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      Some properties of the Catalan–Qi function related to the Catalan numbers

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          Abstract

          In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan–Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function, and inequalities of the Catalan numbers, the Catalan function, and the Catalan–Qi function. As by-products, an exponential expansion and a double inequality for the ratio of two gamma functions are derived.

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          Sharp inequalities for the digamma and polygamma functions

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            A complete monotonicity property of the gamma function

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              A simple proof of logarithmic convexity of extended mean values

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                Author and article information

                Contributors
                qifeng618@gmail.com
                mansour@mans.edu.eg
                xiao-ting.shi@hotmail.com
                fang-liu@qq.com
                Journal
                Springerplus
                Springerplus
                SpringerPlus
                Springer International Publishing (Cham )
                2193-1801
                19 July 2016
                19 July 2016
                2016
                : 5
                : 1
                : 1126
                Affiliations
                [ ]Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, 454010 Henan Province China
                [ ]College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, 028043 Inner Mongolia Autonomous Region China
                [ ]Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516 Egypt
                [ ]Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387 China
                Author information
                http://orcid.org/0000-0001-6239-2968
                Article
                2793
                10.1186/s40064-016-2793-1
                4951397
                27478743
                cc880d0e-e1c1-4dac-ad6c-9785a48994e4
                © The Author(s) 2016

                Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

                History
                : 2 May 2016
                : 7 July 2016
                Categories
                Research
                Custom metadata
                © The Author(s) 2016

                Uncategorized
                property,catalan number,catalan function,catalan–qi function,asymptotic expansion,integral representation,logarithmic convexity,complete monotonicity,logarithmically complete monotonicity,minimality,inequality,ratio of gamma functions,primary 33b15; secondary 05a10,05a15,05a16,05a19,05a20,11b83,11y55,11y60,26a48,26a51,26d15,33c05,44a20

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