On The Gould’s Formula for Stirling Numbers of The Second Kind

  • A. Zuniga-Segundo ESFM, Instituto Politécnico Nacional, Edif. 9, Dpto. Física, Lindavista 07738, CDMX
  • Jose Luis Lopez Bonilla National Polytechnic Institute
  • S. Vidal-Beltran ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX
Keywords: Stirling numbers, Duality property, Gould’s identity, Nörlund polynomials


We present an alternative deduction of the Gould’s relation for Stirling numbers of the second kind. Our approach is based in the Nörlund polynomials and in the duality property between the Stirling numbers.


H. Gupta, Symmetric functions in the theory of integrals numbers, Lucknow University Studies 14, Allahabad Press (1940)

H. Gupta, Selected topics in number theory, Tunbridge Wells, Abacus Press, England (1980)

D. E. Knuth, Selected papers on Discrete Mathematics, CSLI Lecture Notes, No. 106 (2003)

J. Quaintance, H. W. Gould, Combinatorial identities for Stirling numbers, World Scientific, Singapore (2016)

J. López-Bonilla, J. Yaljá Montiel-Pérez, S. Vidal-Beltrán, Stirling numbers with negative indices, World Scientific News 101 (2018) 217-221

H. W. Gould, Stirling number representation problems, Proc. Amer. Math. Soc. 11, No. 3 (1960) 447-451

L. Schläfli, Sur les coefficients du développement du produit 1(1+x)(1+2x)…(1+(n-1)x) suivant les puissance ascendantes de x, Crelle’s Journal 43 (1852) 1-22

J. López-Bonilla, R. López-Vázquez, Harmonic numbers in terms of Stirling numbers of the second kind, Prespacetime Journal 8, No. 2 (2017) 233-234

L. Carlitz, Note on Nörlund polynomials B_n^((z)), Proc. Amer. Math. Soc. 11, No. 3 (1960) 452-455

I. M. Gessel, On Miki’s identity for Bernoulli numbers, J. Number Theory 110, No. 1 (2005) 75-82

N. E. Nörlund, Vorlesungen über differenzenrechnung, Springer-Verlag, Berlin (1924)

I. M. Gessel, R. P. Stanley, Stirling polynomials, J. of Combinatorial Theory A24 (1978) 24-33

J. López-Bonilla, S. Yáñez-San Agustín, A. Zaldívar-Sandoval, On the Guo-Qi formula involving Bernoulli and Stirling numbers, Prespacetime Journal 7, No. 12 (2016) 1672-1673

H. M. Srivastava, J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier,London 2012)

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