scholarly journals On a system of partial differential equations and the bivariate Hermite polynomials

2017 ◽  
Vol 454 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hermite Polynomials ◽  
Partial Differential

scholarly journals On the complex Hermite polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 409-420
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generating Function ◽  
Poisson Kernel ◽  
Hermite Polynomials ◽  
Addition Formula ◽  
Partial Differential ◽  
Expansion Theorem ◽  
New Approach ◽  
Complex Hermite Polynomials

In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite polynomials. Using this expansion, we derive the Poisson Kernel, the Nielsen type formula, the addition formula for the complex Hermite polynomials with ease. A multilinear generating function for the complex Hermite polynomials is proved.

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.

Partial Differential Equations

2020 ◽  
Author(s):  
A. K. Nandakumaran ◽  
P. S. Datti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential
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