scholarly journals Some Properties of the Fuss–Catalan Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 277 ◽  
Author(s):  
Feng Qi ◽  
Pietro Cerone
Keyword(s):  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Logarithmic Convexity ◽  
Double Inequality

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

scholarly journals Some Properties of the Fuss–Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Pietro Cerone
Keyword(s):  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Logarithmic Convexity ◽  
Double Inequality

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

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 Sharp Inequalities for Polygamma Functions

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Bai-Ni Guo ◽  
Feng Qi ◽  
Jiao-Lian Zhao ◽  
Qiu-Ming Luo
Keyword(s):  
Rational Functions ◽  
Complete Monotonicity ◽  
Double Inequality ◽  
Polygamma Function ◽  
Polygamma Functions

AbstractIn the paper, the authors review some inequalities and the (logarithmically) complete monotonicity concerning the gamma and polygamma functions and, more importantly, present a sharp double inequality for bounding the polygamma function by rational functions.

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 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 Complete monotonicity of a difference defined by four derivatives of a function containing trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Integral Inequality ◽  
Sufficient Conditions ◽  
Complete Monotonicity ◽  
Logarithmic Convexity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, logarithmic convexity of the gamma function, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Moreover, by virtue of Cebysev integral inequality, the author presents logarithmic convexity of the sequence of polygamma functions.

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 A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers

2019 ◽  
Vol 24 (2) ◽  
pp. 49
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Pietro Cerone
Keyword(s):  
Second Order ◽  
Catalan Numbers ◽  
Integral Representations ◽  
Complete Monotonicity ◽  
Product Ratio ◽  
Special Case

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan–Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization.

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

2016 ◽  
Vol 3 (1) ◽  
pp. 1179379 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo ◽  
Lishan Liu
Keyword(s):  
Catalan Numbers ◽  
Complete Monotonicity
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