The determination of generating functions and combinatorial sums by multidimensional residues

1975 ◽  
Vol 15 (5) ◽  
pp. 740-747 ◽  
Author(s):  
G. P. Egorychev ◽  
A. P. Yuzhakov
Keyword(s):  
Generating Functions ◽  
Combinatorial Sums ◽  
Multidimensional Residues

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

A Note on Absorption Probabilities for a Random Walk Between a Reflecting and an Absorbing Barrier

1969 ◽  
Vol 6 (01) ◽  
pp. 224-226
Author(s):  
J.C. Hardin ◽  
A.L. Sweet
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Absorption Probabilities ◽  
Explicit Expressions

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

scholarly journals Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
Image Position

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

A Note on Absorption Probabilities for a Random Walk Between a Reflecting and an Absorbing Barrier

1969 ◽  
Vol 6 (1) ◽  
pp. 224-226 ◽  
Author(s):  
J.C. Hardin ◽  
A.L. Sweet
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Absorption Probabilities ◽  
Explicit Expressions

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

scholarly journals A Note on the Difference of Powers and Falling Powers

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Taoufik Sabar
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Mathematical Induction ◽  
Discrete Mathematics ◽  
Special Series ◽  
Induction Principle ◽  
Combinatorial Sums ◽  
The Difference ◽  
Binomial Identities

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.

Micromechanical Modeling of Composites Fracture

1995 ◽  
Author(s):  
V. Kobelev
Keyword(s):  
Generating Functions ◽  
Infinite System ◽  
Mixed Boundary ◽  
Fiber Reinforced Composites ◽  
Micromechanical Modeling ◽  
Reinforced Composites ◽  
System Of Difference Equations ◽  
Design Problems ◽  
Main Emphasis

Abstract The paper considers the new micromechanical models of fracture of fiber-reinforced composites. The main emphasis is made on the determination of the exact analytical solutions of appearing fracture problems, which allow derivation of the closed functional formulas for limit fracture stresses. These expressions are suited for subsequent use in the formulations of optimal design problems. The model presented here describes the deformation and tearing of cloth. The material under study is made up of two families of orthogonal fibers. The object of investigation is the distribution of stresses in the neighborhood of the end of a semi-infinite tear. The description of the stress-strain state in the neighborhood of the end of the tear reduces to the solution of a mixed boundary-value problem for an infinite system of difference equations. When cast in terms of generating functions for an infinite vector of unknowns, the problem reduces to a RIEMANN-HELBERT problem. The analytical solution to the problem shows that the character of tearing of the material depends critically on the stresses in the fibers parallel to the direction of the tear: if the fibers parallel to the tear are stretched, then a finite rip in the material has an elliptical shape and the asymptotic behavior of the stresses around the end of the tear is similar to behavior of a solid elastic body with a crack. Conversely, if there is no stretching of the fibers parallel to the tear, then the sides of the tear intersect at a right angle, and the first unbroken fiber bears a considerably larger load than when stresses in parallel fibers are absent.

scholarly journals On a functional equation arising from two processors with coupled inputs

2015 ◽  
Vol 4 (3) ◽  
pp. 264
Author(s):  
EL-Sayed El-Hady ◽  
Wolfgang Forg-Rob
Keyword(s):  
Functional Equation ◽  
Boundary Value Problem ◽  
Analytic Continuation ◽  
Generating Functions ◽  
Functional Equations ◽  
Elliptic Functions ◽  
Symmetric Case ◽  
The Other ◽  
Interesting Class

<p>During the last few decades, a certain interesting class of functional equations arises when obtaining the generating functions of many system distributions. Such a class of equations has numerous applications in many modern disciplines like wireless networks and communications. This paper has been motivated by an issue considered by Paul E. Wright in [Advances in applied probability, (1992), 986 􀀀 1007]. The functional equation obtained there has been solved using elliptic functions and analytic continuation, which in turn lead to the determination of the main unknown. Unfortunately that solution seems to be a bit too general with many technical assumptions. In this paper on one hand, we introduce a solution in the symmetric case using boundary value problem approach. On the other hand, we investigate the potential singularities of the unknowns of the functional equation giving one possible application, and we compute some expectation of interest using the corresponding generating function.</p>

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