scholarly journals Properties of Partially Degenerate Complex Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1508
Author(s):  
Dojin Kim ◽  
Sangil Kim
Keyword(s):  
Differential Equations ◽  
Exponential Type ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Polynomial Sequences ◽  
General Properties ◽  
Symmetric Identities ◽  
Degenerate Version

Degenerate versions of polynomial sequences have been recently studied to obtain useful properties such as symmetric identities by introducing degenerate exponential-type generating functions. As part of our continued work in degenerate versions of generating functions, we subsequently present our study on degenerate complex Appell polynomials by considering a partially degenerate version of the generating functions of ordinary complex Appell polynomials in this paper. We only consider partially degenerate generating functions to retain the crucial properties of the Appell sequence, and we present useful identities and general properties by splitting complex values into their real and imaginary parts; moreover, we provide several explicit examples. Additionally, the differential equations satisfied by degenerate complex Bernoulli and Euler polynomials are derived by the quasi-monomiality principle using Appell-type polynomials.

scholarly journals Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 695-708 ◽  
Author(s):  
H.M. Srivastava ◽  
M.A. Özarslan ◽  
Banu Yılmaz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Bernoulli Polynomials ◽  
Factorization Method ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Partial Differential ◽  
Genocchi Polynomials

Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and Applications, J. Math. Anal. Appl. 351 (2009), 756-764] defined the Hermite-based Appell polynomials by G(x, y, z, t) := A(t)?exp(xt + yt2 + zt3) = ??,n=0 HAn(x, y, z) tn/n! and investigated their many interesting properties and characteristics by using operational techniques combined with the principle of monomiality. Here, in this paper, we find the differential, integro-differential and partial differential equations for the Hermite-based Appell polynomials via the factorization method. Furthermore, we derive the corresponding equations for the Hermite-based Bernoulli polynomials and the Hermite-based Euler polynomials. We also indicate how to deduce the corresponding results for the Hermite-based Genocchi polynomials from those involving the Hermite-based Euler polynomials.

scholarly journals Differential Equations Associated with Two Variable Degenerate Hermite Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 228 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Symmetric Identities

In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations.

scholarly journals General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 964
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Linear Functional ◽  
Recurrence Relations ◽  
Linear Interpolation ◽  
Appell Polynomials ◽  
Polynomial Sequences ◽  
Matrix Calculus

An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given.

scholarly journals Generalizations of the Bernoulli and Appell polynomials

2004 ◽  
Vol 2004 (7) ◽  
pp. 613-623 ◽  
Author(s):  
Gabriella Bretti ◽  
Pierpaolo Natalini ◽  
Paolo E. Ricci
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Factorization Method ◽  
Bernoulli Numbers ◽  
Appell Polynomials

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

scholarly journals A Note on the Degenerate Type of Complex Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1339 ◽  
Author(s):  
Dojin Kim
Keyword(s):  
Imaginary Part ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
The Real ◽  
General Properties ◽  
Appell Sequences ◽  
Real Value ◽  
Degenerate Type

In this paper, complex Appell polynomials and their degenerate-type polynomials are considered as an extension of real-valued polynomials. By treating the real value part and imaginary part separately, we obtained useful identities and general properties by convolution of sequences. To justify the obtained results, we show several examples based on famous Appell sequences such as Euler polynomials and Bernoulli polynomials. Further, we show that the degenerate types of the complex Appell polynomials are represented in terms of the Stirling numbers of the first kind.

scholarly journals Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

2020 ◽  
Author(s):  
Cheon Seoung Ryoo
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Symmetric Identities

In this chapter, we introduce the 2-variable modified degenerate Hermite polynomials and obtain some new symmetric identities for 2-variable modified degenerate Hermite polynomials. In order to give explicit identities for 2-variable modified degenerate Hermite polynomials, differential equations arising from the generating functions of 2-variable modified degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the 2-variable modified degenerate Hermite equations.

scholarly journals Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 47 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernstein Polynomials ◽  
Umbral Calculus ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials ◽  
Degenerate Bernstein Polynomials ◽  
Combinatorial Methods

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.

scholarly journals Some Properties for Multiple Twisted (p, q)-L-Function and Carlitz’s Type Higher-Order Twisted (p, q)-Euler Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1205 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Generating Functions ◽  
Higher Order ◽  
Integral Representations ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Mellin Transformation ◽  
Euler Numbers And Polynomials ◽  
Symmetric Identities ◽  
Positive Integers ◽  
Symmetric Property

The main goal of this paper is to study some interesting identities for the multiple twisted ( p , q ) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted ( p , q ) -Euler zeta function and multiple twisted ( p , q ) -L-function, which interpolate the Carlitz-type higher order twisted ( p , q ) -Euler numbers and Carlitz-type higher order twisted ( p , q ) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted ( p , q ) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted ( p , q ) -Euler numbers and polynomials by using the symmetric property for the multiple twisted ( p , q ) -L-function.

scholarly journals Multiplication formulas for q-Appell polynomials and the multiple q-power sums

2016 ◽  
Vol 70 (1) ◽  
pp. 1
Author(s):  
Thomas Ernst
Keyword(s):  
Generating Functions ◽  
Rational Numbers ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Power Sums ◽  
Special Cases

In the first article on q-analogues of two Appell polynomials, the generalized Apostol-Bernoulli  and Apostol-Euler  polynomials, focus was on generalizations, symmetries, and complementary argument theorems. In this second article, we focus on a recent paper by Luo, and one paper on power sums by Wang and Wang. Most of the proofs are made by using generating functions, and the (multiple) q-addition plays a fundamental role. The introduction of the q-rational numbers in formulas with q-additions enables natural q-extension of vector forms of Raabes multiplication formulas. As special cases, new formulas for q-Bernoulli and q-Euler polynomials are obtained.

scholarly journals Explicit Identities for 3-Variable Degenerate Hermite Kampé de Fériet Polynomials and Differential Equation Derived from Generating Function

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 7
Author(s):  
Kyung-Won Hwang ◽  
Young-Soo Seol ◽  
Cheon-Seoung Ryoo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Symmetric Identities

We get the 3-variable degenerate Hermite Kampé de Fériet polynomials and get symmetric identities for 3-variable degenerate Hermite Kampé de Fériet polynomials. We make differential equations coming from the generating functions of degenerate Hermite Kampé de Fériet polynomials to get some identities for 3-variable degenerate Hermite Kampé de Fériet polynomials,. Finally, we study the structure and symmetry of pattern about the zeros of the 3-variable degenerate Hermite Kampé de Fériet equations.

Sign in / Sign up

Export Citation Format

Share Document