PL EN


Preferences help
enabled [disable] Abstract
Number of results
2018 | 107 | 201-206
Article title

Todorov’s formula involving Stirling numbers of the second kind and Nörlund polynomials

Content
Title variants
Languages of publication
EN
Abstracts
EN
We use the Todorov’s formula for the generalized Bernoulli numbers, in terms of Stirling numbers of the second kind, to deduce the Guo-Qi and Schläfli identities. Our approach is based in the duality property between the Stirling numbers, and in the Nörlund polynomials. We also consider the Janjic’s definitions for the Stirling numbers.
Year
Volume
107
Pages
201-206
Physical description
Contributors
  • Depto. Física, ESFM, Instituto Politécnico Nacional, Edif. 9, Col. Lindavista CP 07738, CDMX, México
  • ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, Col. Lindavista CP 07738, CDMX, México
  • ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, Col. Lindavista CP 07738, CDMX, México
References
  • P. G. Todorov, Une formule simple explicite des nombres de Bernoulli généralisés, C. R. Acad. Sci. Paris Sér. I Math 301 (1985) 665-666.
  • H. M. Srivastava, J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier, London (2012).
  • M. Aigner, A course in enumeration, Springer-Verlag, Berlin (2007).
  • J. Quaintance, H. W. Gould, Combinatorial identities for Stirling numbers, World Scientific, Singapore (2016).
  • V. Barrera-Figueroa, J. López-Bonilla, R. López-Vázquez, On Stirling numbers of the second kind, Prespacetime Journal 8(5) (2017) 572-574.
  • V. Barrera-Figueroa, J. López-Bonilla, R. López-Vázquez, S. Vidal-Beltrán, On Stirling numbers, World Scientific News 98 (2018) 228-232.
  • N. E. Nörlund, Vorlesungen über differenzenrechnung, Springer-Verlag, Berlin (1924).
  • I. M. Gessel, On Miki’s identity for Bernoulli numbers, J. Number Theory 110(1) (2005) 75-82.
  • J. Faulhaber, Academiae algebrae, Ulm, Germany (1631).
  • Jakob Bernoulli, Ars conjectandi, Thurneysen, Basel (1713) [The art of conjecturing, Johns Hopkins University Press, Baltimore (2005)].
  • T. Arakawa, T. Ibukiyama, M. Kaneko, Bernoulli numbers and zeta functions, Springer, Japan (2014).
  • A. Iturri, J. López-Bonilla, R. López-Vázquez, Harmonic, Stirling, and Bernoulli numbers, Prespacetime Journal 8(9) (2017) 1173-1175.
  • Bai-Ni Guo, Feng Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. of Analysis & Number Theory 3(1) (2015) 27-30.
  • J. López-Bonilla, S. Yáñez-San Agustín, A. Zaldívar-Sandoval, On the Guo-Qi formula involving Bernoulli and Stirling numbers, Prespacetime Journal 7(12) (2016) 1672-1673.
  • Ch. Jordan, Calculus of finite differences, Chelsea Pub., New York (1957).
  • S. Shirai, K. I. Sato, Some identities involving Bernoulli and Stirling numbers, J. Number Theory 90(1) (2001) 130-142.
  • S. Jeong, M. S. Kim, J. W. Son, On explicit formulae for Bernoulli numbers and their counterparts in positive characteristic, J. Number Theory 113(1) (2005) 53-68.
  • L. Carlitz, Note on Nörlund polynomials B_n^((z)), Proc. Amer. Math. Soc. 11 (1960) 452-455.
  • H. Gupta, Symmetric functions in the theory of integrals numbers, Lucknow University Studies 14, Allahabad Press (1940).
  • I. M. Gessel, R. P. Stanley, Stirling polynomials, J. of Combinatorial Theory A24 (1978) 24-33.
  • 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. López-Bonilla, J. Yaljá Montiel-Pérez, S. Vidal-Beltrán, Stirling numbers with negative indices, World Scientific News 101 (2018) 217-221.
  • A. Iturri, J. López-Bonilla, R. López-Vázquez, O. Salas-Torres, Bernoulli and Stirling numbers, BAOJ Physics 2(1) (2017) 3-5.
  • 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.
  • H. W. Gould, Stirling number representation problems, Proc. Amer. Math. Soc. 11(3) (1960) 447-451.
  • J. López-Bonilla, S. Vidal-Beltrán, A. Zúñiga-Segundo, On the Gould’s formula for Stirling numbers of the second kind, MathLAB Journal 1(2) (2018) to appear.
  • J. Choi, Explicit formulas for the Bernoulli polynomials of order n, Indian J. Pure Appl. Math. 27 (1996) 667-674.
  • M. Janjic, On a non-combinatorial definition of Stirling numbers, arXiv: 0806.2366v1 [math.CO] 14 Jun 2008.
  • H. W. Gould, The generalized chain rule of differentiation with historical notes, Utilitas Mathematica 61 (2002) 97-106.
  • W. P. Johnson, The curious history of Faà di Bruno’s formula, The Math. Assoc. of America 109 (2002) 217-234.
  • Z-H. Sun, Some inversion formulas and formulas for Stirling numbers, Graphs and Combinatorics 29(4) (2013) 1087-1100.
Document Type
short_communication
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.psjd-505ac210-98e4-4b61-a2fa-2beebb41a07c
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.