Related papers: The asymptotic expansion of the Humbert hyper-Bess…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[F(a+\epsilon\lambda,m;c+\lambda;x),\qquad \lambda\to+\infty\] for $x<1$ and positive integer $m$ when the parameter $\epsilon>1$ and the constants $a$ and…

We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{\Gamma(1+\nu)|J_\nu(x)|}{(x/2)^\nu}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_\nu(x)$ is replaced by the…

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $\Phi_1[a,b;c;x, y]$. Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) $x\to\infty$; (ii)…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and…

Recently, Wald and Henkel (2018) derived the leading-order estimate of the Humbert functions $\Phi_2$, $\Phi_3$ and $\Xi_2$ for two large arguments, but their technique cannot handle the Humbert function $\Psi_1$. In this paper, we…

In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function $\Psi_1[a,b;c,c';x,y]$ when $x\to\infty$ and $y\to +\infty$ has been derived in a direct and simple manner. In this paper, we obtain…

We consider the asymptotic behaviour of the generalised hypergeometric function \[{}_3F_2\bl(\!\!\begin{array}{c} 1, (1+t)k/2, (1+t)k/2+1/2\\tk+1, k+1\end{array}\!\!; x\br),\qquad 0<x,t\leq 1\] as the parameter $k\to+\infty$. Numerical…

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found…

We introduce a version of the asymptotic expansions for Bessel functions $J_\nu(z)$, $Y_\nu(z)$ that is valid whenever $|z| > \nu$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the…

In this paper we study the following Bessel series $\sum _{l=1}^{\infty } {J_{l+m'}(r)J_{l+m}(r)}{(l+\beta)^\alpha}$ for any $m,m'\in\mathbb{Z}$, $\alpha\in\mathbb{R}$ and $\beta>-1$. They are a particular case of the second type Neumann…

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where…

We obtain the asymptotic expansion for the Gauss hypergeometric function \[F(a-\lambda,b+\lambda;c+i\alpha\lambda;z)\] for $\lambda\rightarrow+\infty$ with $a$, $b$ and $c$ finite parameters by application of the method of steepest…

A recent asymptotic expansion for the positive zeros $x=j_{\nu,m}$ ($m=1,2,3,\ldots$) of the Bessel function of the first kind $J_{\nu}(x)$ is studied, where the order $\nu$ is positive. Unlike previous well-known expansions in the…

We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed,…

Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals \[\frac{2}{\pi}\int_0^{\pi/2} \!\!\!\begin{array}{c} \cos\\\sin\end{array}\!(x\tan u-\nu u)\,du\] are obtained for large real $x$ and large…

We obtain the asymptotic expansion for large integer $n$ of a generalised sine-integral \[\int_0^\infty\left(\frac{\sin\,x}{x}\right)^{n}dx\] by utilising the saddle-point method. This expansion is shown to agree with recent results of J.…

The leading asymptotic behaviour of the Humbert functions $\Phi_2$, $\Phi_3$, $\Xi_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these…

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…

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 examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind…