An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions

  • Authors

    • Bai-Ni Guo
    • Feng Qi Department of Mathematics, College of Science, Tianjin Polytechnic University Tianjin City, 300160, China
    2014-09-08
    https://doi.org/10.14419/gjma.v2i4.3310

  • Abstract

    In the paper, by two methods, the authors find an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind. Moreover, the authors supply an alternative proof of the well-known ``triangular'' recurrence relation for Stirling numbers of the second kind. In a remark, the authors reveal the combinatorial interpretation of the special values for Kummer confluent hypergeometric functions and the total sum of Lah numbers.

    Keywords: explicit formula; Bell number; conuent hypergeometric function of the first kind; Stirling number of the second kind; combinatorial interpretation; alternative proof; recurrence relation; polylogarithm

    Author Biography

    • Feng Qi, Department of Mathematics, College of Science, Tianjin Polytechnic University Tianjin City, 300160, China

  • References

    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, 1972. [2] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. [3] B.-N. Guo and F. Qi, Some integral representations and properties of Lah numbers, available online at http://arxiv.org/abs/1402.2367. [4] F. Qi, An explicit formula for computing Bell numbers in terms of Lah and Stirling numbers, available online at http://arxiv.org/abs/1401.1625. [5] F. Qi, An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, available online at http://arxiv.org/abs/1402.2361. [6] F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl. 18 (2015), in press; Available online at http://arxiv.org/abs/1302.6731. [7] 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. [8] 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. [9] C.-F. Wei and B.-N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal. 2014 (2014), Article ID 851213, 5 pages; Available online at http://dx.doi.org/10.1155/2014/851213. [10] E. W. Weisstein, Polylogarithm, MathWorld-A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/Polylogarithm.html. [11] E. W. Weisstein, Stirling Number of the Second Kind, MathWorld-A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html.

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  • How to Cite

    Guo, B.-N., & Qi, F. (2014). An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions. Global Journal of Mathematical Analysis, 2(4), 243-248. https://doi.org/10.14419/gjma.v2i4.3310

    Received date: 2014-08-03

    Accepted date: 2014-09-06

    Published date: 2014-09-08