Related papers: Uniform asymptotics for the incomplete gamma funct…

In this work we study complete asymptotic expansions for the q-series $\sum_{n=1}^{\infty}\frac{1}{n^{b}}q^{n^{a}}$ and $\sum_{n=1}^{\infty}\frac{\sigma_{\alpha}(n)}{n^{b}}q^{n^{a}}$ in the scale function $(\log q)^{n}$ as $q\to1^{-}$,…

We study the resurgence properties of the coefficients $C_n(\tau)$ appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of $C_n(\tau)$ as…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $\Lambda \subseteq \mathbb{R}^d$ inside the $d$-sphere of radius $R$. In contrast to…

An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…

Considered herein are the family of nonlinear equations with both dispersive and dissipative homogeneous terms appended. Solutions of these equations that start with finite energia decay to zero as time goes to infinity. We present an…

When are asymptotic approximations using the delta-method uniformly valid? We provide sufficient conditions as well as closely related necessary conditions for uniform negligibility of the remainder of such approximations. These conditions…

Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series,…

Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of…

We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel function \[{}_0\Psi_1(z)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(an+b) n!},\] where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the…

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations.…

Using a recently derived integral in terms of elementary functions, we derive new asymptotic expansions of the normal inverse Gaussian cumulative distribution function. One of the asymptotic representations is in terms of the normal…

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…

We will use a discrete analogue of the classical \emph{Laplace} method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansions of scaled confluent basic hypergeometric functions, including the…

Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These…

Our main aim is to apply the theory of regularly varying functions to the asymptotical analysis at infinity of solutions of Friedmann cosmological equations. A new constant $\Gamma$ is introduced related to the Friedmann cosmological…

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…

In Slater's 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large…

In this work we construct uniform asymptotic expansion of $\sn(t|m)$ - Jacobi when $m\to1-0$. The constructed expansion is valid over more than a half of period. The turning point is included into the interval of validity for the…