Ordinary generating function for a class of Appell polynomials and Stirling symmetric polynomials

Author(s):  
Hacène Belbachir
Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 258 ◽  
Author(s):  
Subuhi Khan ◽  
Tabinda Nahid

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations are established. Examples providing the corresponding results for certain members belonging to this q-Hermite-Appell family are considered. In addition, graphs of certain q-special polynomials are demonstrated using computer experiment. Thereafter, distribution of zeros of these q-special polynomials is displayed.


Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 159 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi ◽  
Serkan Araci

In this paper, the class of q -Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of q -Sheffer–Appell polynomials are investigated and some properties of these members are derived. In addition, the class of 2D q -Sheffer–Appell polynomials is introduced. Further, the graphs of some members of q -Sheffer–Appell polynomials and 2D q -Sheffer–Appell polynomials are plotted for different values of indices by using Matlab.


Author(s):  
Thomas Ernst

AbstractWe study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.


2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Subuhi Khan ◽  
Nusrat Raza

A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). The generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. The differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. This paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases.


Author(s):  
Thomas Ernst

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.


2022 ◽  
Vol 40 ◽  
pp. 1-15
Author(s):  
Subuhi Khan ◽  
Tabinda Nahid

The intended objective of this paper is to introduce a new class of the hybrid q-Sheffer polynomials by means of the generating function and series definition. The determinant definition and other striking properties of these polynomials are established. Certain results for the continuous q-Hermite-Appell polynomials are obtained. The graphical depictions are performed for certain members of the hybrid q-Sheffer family. The zeros of these members are also explored using numerical simulations. Finally, the orthogonality condition for the hybrid q-Sheffer polynomials is established.


2021 ◽  
Vol 45 (03) ◽  
pp. 409-426
Author(s):  
GHAZALA YASMIN ◽  
ABDULGHANI MUHYI

In this paper, the class of (p,q)-Bessel-Appell polynomials is introduced. The generating function, series definition and determinant definition of this class are established. Certain members of (p,q)-Bessel-Appell polynomials are considered and some properties of these members are also derived. Further, the class of 2D (p,q)-Bessel-Appell polynomials is introduced by means of the generating function and series definition. In addition, the graphical representations of some members of (p,q)-Bessel-Appell polynomials and 2D (p,q)-Bessel-Appell polynomials are plotted with the help of Matlab.


Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1307 ◽  
Author(s):  
Hari M. Srivastava ◽  
Ghazala Yasmin ◽  
Abdulghani Muhyi ◽  
Serkan Araci

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated.


2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Subuhi Khan ◽  
Shahid Ahmad Wani

Abstract In this article, an operational definition, generating function, explicit summation formula, determinant definition and recurrence relations of the generalized families of Hermite–Appell polynomials are derived by using integral transforms and some known operational rules. An analogous study of these results is also carried out for the generalized forms of the Hermite–Bernoulli and Hermite–Euler polynomials.


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