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 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 Involving Two-Variable Partially Degenerate Hermite Polynomials Induced from Differential Equations and Structure of Their Roots

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 632
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Symmetric Identities ◽  
Symmetric Properties

In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

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 Generalized q-Laguerre type polynomials and q-partial differential equations

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1403-1415
Author(s):  
Da-Wei Niu
Keyword(s):  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Generating Functions ◽  
Final Section ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Partial Differential ◽  
Integral Formulas ◽  
Special Cases

In this paper we define the q-Laguerre type polynomials Un(x; y; z; q), which include q-Laguerre polynomials, generalized Stieltjes-Wigert polynomials, little q-Laguerre polynomials and q-Hermite polynomials as special cases. We also establish a generalized q-differential operator, with which we build the relations between analytic functions and Un(x; y; z; q) by using certain q-partial differential equations. Therefore, the corresponding conclusions about q-Laguerre polynomials, little q-Laguerre polynomials and q-Hermite polynomials are gained as corollaries. As applications, some generating functions and generalized Andrews-Askey integral formulas are given in the final section.

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.

scholarly journals On the three families of extended Laguerre-based Apostol-type polynomials

2021 ◽  
Vol 40 (2) ◽  
pp. 313-334
Author(s):  
M. A. Pathan ◽  
Waseem A. Khan
Keyword(s):  
Generating Functions ◽  
Special Functions ◽  
Hermite Polynomials ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae ◽  
Symmetric Identities

In this paper, we introduce a new class of generalized extended Laguerre-based Apostol-type-Bernoulli, Apostol-type-Euler and Apostoltype-Genocchi polynomials. These Apostol type polynomials are used to connect Fubini-Hermite and Bell-Hermite polynomials and to find new representations. We derive some implicit summation formulae and symmetric identities for these families of special functions by applying the generating functions.

scholarly journals Some Properties Involving q-Hermite Polynomials Arising from Differential Equations and Location of Their Zeros

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1168
Author(s):  
Cheon Seoung Ryoo ◽  
Jung Yoog Kang
Keyword(s):  
Differential Equations ◽  
Hermite Polynomials ◽  
Real Numbers ◽  
Different Types

Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of q-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of q-Hermite polynomials according to the conditions of q-numbers, and look for values which have approximate roots that are real numbers.

scholarly journals A STUDY ON GENERALIZED HERMITE POLYNOMIALS

2017 ◽  
Vol 4 (37) ◽  
Author(s):  
Kamal Gupta
Keyword(s):  
Generating Functions ◽  
Hermite Polynomials ◽  
Type Formula

In this paper, we obtain generating functions involving hyper geometric functions. Rodrigues type formula of Hermite polynomials which is closely related to generalized Hermite polynomials of Dattoli et. al. These results provide useful extensions of the well known results of classical Hermite polynomials Hn(x).

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

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