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Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument…

We prove asymptotics for Serre's problem on the number of diagonal planar conics with a rational point and use this to put forward a new conjecture on counting the number of varieties in a family which are everywhere locally soluble.

For a family of second-order parabolic systems with rapidly oscillating and time-dependent periodic coefficients, we investigate the asymptotic behavior of fundamental solutions and establish sharp estimates for the remainders.

This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…

We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and…

In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general…

Asymptotic expansion is constructed and justified for the solution to a nonuniform Neumann boundary-value problem for the Poisson equation with the right-hand side that depends both on longitudinal and transversal variables in a thin…

Necessary and sufficient conditions for the solvability of boundary value problems for a family of functional differential equations with a non-integrable singularity are obtained.

Summation arithmetic functions with asymptotically independent terms are studied in the paper, the limit of which is the law of normal distribution. Assertions about the asymptotic behavior of the indicated functions are proved.

We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization…

We present an asymptotic expansion for a family of multiple integrals connected with relatives of the Dickman function. The coefficients of this expansion have a similar arithmetic structure as those appearing in work of Soundararajan on an…

This paper reports on a new algorithm to compute the asymptotic solutions of a linear differential system. A feature of the algorithm is the ability to accommodate periodic coefficients.

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums…

For homogeneous difference equation of the second order we study the analogy of Hartman-Wintner problem on asymptotic integration of fundamental system of solutions as argument tends to infinity.

In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic…

We provide an elementary proof of the asymptotic behavior of solutions of second order differential equations.

In this paper, we establish the existence of large solutions of Hessian equations and obtain a new boundary asymptotic behavior of solutions.

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour…