Open Access

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

 and    | Aug 02, 2016

Cite

[1] M. Abramowitz, I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.Search in Google Scholar

[2] N. Alexeev, F. Götze, A. Tikhomirov, Asymptotic distribution of singular values of powers of random matrices, Lith. Math. J., 50 (2010), 121–132; Available online at http://dx.doi.org/10.1007/s10986-010-9074-4.Search in Google Scholar

[3] R. D. Atanassov, U. V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci., 41 (1988), 21–23.Search in Google Scholar

[4] J. -C. Aval, Multivariate Fuss-Catalan numbers, Discrete Math., 308 (2008), 4660–4669; Available online at http://dx.doi.org/10.1016/j.disc.2007.08.100.Search in Google Scholar

[5] D. Bisch, V. Jones, Algebras associated to intermediate subfactors, Invent. Math., 128 (1997), 89–157; Available online at http://dx.doi.org/10.1007/s002220050137.Search in Google Scholar

[6] N. I. Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Nova Acta Academiae Sci. Petropolitanae, 9 (1791), 243–251.Search in Google Scholar

[7] I. G. Gordon, S. Griffeth, Catalan numbers for complex reection groups, Amer. J. Math., 134 (2012), 1491–1502; Available online at http://dx.doi.org/10.1353/ajm.2012.0047.Search in Google Scholar

[8] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics—A Foundation for Computer Science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994.Search in Google Scholar

[9] P. Hilton, J. Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer, 13 (1991), 64–75; Available online at http://dx.doi.org/10.1007/BF03024089.Search in Google Scholar

[10] D. A. Klarner, Correspondences between plane trees and binary sequences, J. Combinatorial Theory, 9 (1970), 401–411.10.1016/S0021-9800(70)80093-XSearch in Google Scholar

[11] T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009.10.1093/acprof:oso/9780195334548.001.0001Search in Google Scholar

[12] C. -H. Lin, Some combinatorial interpretations and applications of Fuss-Catalan numbers, ISRN Discrete Math., 2011 (2011), Article ID 534628, 8 pages; Available online at http://dx.doi.org/10.5402/2011/534628.Search in Google Scholar

[13] F. -F. Liu, X.-T. Shi, F. Qi, A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function, Glob. J. Math. Anal., 3 (2015), 140–144; Available online at http://dx.doi.org/10.14419/gjma.v3i4.5187.Search in Google Scholar

[14] D. -Z. Liu, C. -W. Song, Z. -D. Wang, On explicit probability densities associated with Fuss-Catalan numbers, Proc. Amer. Math. Soc., 139 (2011), 3735–3738; Available online at http://dx.doi.org/10.1090/S0002-9939-2011-11015-3.Search in Google Scholar

[15] M. Mahmoud, F. Qi, Three identities of Catalan-Qi numbers, Mathematics, 4 (2) (2016), Article 35, 7 pages; Available online at http://dx.doi.org/10.3390/math4020035.Search in Google Scholar

[16] J. McCarthy, Catalan numbers. Letter to the editor: “Catalan numbers, their generalization, and their uses” [Math. Intelligencer, 13 (1991), 64–75] by P. Hilton and J. Pedersen, Math. Intelligencer, 14 (1992), 5.Search in Google Scholar

[17] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993; Available online at http://dx.doi.org/10.1007/978-94-017-1043-5.Search in Google Scholar

[18] W. Młotkowski, Fuss-Catalan numbers in noncommutative probability, Doc. Math., 15 (2010), 939–955.10.4171/dm/318Search in Google Scholar

[19] W. Młotkowski, K. A. Penson, K. Ż yczkowski, Densities of the Raney distributions, Doc. Math., 18 (2013), 1573–1596.10.4171/dm/437Search in Google Scholar

[20] I. Pak, Catalan Numbers Page, available online at http://www.math.ucla.edu/~pak/lectures/Cat/pakcat.htm.Search in Google Scholar

[21] J. H. Przytycki, A. S. Sikora, Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers, J. Combin. Theory Ser. A, 92 (2000), 68–76; Available online at http://dx.doi.org/10.1006/jcta.1999.3042.Search in Google Scholar

[22] F. Qi, Asymptotic expansions, complete monotonicity, and inequalities of the Catalan numbers, ResearchGate Technical Report, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.4371.6321.Search in Google Scholar

[23] F. Qi, B. -N. Guo, Logarithmically complete monotonicity of Catalan-Qi function related to Catalan numbers, Cogent Math., (2016), 3:1179379, 6 pages; Available online at http://dx.doi.org/10.1080/23311835.2016.1179379.Search in Google Scholar

