The Monotonicity and Log-Behaviour of Some Functions Related to the Euler Gamma Function

2016 ◽  
Vol 60 (2) ◽  
pp. 527-543
Author(s):  
Bao-Xuan Zhu
Keyword(s):  
Gamma Function ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Binomial Coefficients ◽  
Euler Gamma Function ◽  
Continuous Analogue ◽  
Riemann Zeta ◽  
The One ◽  
Image Position ◽  
Positive Integers

AbstractThe aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals Analogues of the binomial coefficient theorems of Gauss and Jacobi

2016 ◽  
Vol 12 (08) ◽  
pp. 2125-2145
Author(s):  
Abdullah Al-Shaghay ◽  
Karl Dilcher
Keyword(s):  
Gamma Function ◽  
Simple Form ◽  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients

The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

scholarly journals Proof of two Divisibility Properties of Binomial Coefficients Conjectured by Z.-W. Sun

10.37236/4258 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Victor J. W. Guo
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Divisibility Properties ◽  
Positive Integers

For all positive integers $n$, we prove the following divisibility properties:\[ (2n+3){2n\choose n}  \left|3{6n\choose 3n}{3n\choose n},\right. \quad\text{and}\quad(10n+3){3n\choose n} \left|21{15n\choose 5n}{5n\choose n}.\right. \]This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers $m$ and $n$, the product $am{am+bm-1\choose am}{an+bn\choose an}$ is divisible by $m+n$. In fact, the latter result can be further generalized to the $q$-binomial coefficients and $q$-integers case, which generalizes the positivity of $q$-Catalan numbers. We also propose several related conjectures.

Tribinomial coefficients and Catalan numbers

2009 ◽  
Vol 93 (528) ◽  
pp. 449-455 ◽  
Author(s):  
Thomas Koshy ◽  
Mohammad Salmassi
Keyword(s):  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Triangular Numbers ◽  
Interesting Family ◽  
Positive Integers

The concept of the ordinary binomial coefficientcan be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

scholarly journals Continous analogues for the binomial coefficients and the Catalan numbers

2019 ◽  
Vol 13 (2) ◽  
pp. 542-568
Author(s):  
Rafael Díaz ◽  
Leonardo Cano
Keyword(s):  
Binomial Distribution ◽  
Convex Polytopes ◽  
Catalan Numbers ◽  
Lattice Paths ◽  
Binomial Coefficients ◽  
Continuous Analogue

Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an application we develop a continuous analogue for the binomial distribution.

Γ2 (½)is more than just π

2013 ◽  
Vol 97 (540) ◽  
pp. 430-434
Author(s):  
Samuel G. Moreno ◽  
Esther M. García-Caballero
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Infinite Product ◽  
Positive Real ◽  
Improper Integral ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Positive Integers

For a fixed positive integer m, factorial m is defined byThe problem of finding a formula extending the factorial m! to positive real values of m was posed by D. Bernoulli and C. Goldbach and solved by Euler. In his letter of 13 October 1729 to Goldbach [1], Euler defined a function (which we denote as Γ (x + 1)) by means ofand showed that Γ (m + 1) = m! for positive integers m. After that, Euler found representations for the so-called gamma function (1) in terms of either an infinite product or an improper integral. We refer the reader to the classical (and short) treatise [2] for a brief introduction and main properties of the gamma function.

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