Related papers: Asymptotics of multivariate sequences, part III: q…
We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…
We compute the integrals of products of Hermite functions using the generating functions. The precise asymptotics of 4 Hermite functions are presented below. This estimate is relevant for the corresponding cubic nonlinear equation.
The right tail asymptotic series consisting of attenuating exponential terms are derived for the densities of Galton-Watson processes with fractional probability generating functions. The frequencies in the exponential factors form fractal…
The generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera…
The results of part I (hep-ph/9612284) are used to obtain full asymptotic expansions of Feynman diagrams renormalized within the MS-scheme in the regimes when some of the masses and external momenta are large with respect to the others. The…
We show how the Riemann-Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptotic combinatorics and representation theory. The Whittaker kernel and the discrete…
Consider a non-negative sequence $c_n = h(n) \cdot n^{\alpha-1} \cdot \rho^{-n}$, where $h$ is slowly varying, $\alpha>0$, $0<\rho<1$ and $n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) = \prod_{k\ge1}(1-x^ky)^{-c_k}$, which is…
This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…
Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…
The problem of asymptotic expansions of Green functions in perturbative QFT is studied for the class of Euclidean asymptotic regimes. Phenomenological applications are analyzed to obtain a meaningful mathematical formulation of the problem.…
We define the asymptotic behavior "almost everywhere" of additive and multiplicative arithmetic functions in the paper. Classes of additive and multiplicative arithmetic functions are singled out for which the asymptotics coincides "almost…
The system of equations for parametric sub-resonant growth of the amplitude of oscillations was obtained. The time of turning point from the growing of the amplitude to the bounded oscillations in the slow variable was found. The comparison…
Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…
We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.
We study the questions of determining the asymptotics of the probabilistic characteristics of additive arithmetic functions in the paper, regardless of whether they have a limit distribution or not. Several assertions are proved about the…
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves…
We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…
We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case…
In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.
We consider the $q$th root number function for the symmetric group. Our aim is to develop an asymptotic formula for the multiplicities of the $q$th root number function as $q$ tends to $\infty$. We use character theory, number theory and…