[24] F. Qi, Some properties and generalizations of the Catalan, Fuss, and Fuss-Catalan numbers, ResearchGate Research, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.1778.3128.Search in Google Scholar

[25] F. Qi, Two product representations and several properties of the Fuss-Catalan numbers, ResearchGate Research, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.1655.6004.Search in Google Scholar

[26] F. Qi, C. -P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl., 296 (2004), 603–607; Available online at http://dx.doi.org/10.1016/j.jmaa.2004.04.026.Search in Google Scholar

[27] F. Qi, S. -X. Chen, Complete monotonicity of the logarithmic mean, Math. Inequal. Appl., 10 (2007), 799–804; Available online at http://dx.doi.org/10.7153/mia-10-73.Search in Google Scholar

[28] F. Qi, B. -N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll., 7 (2004), 63–72; Available online at http://rgmia.org/v7n1.php.Search in Google Scholar

[29] F. Qi, M. Mahmoud, X. -T. Shi, F. -F. Liu, Some properties of the Catalan-Qi function related to the Catalan numbers, SpringerPlus, (2016), 5:1126, 20 pages; Available online at http://dx.doi.org/10.1186/s40064-016-2793-1.Search in Google Scholar

[30] F. Qi, X. -T. Shi, P. Cerone, A unified generalization of the Catalan, Fuss, Fuss-Catalan numbers and Catalan-Qi function, ResearchGate Working Paper, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.3198.6000.Search in Google Scholar

[31] F. Qi, X. -T. Shi, F. -F. Liu, An exponential representation for a function involving the gamma function and originating from the Catalan numbers, ResearchGate Research, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.1086.4486.Search in Google Scholar

[32] F. Qi, X. -T. Shi, F. -F. Liu, An integral representation, complete monotonicity, and inequalities of the Catalan numbers, ResearchGate Technical Report, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.3754.4806.Search in Google Scholar

[33] F. Qi, X. -T. Shi, F. -F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, ResearchGate Technical Report, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.3230.1927.Search in Google Scholar

[34] F. Qi, X. -T. Shi, M. Mahmoud, F. -F. Liu, Schur-convexity of the Catalan-Qi function, ResearchGate Technical Report, (2015); Available online at http://dx.doi.org/10.13140/RG.2.1.2434.4802.Search in Google Scholar

[35] F. Qi, X. -J. Zhang, W. -H. Li, An integral representation for the weighted geometric mean and its applications, Acta Math. Sin. (Engl. Ser.), 30 (2014), 61–68; Available online at http://dx.doi.org/10.1007/s10114-013-2547-8.Search in Google Scholar

[36] F. Qi, X. -J. Zhang, W. -H. Li, Lévy-Khintchine representation of the geometric mean of many positive numbers and applications, Math. Inequal. Appl., 17 (2014), 719–729; Available online at http://dx.doi.org/10.7153/mia-17-53.Search in Google Scholar

[37] F. Qi, X. -J. Zhang, W. -H. Li, Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean, Mediterr. J. Math., 11 (2014), 315–327; Available online at http://dx.doi.org/10.1007/s00009-013-0311-z.Search in Google Scholar

[38] R. L. Schilling, R. Song, Z. Vondraček, Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012; Available online at http://dx.doi.org/10.1515/9783110269338.Search in Google Scholar

[39] X. -T. Shi, F. -F. Liu, F. Qi, An integral representation of the Catalan numbers, Glob. J. Math. Anal., 3 (2015), 130–133; Available online at http://dx.doi.org/10.14419/gjma.v3i3.5055.Search in Google Scholar

[40] R. Stanley, E. W. Weisstein, Catalan number, From MathWorld–A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/CatalanNumber.html.Search in Google Scholar

[41] C. Stump, q, t-Fuß-Catalan numbers for complex reection groups, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics, 2008, 295–306, Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.10.46298/dmtcs.3639Search in Google Scholar

[42] C. Stump, q, t-Fuß-Catalan numbers for finite reection groups, J. Algebraic Combin., 32 (2010), 67–97; Available online at http://dx.doi.org/10.1007/s10801-009-0205-0.Search in Google Scholar

[43] I. Vardi, Computational Recreations in Mathematica, Addison-Wesley, Redwood City, CA, 1991.Search in Google Scholar

[44] D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, N. J., 1941.Search in Google Scholar

[45] Wikipedia, Fuss-Catalan number, From the Free Encyclopedia; Available online at https://en.wikipedia.org/wiki/Fuss-Catalan_number.Search in Google Scholar

eISSN:
2066-7752
Language:
English
Publication timeframe:
2 times per year
Journal Subjects:
Mathematics, General Mathematics