Abstract
Factorial transformation known from Euler’s time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Padé approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. K. Lando, Lectures on Generating Functions (Am. Math. Soc., Providence, R.I., 2003; MKhAMO, Moscow, 2004).
T. Arakawa, T. Ibukiyama, and M. Kaneko, Bernoulli Numbers and Zeta Functions (Springer, Japan, 2014).
J. Frame, “The Hankel power sum matrix inverse and the Bernoulli continued fraction,” Math. Comput. 33 (146), 815–826 (1979).
L. Euler, “De seriebus divergentibus,” Novi Comment. Acad. Sci. Petropolitanae 5, 205–237 (1754/1755). = Opera Omnia, Ser. I, Teubner, Leipzig 14, 585–617 (1925).
N. Nielsen, Die Gammafunktion (Teubner, Leipzig, 1906).
E. J. Weniger, “Summation of divergent power series by means of factorial series” (2010). http:// arxiv.org/abs/1005.0466v1.
K. Knopp, Theory and Applications of Infinite Series (Blackie & Son, London, 1946).
Sloane Online Encyclopedia of Integer Sequences. http://oeis.org.
S. Khrushchev, Orthogonal Polynomials and Continued Fractions (Cambridge Univ. Press, Cambridge, 2008).
R. Apéry, “Irrationalité de \(\zeta (2)\) et \(\zeta (3)\),” Astérisque 61, 11–13 (1979).
S. R. Finch, Mathematical Constants (Cambridge Univ. Press, Cambridge, 2003).
B. Candelpergher, Ramanujan Summation of Divergent Series (Springer, Berlin, 2017).
G. N. Watson, “The transformation of an asymptotic series into a convergent series of inverse factorials,” Rend. Circ. Mat. Palermo 34, 41–88 (1912).
W. Petkovšek, H. S. Wilf, and D. Zeilberger, A = B (Taylor and Francis, London, 1996).
M. Bernstein and N. J. A. Sloane, “Some canonical sequences of integers,” Linear Algebra Appl. 226–228, 57–72 (1995).
G. H. Hardy, Divergent Series (Chelsea, New York, 1992).
J. Glimm and A. Jaffe, Quantum Physics, 2nd ed. (Springer, Berlin, 1987).
C. Bender and C. Heissenberg, “Convergent and divergent series in physics” (2016). https:// arxiv.org/abs/1703.05164v2.
Author information
Authors and Affiliations
Corresponding author
Additional information
The article is published in the original.
Rights and permissions
About this article
Cite this article
Varin, V.P. Factorial Transformation for Some Classical Combinatorial Sequences. Comput. Math. and Math. Phys. 58, 1687–1707 (2018). https://doi.org/10.1134/S0965542518110155
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542518110155