scholarly journals A further generalization of the Catalan numbers and its explicit formula and integral representation

2021 ◽  
Author(s):  
Wen-Hui Li ◽  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Number Theory ◽  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Complex Analysis ◽  
Integral Formula ◽  
Catalan Numbers ◽  
Combinatorial Number Theory ◽  
Integral Formulas ◽  
Cauchy’S Integral Formula

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

scholarly journals A further generalization of the Catalan numbers and its explicit formula and integral representation

2020 ◽  
Author(s):  
Wen-Hui Li ◽  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Number Theory ◽  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Complex Analysis ◽  
Integral Formula ◽  
Catalan Numbers ◽  
Combinatorial Number Theory ◽  
Integral Formulas ◽  
Cauchy’S Integral Formula

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy's integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

scholarly journals An integral representation, complete monotonicity, and inequalities of the Catalan numbers

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Fang-Fang Liu
Keyword(s):  
Integral Representation ◽  
Generating Function ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Cauchy Integral ◽  
Complex Functions

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

scholarly journals Deriving Cauchy's Integral Formula Using Division Method

2019 ◽  
Vol 1 ◽  
pp. 259-264
Author(s):  
M Egahi ◽  
I O Ogwuche ◽  
J Ode
Keyword(s):  
Analytic Functions ◽  
Complex Analysis ◽  
Integral Formula ◽  
Integral Theorem ◽  
Cauchy’S Integral Formula ◽  
Cauchy’S Integral Theorem

Cauchy's integral theorem and formula which holds for analytic functions is proved in most standard complex analysis texts. The nth derivative form is also proved. Here we derive the nth derivative form of Cauchy's integral formula using division method and showed its link with Taylor's theorem and demonstrate the result with some polynomials.

scholarly journals The Translated Dowling Polynomials and Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mahid M. Mangontarum ◽  
Amila P. Macodi-Ringia ◽  
Normalah S. Abdulcarim
Keyword(s):  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Hankel Transform ◽  
Bell Polynomials ◽  
Basic Properties ◽  
Whitney Numbers ◽  
Exponential Generating Function

More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.

scholarly journals Integral Representations for Multivariate Logarithmic Polynomials

2017 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Integral Representation ◽  
Uniform Convergence ◽  
Generating Function ◽  
Integral Representations ◽  
Monotonic Function ◽  
Bernstein Function ◽  
Completely Monotonic Function ◽  
Bernstein Functions ◽  
Completely Monotonic ◽  
Integral Formulas

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

scholarly journals Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Guy Louchard
Keyword(s):  
Saddle Point ◽  
Generating Function ◽  
Central Region ◽  
Integral Formula ◽  
Stirling Numbers ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Central Regions ◽  
Cauchy’S Integral Formula ◽  
International Audience

International audience Using the saddle point method, we obtain from the generating function of the Stirling numbers of the first kind [n j] and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the region j = n - n(alpha); alpha > 1/2, we analyze the dependence of [n j] on alpha.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Qing Zou ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

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