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

This article is cited in 8 scientific papers (total in 8 papers)

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 $N^{-1}$.


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

Bibliographic databases:

UDC: 519.2
Received: 21.04.1997
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
\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
\jour Discrete Math. Appl.
\yr 1998
\vol 8
\issue 5
\pages 533--544

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    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    3. Kolchin A.V., “Random partitions of a set and the generalised allocation scheme”, Probabilistic Methods in Discrete Mathematics, 2002, 215–218  isi
    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  mathnet  crossref  crossref  mathscinet  zmath
    5. Mezo I., “New properties of r-Stirling series”, Acta Mathematica Hungarica, 119:4 (2008), 341–358  crossref  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  elib  scopus
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