Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers

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

[2] Á. Besenyei, On complete monotonicity of some functions related to means, Math. Inequal. Appl. 16 (2013), no. 1, 233–239; Available online at http://dx.doi.org/10.7153/mia-16-17.10.7153/mia-16-17Search in Google Scholar

[3] N. Bourbaki, Functions of a Real Variable—Elementary Theory, Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004; Available online at http://dx.doi.org/10.1007/978-3-642-59315-4.10.1007/978-3-642-59315-4Search in Google Scholar

[4] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974.Search in Google Scholar

[5] B.-N. Guo and F. Qi, A completely monotonic function involving the trigamma function and with degree one, Appl. Math. Comput., 218 (2012), no. 19, 9890–9897; Available online at http://dx.doi.org/10.1016/j.amc.2012.03.075.10.1016/j.amc.2012.03.075Search in Google Scholar

[6] B.-N. Guo and F. Qi, A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers, Glob. J. Math. Anal., 3 (2015), no. 1, 33–36; Available online at http://dx.doi.org/10.14419/gjma.v3i1.4168.10.14419/gjma.v3i1.4168Search in Google Scholar

[7] B.-N. Guo and F. Qi, Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis (Berlin) 34 (2014), no. 2, 187–193; Available online at http://dx.doi.org/10.1515/anly-2012-1238.10.1515/anly-2012-1238Search in Google Scholar

[8] B.-N. Guo and F. Qi, An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, Glob. J. Math. Anal., 2 (2014), no. 4, 243–248; Available online at http://dx.doi.org/10.14419/gjma.v2i4.3310.10.14419/gjma.v2i4.3310Search in Google Scholar

[9] B.-N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Anal. Number Theory, 3 (2015), no. 1, 27–30; Available online at http://dx.doi.org/10.12785/jant/030105.Search in Google Scholar

[10] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272 (2014), 251–257; Available online at http://dx.doi.org/10.1016/j.cam.2014.05.018.10.1016/j.cam.2014.05.018Search in Google Scholar

[11] B.-N. Guo and F. Qi, Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/2.1.3794.8808.Search in Google Scholar

[12] B.-N. Guo and F. Qi, On the degree of the weighted geometric mean as a complete Bernstein function, Afr. Mat., 26 (2015), no. 7, 1253–1262; Available online at http://dx.doi.org/10.1007/s13370-014-0279-2.10.1007/s13370-014-0279-2Search in Google Scholar

[13] B.-N. Guo and F. Qi, Six proofs for an identity of the Lah numbers, OnlineJ. Anal. Comb., 10 (2015), 5 pages.Search in Google Scholar

[14] B.-N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., 255 (2014), 568–579; Available online at http://dx.doi.org/10.1016/j.cam.2013.06.020.10.1016/j.cam.2013.06.020Search in Google Scholar

[15] B.-N. Guo and F. Qi, Some integral representations and properties of Lah numbers, J. Algebra Number Theory Acad., 4 (2014), no. 3, 77–87.Search in Google Scholar

[16] F. Qi, An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers, Mediterr. J. Math., 13 (2016) no. 5, 2795–2800; Available online at http://dx.doi.org/10.1007/s00009-015-0655-7.10.1007/s00009-015-0655-7Search in Google Scholar

[17] F. Qi, A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, Publ. Inst. Math. (Beograd) (N.S.) 100 (114) (2016), 243–249; Available online at http://dx.doi.org/10.2298/PIM150501028Q.10.2298/PIM150501028QSearch in Google Scholar

[18] F. Qi, An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind, J. Number Theory, 144 (2014), 244–255; Available online at http://dx.doi.org/10.1016/j.jnt.2014.05.009.10.1016/j.jnt.2014.05.009Search in Google Scholar

[19] F. Qi, Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind, Math. Inequal. Appl. 19 (2016), no. 1, 313–323; Available online at http://dx.doi.org/10.7153/mia-19-23.10.7153/mia-19-23Search in Google Scholar

[20] F. Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contrib. Discrete Math., 11 (2016), no. 1, 22–30; Available online at http://hdl.handle.net/10515/sy5wh2dx6.Search in Google Scholar

[21] F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat, 28 (2014), no. 2, 319–327; Available online at http://dx.doi.org/10.2298/FIL1402319O.10.2298/FIL1402319OSearch in Google Scholar

[22] F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory, 133 (2013), no. 7, 2307–2319; Available online at http://dx.doi.org/10.1016/j.jnt.2012.12.015.10.1016/j.jnt.2012.12.015Search in Google Scholar

[23] F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat, 27 (2013), no. 4, 601–604; Available online at http://dx.doi.org/10.2298/FIL1304601Q.10.2298/FIL1304601QSearch in Google Scholar

