RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Diskr. Mat.: Year: Volume: Issue: Page: Find

 Diskr. Mat., 1998, Volume 10, Issue 3, Pages 148–159 (Mi dm440)

On asymptotic expansions of Stirling numbers of the first and second kinds

A. N. Timashev

Abstract: We consider the problem of asymptotic estimation of Stirling numbers $s(n,N)$ of the first kind and Stirling numbers $\sigma(n,N)$ of the second kind under the condition that $n,N\to\infty$ so that
$$1<\alpha_0\le \alpha=\frac{n}{N}\le \alpha_1<\infty,$$
where $\alpha_0$, $\alpha_1$ are some constants. Under this condition, by making use of the saddle point method, we demonstrate that the coefficients of the negative powers of the form $N^{-m}$, $m=1,2,…$, in asymptotic expansions of the numbers $s(n,N)$ and $\sigma(n,N)$ in powers of $N^{-1}$ are determined from the representation in the form of a power series of a certain function that depends on the solution of a given non-linear differential equation of the first order with a given initial condition. These results allow us to show that these coefficients obey some linear recurrence relations in the complex plane. As corollaries, we give explicit formulas for the coefficient of $N^{-1}$.

DOI: https://doi.org/10.4213/dm440

Full text: PDF file (847 kB)

English version:
Discrete Mathematics and Applications, 1998, 8:5, 533–544

Bibliographic databases:

UDC: 519.2
Revised: 18.05.1998

Citation: A. N. Timashev, “On asymptotic expansions of Stirling numbers of the first and second kinds”, Diskr. Mat., 10:3 (1998), 148–159; Discrete Math. Appl., 8:5 (1998), 533–544

Citation in format AMSBIB
\Bibitem{Tim98} \by A.~N.~Timashev \paper On asymptotic expansions of Stirling numbers of the first and second kinds \jour Diskr. Mat. \yr 1998 \vol 10 \issue 3 \pages 148--159 \mathnet{http://mi.mathnet.ru/dm440} \crossref{https://doi.org/10.4213/dm440} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1673690} \zmath{https://zbmath.org/?q=an:0973.11022} \transl \jour Discrete Math. Appl. \yr 1998 \vol 8 \issue 5 \pages 533--544 

• http://mi.mathnet.ru/eng/dm440
• https://doi.org/10.4213/dm440
• http://mi.mathnet.ru/eng/dm/v10/i3/p148

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. N. Timashev, “On asymptotic expansions in local limit theorems for equiprobable schemes of allocating particles to distinguishable cells”, Discrete Math. Appl., 10:1 (2000), 63–73
2. A. N. Timashev, “On the distribution of the number of cycles of a given length in the class of permutations with known number of cycles”, Discrete Math. Appl., 11:5 (2001), 471–483
3. Kolchin A.V., “Random partitions of a set and the generalised allocation scheme”, Probabilistic Methods in Discrete Mathematics, 2002, 215–218
4. A. N. Timashev, “On asymptotic expansions for the distribution of the number of cycles in a random permutation”, Discrete Math. Appl., 13:5 (2003), 417–427
5. Mezo I., “New properties of r-Stirling series”, Acta Mathematica Hungarica, 119:4 (2008), 341–358
6. Louchard G., “Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach”, Discrete Mathematics and Theoretical Computer Science, 12:2 (2010), 167–184
7. Blagouchine I.V., “Expansions of Generalized Euler'S Constants Into the Series of Polynomials in Pi(-2) and Into the Formal Enveloping Series With Rational Coefficients Only”, J. Number Theory, 158 (2016), 365–396
8. Blagouchine I.V., “Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1”, J. Math. Anal. Appl., 442:2 (2016), 404–434
•  Number of views: This page: 752 Full text: 195 First page: 2