相关论文: The asymptotic expansion for the factorial and Lag…

Many asymptotic formulas exist for unrestricted integer partitions as well as for distinct partitions of integers into a finite number of parts. Szekeres and Canfield have derived an asymptotic formula for the number of partitions that is…

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…

We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L-functions over rational function fields. Our prediction is motivated by two natural conjectures that provide…

We introduce a new sequence of unsigned degenerate Stirling numbers of the first kind. Following the work of Adell-Lekuona, who represented unsigned Stirling numbers of the first kind as multiples of the expectations of specific random…

This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…

We provide a combinatorial derivation of an asymptotic formula for averages of mixed ratios of characteristic polynomials over the unitary group, where mixed ratios are products of ratios and/or logarithmic derivatives. Our proof of this…

In this paper, we discuss asymptotic relations for the approximation of $\left\vert x\right\vert ^{\alpha},\alpha>0$ in $L_{\infty}\left[ -1,1\right] $ by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of…

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion…

Lieanders are special cases of meanders and first appeared in connection with Lie algebras. Using the results from the author with E. Goujard, P. Zograf and A. Zorich, we prove a polynomial asymptotics for the number of lieanders with fixed…

We give a new combinatorial proof of Parraud's formula for the asymptotic expansion in powers of $1/N^2$ for the expected trace of polynomials of several independent $N \times N$ GUE matrices, which expresses the result using a mixture of…

In this note, we consider asymptotic products of binomial and multinomial coefficients and determine their asymptotic constants and formulas. Among them, special cases are the central binomial coefficients, the related Catalan numbers, and…

We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the…

Through a brute-force approach to calculating the higher derivatives of the falling factorial function, a number of interesting quantities were obtained and analyzed. In particular, it was found that a quantity that can be described as the…

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author…

Purpose of writing this paper is to solve a transcendental function containing a product of a variable and its double exponential by a unique method of approximation. If the value of the said product is given, then its inverse function is…

The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised,…

We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\infty$-Laplacian $\Delta_\infty^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in…

The aim of this paper consists of providing summation formulas for the $k$-Fibonacci numbers ($k \in \mathbb{Z}$, $k \geq 2$) and their asymptotic equivalents in terms of generalized binomial coefficients. Our main tools are the Lagrange…

In the paper, with the aid of the Fa\`a di Bruno formula, in terms of central factorial numbers of the second kind, and with the terminology of the Stirling numbers of the second kind, the authors derive several series expansions for any…