Skip to main content
Have a personal or library account? Click to login
Logarithmically complete monotonicity of a function related to the Catalan-Qi function Cover

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

Acta Universitatis Sapientiae, Mathematica
Volume 8 (2016): Issue 1 (August 2016)
By:  and    
Open Access
|Aug 2016
References

References

  1. [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. [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. [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. [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. [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. [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. [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. [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. [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. [10] D. A. Klarner, Correspondences between plane trees and binary sequences, J. Combinatorial Theory, 9 (1970), 401–411.10.1016/S0021-9800(70)80093-X
    Open DOISearch in Google Scholar
  11. [11] T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009.10.1093/acprof:oso/9780195334548.001.0001
    Open DOISearch in Google Scholar
  12. [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. [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. [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. [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. [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. [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. [18] W. Młotkowski, Fuss-Catalan numbers in noncommutative probability, Doc. Math., 15 (2010), 939–955.10.4171/dm/318
    Open DOISearch in Google Scholar
  19. [19] W. Młotkowski, K. A. Penson, K. Ż yczkowski, Densities of the Raney distributions, Doc. Math., 18 (2013), 1573–1596.10.4171/dm/437
    Open DOISearch in Google Scholar
  20. [20] I. Pak, Catalan Numbers Page, available online at http://www.math.ucla.edu/~pak/lectures/Cat/pakcat.htm.
    Search in Google Scholar
  21. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.3639
    Open DOISearch in Google Scholar
  42. [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. [43] I. Vardi, Computational Recreations in Mathematica, Addison-Wesley, Redwood City, CA, 1991.
    Search in Google Scholar
  44. [44] D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, N. J., 1941.
    Search in Google Scholar
  45. [45] Wikipedia, Fuss-Catalan number, From the Free Encyclopedia; Available online at https://en.wikipedia.org/wiki/Fuss-Catalan_number.
    Search in Google Scholar
DOI: https://doi.org/10.1515/ausm-2016-0006 | Journal eISSN: 2066-7752
Journal RSS Feed
Language: English
Page range: 93 - 102
Submitted on: Oct 22, 2015
Published online: Aug 2, 2016
Published in print: Aug 1, 2016
Published by: Sapientia Hungarian University of Transylvania
In partnership with: Paradigm Publishing Services
Keywords:
necessary and sufficient condition,
logarithmically complete monotonicity,
Catalan number,
Catalan-Qi function
MSC:
11B75,
11B83,
11Y35,
11Y55,
11Y60,
26A48,
33B15
Related subjects:
Mathematics,
General mathematics

© 2016 Feng Qi, Bai -Ni Guo, published by Sapientia Hungarian University of Transylvania
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 8 (2016): Issue 1 (August 2016)
Previous article
Next article