An integral representation of the Catalan numbers

2015 ◽  
Vol 3 (3) ◽  
pp. 130 ◽  
Author(s):  
Xiao-Ting Shi ◽  
Fang-Fang Liu ◽  
Feng Qi
Keyword(s):  
Integral Representation ◽  
Open Problem ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Gamma Functions

<p><span>In the paper, the authors establish an integral representation of the Catalan numbers, connect the Catalan numbers with the (logarithmically) complete monotonicity, and pose an open problem on the logarithmically complete monotonicity of a function involving ratio of gamma functions.</span></p>

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 A Solution to Qi’s Conjecture on a Double Inequality for a Function Involving the Tri- and Tetra-Gamma Functions

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1098 ◽  
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Open Problem ◽  
Complete Monotonicity ◽  
Double Inequality ◽  
Gamma Functions

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper “Complete monotonicity of a function involving the tri- and tetragamma functions” (2015) and repeated in the seventh open problem of the paper “On complete monotonicity for several classes of functions related to ratios of gamma functions” (2019).

scholarly journals On the complete monotonicity of quotient of gamma functions

2012 ◽  
pp. 395-402 ◽  
Author(s):  
Yi-Chao Chen ◽  
Toufik Mansour ◽  
Qian Zou
Keyword(s):  
Complete Monotonicity ◽  
Gamma Functions

scholarly journals COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE DIVIDED DIFFERENCE OF PSI FUNCTIONS

2013 ◽  
Vol 88 (2) ◽  
pp. 309-319 ◽  
Author(s):  
FENG QI ◽  
PIETRO CERONE ◽  
SEVER S. DRAGOMIR
Keyword(s):  
Divided Difference ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Divided Differences ◽  
Psi Function ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

scholarly journals Some new properties of the Barnes G-function and related results

2020 ◽  
pp. 56-56
Author(s):  
Chao-Ping Chen ◽  
H.M. Srivastava
Keyword(s):  
Integral Representation ◽  
Continued Fraction ◽  
Complete Monotonicity

In this paper, we present several potentially useful properties of the Barnes G-function. The properties considered here include, for example, its integral representation, complete monotonicity, and continued-fraction approximation. We also derive continued-fraction approximations of the Glaisher-Kinkelin constant and the Choi-Srivastava constants.

scholarly journals Logarithmically complete monotonicity of a function related to the Catalan-Qi function

2016 ◽  
Vol 8 (1) ◽  
pp. 93-102 ◽  
Author(s):  
Feng Qi ◽  
Bai -Ni Guo
Keyword(s):  
Sufficient Conditions ◽  
Catalan Numbers ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Necessary And Sufficient

AbstractIn the paper, the authors find necessary and sufficient conditions such that a function related to the Catalan-Qi function, which is an alternative generalization of the Catalan numbers, is logarithmically complete monotonic.

scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

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