degenerate Hermite appell polynomials in three variables

Baran,

Özat,

Çekim

et al. 2023

Filomat

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This study is about degenerate Hermite Appell polynomials in three variables or ?h-Hermite Appell polynomials which include both discrete and degenerate cases. After we recall the definition of these polynomials and special cases, we investigate some properties of them such as recurrence relation, lowering operators (LO), raising operators (RO), difference equation (DE), integro-difference equation (IDE) and partial difference equation (PDE).We also obtain the explicit expression in terms of the Stirling numbers of the first kind. Moreover, we introduce 3D- ?h-Hermite ?-Charlier polynomials, 3D-?h-Hermite degenerate Apostol-Bernoulli polynomials, 3D-?h-Hermite degenerate Apostol-Euler polynomials and 3D-?h-Hermite ?-Boole polynomials as special cases of ?h-Hermite Appell polynomials. Furthermore, wederive the explicit representation, determinantal form, recurrence relation, LO, RO and DE for these special cases. Finally, we introduce new approximating operators based on h-Hermite polynomials in three variables and examine the weighted Korovkin theorem. The error of approximation is also calculated in terms of the modulus of continuity and Peetre?s K-functional

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Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials

Cited by 6 publications

References 5 publications

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

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

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

Some properties of degenerate Hermite appell polynomials in three variables

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