scholarly journals Combinatorial Identities and Inverse Binomial Coefficients

2002 ◽  
Vol 28 (2) ◽  
pp. 196-202 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Combinatorial Identities ◽  
Binomial Coefficients

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.

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.

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.

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Cosine Function ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Hyperbolic Cosine Function

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Sine Function ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Sine Functions

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Combinatorial sums associated with balancing and Lucas-balancing polynomials

2020 ◽  
Vol Accepted manuscript ◽  
Author(s):  
Frontczak Robert ◽  
Goy Taras
Keyword(s):  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
Combinatorial Sums ◽  
Fibonacci And Lucas Numbers

The aim of the paper is to use some identities involving binomial coefficients to derive new combinatorial identities for balancing and Lucasbalancing polynomials. Evaluating these identities at specific points, we can also establish some combinatorial expressions for Fibonacci and Lucas numbers.

scholarly journals Combinatorial identities associated with Bernstein type basis functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1683-1689 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Catalan Numbers ◽  
Combinatorial Identities ◽  
Basis Functions ◽  
Binomial Coefficients ◽  
Bernstein Basis ◽  
Bernstein Type ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

In this paper, we give some identities and relations for the Bernstein basis functions and the beta type polynomials. Integrating these identities, we derive many identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients and the Catalan numbers. We also give remarks and comments on these identities.

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