Related papers: A survey on recent extensions of the Stirling form…
We present the history and previous approaches to the proof of Stirling's series. We use a different procedure, based on the asymptotic analysis of the difference equation $\Gamma(z+1)=z\Gamma(z)$. The method reproduces Stirling's series…
This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations…
We use the Jacobi-Trudi formula to execute "explicit" evaluation of determinants of Stirling numbers of both kinds. We also offer a Maple package accompanying the paper on the personal websites at the end of the second page.
The Stirling approximation formula for $n!$ dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and…
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…
For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling…
In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $\displaystyle n! = \sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by…
A concise and elementary derivation of the complete asymptotic expansion for the factorial function $n!$ is presented. This treatment produces a new expression for the coefficients, and it brings to light the simple relationship between the…
We study the Cowling approximation by analytical means as applied to a system of linear differential equations arising from models of non-radial stellar pulsation. We consider various asymptotic cases, including those of high harmonic…
We show how the asymptotic expansion for the gamma function $\Gamma(x)$, similar to that obtained by Boyd [Proc. Roy. Soc. London A447 (1994) 609--630], can be obtained by using a form of Lagrange's inversion theorem with a remainder. A…
Transcription into modern notations of the derivation by Stirling and De Moivre of an asymptotic series for $\log(n!)$, usually called Stirling's series. The previous discovery by Wallis of an infinite product for $\pi$, and later results…
We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval.…
In the paper we present some new inversion formulas and two new formulas for Stirling numbers.
We adapt the techniques in Stigler [Ann. Statist. 1 (1973) 472--477] to obtain a new, general asymptotic result for trimmed $U$-statistics via the generalized $L$-statistic representation introduced by Serfling [Ann. Statist. 12 (1984)…
We re-examine the exponentially improved expansion for $\log\,\g(z)$, first considered in Paris and Wood in 1991, to point out that the recent treatment by Kowalenko [Exactification of Stirling's approximation for the logarithm of the gamma…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
We provide completely effective error estimates for Stirling numbers of the first and second kind, denoted by $s(n,m)$ and $S(n,m)$, respectively. These bounds are useful for values of $m \geq n - O(\sqrt{n})$. An application of our Theorem…
In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of…
Stirling numbers of the first kind are used in the derivation of several population genetics statistics, which in turn are useful for testing evolutionary hypotheses directly from DNA sequences. Here, we explore the cumulative distribution…
The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(\alpha,\beta)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian…