scholarly journals On certain generalized q-Appell polynomial expansions

2014 ◽  
Vol 68 (2) ◽  
Author(s):  
Thomas Ernst
Keyword(s):  
Generating Function ◽  
Difference Operators ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Recurrence Formulas ◽  
Symmetry Relations ◽  
Polynomial Expansions

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.

scholarly journals On the complex q-Appell polynomials

2020 ◽  
Vol 74 (1) ◽  
pp. 31
Author(s):  
Thomas Ernst
Keyword(s):  
Imaginary Part ◽  
Generating Function ◽  
Complex Number ◽  
Analytic Functions ◽  
Complex Case ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
The Real ◽  
Cauchy Riemann

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.

scholarly journals Some recurrence formulas for the q-Bernoulli and q-Euler polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 663-669
Author(s):  
Paçin Dere
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Derivative Operator ◽  
Important Place ◽  
Quantum Calculus ◽  
Recurrence Formulas ◽  
Special Polynomials

The recurrence relations have a very important place for the special polynomials such as q-Appell polynomials. In this paper, we give some recurrence formulas that allow us a better understanding of q-Appell polynomials. We investigate the q-Bernoulli polynomials and q-Euler polynomials, which are q-Appell polynomials, and we obtain their recurrence formulas by using the methods of the q-umbral calculus and the quantum calculus. Our methods include some operators which are quite handy for obtaining relations for the q-Appell polynomials. Especially, some applications of q-derivative operator are used in this work.

Fractional calculus and generalized forms of special polynomials associated with Appell sequences

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Subuhi Khan ◽  
Shahid Ahmad Wani
Keyword(s):  
Fractional Calculus ◽  
Generating Function ◽  
Recurrence Relations ◽  
Summation Formula ◽  
Integral Transforms ◽  
Operational Definition ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Special Polynomials ◽  
Appell Sequences

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.

scholarly journals Determinant Forms, Difference Equations and Zeros of the q-Hermite-Appell Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 258 ◽  
Author(s):  
Subuhi Khan ◽  
Tabinda Nahid
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Recurrence Relations ◽  
Computer Experiment ◽  
Appell Polynomials ◽  
Distribution Of Zeros ◽  
Hybrid Form ◽  
Special Polynomials

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.

scholarly journals Some recurrence formulas for the Hermite polynomials and their squares

2018 ◽  
Vol 16 (1) ◽  
pp. 553-560 ◽  
Author(s):  
Yuan He ◽  
Fengzao Yang
Keyword(s):  
Generating Function ◽  
Padé Approximation ◽  
Hermite Polynomials ◽  
Pade Approximation ◽  
Approximation Techniques ◽  
Recurrence Formulas

AbstractIn this paper, by making use of the generating function methods and Padé approximation techniques, we establish some new recurrence formulas for the Hermite polynomials and their squares. These results presented here are the corresponding extensions of some known formulas.

scholarly journals Certain Results of q -Sheffer–Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 159 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi ◽  
Serkan Araci
Keyword(s):  
Generating Function ◽  
Appell Polynomials ◽  
Function Series

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.

scholarly journals -Pascal and -Wronskian Matrices with Implications to -Appell Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Thomas Ernst
Keyword(s):  
Difference Equation ◽  
Recent Article ◽  
Matrix Multiplication ◽  
Euler Polynomials ◽  
Matrix Approach ◽  
Appell Polynomials ◽  
Sum Of Products ◽  
Functional Matrix

We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree .

scholarly journals Identities of Symmetry for Generalized Euler Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Euler Polynomials ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

We derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the -adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.

scholarly journals General-Appell Polynomials within the Context of Monomiality Principle

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Subuhi Khan ◽  
Nusrat Raza
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
General Class ◽  
Recurrence Relations ◽  
Appell Polynomials ◽  
Special Polynomials ◽  
New Family ◽  
Special Polynomial

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.

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