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.

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 New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 264 ◽  
Author(s):  
Can Kızılateş ◽  
Bayram Çekim ◽  
Naim Tuğlu ◽  
Taekyun Kim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generating Functions ◽  
Explicit Representation ◽  
Partial Differential ◽  
Generalized Polynomials ◽  
Special Cases

In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found.

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.

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