scholarly journals Some Inverse Relations Determined by Catalan Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Sheng-liang Yang
Keyword(s):  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Inverse Relation ◽  
Special Cases ◽  
Matrix Identities ◽  
Riordan Array

We use the A-sequence and Z-sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients. Some matrix identities obtained by Shapiro and Radoux are all special cases of our identity. In addition, a unified form of Catalan matrices is introduced.

scholarly journals Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials

10.37236/2476 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Birmajer ◽  
Juan B. Gil ◽  
Michael D. Weiner
Keyword(s):  
Combinatorial Identities ◽  
Convolution Type ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Inverse Relation

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals On a generalization of a waiting time problem and some combinatorial identities

2015 ◽  
Vol 52 (04) ◽  
pp. 981-989
Author(s):  
B. S. El-desouky ◽  
F. A. Shiha ◽  
A. M. Magar
Keyword(s):  
Waiting Time ◽  
Probability Function ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Special Cases ◽  
Time Problem ◽  
Probability Of Success ◽  
Waiting Time Problem ◽  
Finite Memory ◽  
Generalized Harmonic Numbers

In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, …, m. The probability of success pi = ni/N, i = 1, 2, …, m, where ni is a positive integer and Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

On the Solutions of the Matrix Equation f(X,X*)=g(X,X*)

1972 ◽  
Vol 15 (1) ◽  
pp. 45-49
Author(s):  
P. Basavappa
Keyword(s):  
Matrix Equation ◽  
Identity Matrix ◽  
Normal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Matrix Identities ◽  
Square Matrix

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.

Aspects of Lagrangian probability distributions

1994 ◽  
Vol 31 (A) ◽  
pp. 185-197 ◽  
Author(s):  
Masaaki Sibuya ◽  
Norihiko Miyawaki ◽  
Ushio Sumita
Keyword(s):  
Random Walks ◽  
Random Forests ◽  
Busy Period ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Queuing Systems ◽  
Transit System ◽  
Watson Process

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

Aspects of Lagrangian probability distributions

1994 ◽  
Vol 31 (A) ◽  
pp. 185-197 ◽  
Author(s):  
Masaaki Sibuya ◽  
Norihiko Miyawaki ◽  
Ushio Sumita
Keyword(s):  
Random Walks ◽  
Random Forests ◽  
Busy Period ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Queuing Systems ◽  
Transit System ◽  
Watson Process

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks. A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

Sign in / Sign up

Export Citation Format

Share Document