[24] F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl., 18 (2015), no. 2, 493–518; Available online at http://dx.doi.org/10.7153/mia-18-37.10.7153/mia-18-37Search in Google Scholar

[25] F. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math., 10 (2013), no. 4, 1685–1696; Available online at http://dx.doi.org/10.1007/s00009-013-0272-2.10.1007/s00009-013-0272-2Search in Google Scholar

[26] F. Qi and S.-X. Chen, Complete monotonicity of the logarithmic mean, Math. Inequal. Appl., 10 (2007), no. 4, 799–804; Available online at http://dx.doi.org/10.7153/mia-10-73.10.7153/mia-10-73Search in Google Scholar

[27] F. Qi and B.-N. Guo, Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis (Berlin) 34 (2014), no. 3, 311–317; Available online at http://dx.doi.org/10.1515/anly-2014-0003.10.1515/anly-2014-0003Search in Google Scholar

[28] F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput., 5 (2015), no. 4, 626–634; Available online at http://dx.doi.org/10.11948/2015049.10.11948/2015049Search in Google Scholar

[29] F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal., 2 (2014), no. 3, 91–97; Available online at http://dx.doi.org/10.14419/gjma.v2i3.2919.10.14419/gjma.v2i3.2919Search in Google Scholar

[30] F. Qi and X.-J. Zhang, An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind, Bull. Korean Math. Soc., 52 (2015), no. 3, 987–998; Available online at http://dx.doi.org/10.4134/BKMS.2015.52.3.987.10.4134/BKMS.2015.52.3.987Search in Google Scholar

[31] F. Qi and X.-J. Zhang, Complete monotonicity of a difference between the exponential and trigamma functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 21 (2014), no. 2, 141–145; Available online at http://dx.doi.org/10.7468/jksmeb.2014.21.2.141.10.7468/jksmeb.2014.21.2.141Search in Google Scholar

[32] F. Qi, X.-J. Zhang, and W.-H. Li, An elementary proof of the weighted geometric mean being a Bernstein function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 (2015), no. 1, 35–38.Search in Google Scholar

[33] F. Qi, X.-J. Zhang, and W.-H. Li, An integral representation for the weighted geometric mean and its applications, Acta Math. Sin. (Engl. Ser.), 30 (2014), no. 1, 61–68; Available online at http://dx.doi.org/10.1007/s10114-013-2547-8.10.1007/s10114-013-2547-8Search in Google Scholar

[34] F. Qi, X.-J. Zhang, and W.-H. Li, Lévy-Khintchine representation of the geometric mean of many positive numbers and applications, Math. Inequal. Appl., 17 (2014), no. 2, 719–729; Available online at http://dx.doi.org/10.7153/mia-17-53.10.7153/mia-17-53Search in Google Scholar

[35] F. Qi, X.-J. Zhang, and W.-H. Li, Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean, Mediterr. J. Math., 11 (2014), no. 2, 315–327; Available online at http://dx.doi.org/10.1007/s00009-013-0311-z.10.1007/s00009-013-0311-zSearch in Google Scholar

[36] F. Qi, X.-J. Zhang, and W.-H. Li, The harmonic and geometric means are Bernstein functions, Bol. Soc. Mat. Mex., (3) 22 (2016), in press; Available online at http://dx.doi.org/10.1007/s40590-016-0085-y.10.1007/s40590-016-0085-ySearch in Google Scholar

[37] F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput., 258 (2015), 597–607; Available online at http://dx.doi.org/10.1016/j.amc.2015.02.027.10.1016/j.amc.2015.02.027Search in Google Scholar

[38] R. L. Schilling, R. Song, and Z. Vondraćek, Bernstein Functions—Theory and Applications, de Gruyter Studies in Mathematics 37, De Gruyter, Berlin, Germany, 2010.10.1515/9783110215311Search in Google Scholar

[39] C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl., 2015, 2015:219, 8 pages; Available online at http://dx.doi.org/10.1186/s13660-015-0738-9.10.1186/s13660-015-0738-9Search in Google Scholar

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

[41] A.-M. Xu and Z.-D. Cen, Some identities involving exponential functions and Stirling numbers and applications, J. Comput. Appl. Math., 260 (2014), 201–207; Available online at http://dx.doi.org/10.1016/j.cam.2013.09.077.10.1016/j.cam.2013.09.077Search in Google Scholar

[42] X.-J. Zhang, Integral Representations, Properties, and Applications of Three Classes of Functions, Thesis supervised by Professor Feng Qi and submitted for the Master Degree of Science in Mathematics at Tianjin Polytechnic University in January 2013. (Chinese)Search in Google Scholar