On an Identity Involving Stirling Numbers of the Second Kind



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Submitted May 19, 2023
Published Jul 28, 2023
10.24018/ejmath.2023.4.4.255

Abstract viewed = 144 times

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We investigate two generalized forms for the recurrence relation S(n, k) = kS(n - 1, k) + S(n - 1, k - 1). From these generalized forms, we derive a new identity, for which a proof of the identity is given.

Keywords: Stirling numbers of the Second Kind recurrence relations identities proofs

References

  1. Moser L, Wyman M. Stirling numbers of the second kind. Duke Mathematical Journal. 1958; 25:29 – 43.
     Google Scholar
  2. Moser L, Wyman M. Asymptotic development of the Stirling numbers of the first kind. Journal of the London Mathematical Society. 1958; 33:133–146.
     Google Scholar
  3. Quaintance J, Gould HW. Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H. W. Gould. With a foreward by George E. Andrews. Singapore: World Scientific Publishing Co. Pte. Ltd., 2016.
     Google Scholar
  4. Knessl C, Keller JB. Stirling number asymptotics from recursion equations using the ray method. Studies in Applied Mathematics. 1991; 84:43–56.
     Google Scholar
  5. Gould HW. Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations, West Virginia: Morgantown, 1972
     Google Scholar
  6. Wilf HS. Generatingfunctionology. Boston: Academic Press, 2005.
     Google Scholar
  7. Wyman M. Asymptotic behaviour of Laurent coefficients. Canadian Journal of Mathematics. 1959; 11:534–555.
     Google Scholar
  8. Bressoud DM, Zeilberger D. Generalized Rogers-Ramanujan bijections. Advances in Mathematics. 1989; 78(1):42–75.
     Google Scholar
  9. Bressoud DM. Almost poised basic hypergeometric series. Proceedings of the Indian Academy of Sciences-Mathematical Sciences. 1987; 97:61–66.
     Google Scholar
  10. Bressoud DM. A generalization of the Rogers-Ramanujan identities for all moduli. Journal of Combinatorial Theory. 1979; Series A 27 (1):64–68.
     Google Scholar
  11. Canfield ER, Pomerance C. On the problem of uniqueness for the maximum Stirling number(s) of the second kind. Integers. 2002; vol. 2.
     Google Scholar
  12. Bressoud DM. Lattice paths and the Rogers-Ramanujan identities. In: Number Theory, Madras 1987: Proceedings of the International Ramanujan Centenary Conference held at Anna University, Madras, India, Dec. 21, 1987, Springer, 2006, pp. 140–172.
     Google Scholar
  13. Gould HW. Euler’s Formula for nth Differences of Powers. The American Mathematical Monthly. 1978; 85(6):450–467.
     Google Scholar
  14. Gould HW, Kaucky J. Evaluation of a class of binomial coefficient summations. Journal of Combinatorial Theory. 1966; 1, 233–247.
     Google Scholar
  15. Gould HW, Squire W. Maclaurin’s second formula and its generalization. The American Mathematical Monthly. 1963; 70:44–52.
     Google Scholar
  16. Garrett K, Ismail MEH, Stanton D. Variants of the Rogers-Ramanujan identities. Advances in Applied Mathematics. 1999; 23(3):274–299.
     Google Scholar
  17. Hsu LC, Shiue PJS. A unified approach to generalized Stirling numbers. Advances in Applied Mathematics. 1988; 20:366–384.
     Google Scholar
  18. Gould HW. A new symmetrical combinatorial identity. Journal of Combinatorial Theory. 1972; Series A 13:278–286.
     Google Scholar
  19. Arratia A, DeSalvo S. Completely effective error bounds for Stirling Numbers of the first and second kinds via Poisson approximation. Annals of Combinatorics. 2017; 21:1–24.
     Google Scholar
  20. Alsardary SY, Gould HW. Special Function Inverse Series Pairs. KYUNGPOOK Math Journal. 2010; 50:177–193
     Google Scholar
  21. Bressoud DM. Some identities for terminating q-series. Mathematical Proceedings of the Cambridge Philosophical Society. 1981; 89:211–223.
     Google Scholar
  22. Butzer PL, Hauss M. On Stirling functions of the second kind. Studies in Applied Mathematics. 1991;84 (1):71–91.
     Google Scholar
  23. Bressoud DM. An easy proof of the Rogers-Ramanujan identities. Journal of Number Theory. 1983; 16(2):235–241.
     Google Scholar
  24. Gessel I, Stanton D. Strange evaluations of hypergeometric series. SIAM Journal on Mathematical Analysis. 1982; 13:295–308.
     Google Scholar
  25. Gould HW. Which numbers are interesting? The Mathematics Teacher. 1980; 73(6):408.
     Google Scholar
  26. Ismail MEH, Stanton D. On the Askey-Wilson and Rogers polynomials. Canadian Journal of Mathematics. 1988; 40:1025–1045.
     Google Scholar