Related papers: Uniform asymptotics for the incomplete gamma funct…

We consider the asymptotic expansion of the Wright function \[W_{\lambda,\mu}(z)=\sum_{n=0}^\infty\frac{z^n}{n! \Gamma(\lambda n+\mu)}\qquad (\lambda>-1)\] for large (positive and negative) variable and large parameter $\mu$. The analysis…

We consider a function $G(\lambda, z)$, entire in $\lambda$, which interpolates the derivatives of the Gamma function in the sense that $G(m, z) = \Gamma^{(m)}(z)$ for any integer $m \geq 0$ and we calculate the asymptotics of $G(\lambda,…

We consider differences between $\log \Gamma(x)$ and truncations of certain classical asymptotic expansions in inverse powers of $x-\lambda$ whose coefficients are expressed in terms of Bernoulli polynomials $B_n(\lambda)$, and we obtain…

We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)\] when the parameters $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector…

We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$ in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that many properties which $\Ga(z)$ enjoys extend in a…

We consider the asymptotic expansion of the single-parameter Mittag-Leffler function $E_a(-x)$ for $x\to+\infty$ as the parameter $a\to1$. The dominant expansion when $0<a<1$ consists of an algebraic expansion of $O(x^{-1})$ (which vanishes…

We define an absolutely convergent series for the upper incomplete Gamma function $\Gamma(s,z)$ for $z\geq 1$ and $s\in \mathbb{C}$. We express this series using certain polynomials which we define using the Stirling numbers of the first…

An asymptotic expansion for a ratio of products of gamma functions is derived.

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma…

In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a resurgence representation for the gamma function, exploiting the reformulation of the method of steepest descents by M. Berry and C. Howls (Berry and Howls, Proc.…

Asymptotic expansions are obtained for contour integrals of the form \[ \int_a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right)q(t)dt, \] in which $z$ is a large real or complex parameter, $p(t)$, $q(t)$ and $r(t)$ are analytic functions…

Lower and upper bounds $B_a(x)$ on the incomplete gamma function $\Gamma(a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\underset{x\downarrow0}\sim\Gamma(a,x)$ and…

We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta(s)$, $s=\sigma+i t$, $0\leq \sigma \leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical…

We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function $h(u)$ as $u\to 0^+$ or $1^-$. This is focussed on important univariate distributions…

Bessel and modified Bessel functions of imaginary order $i\nu$ ($\nu >0$) are studied. Asymptotic expansions are derived as $\nu \to \infty$ that are uniformly valid in unbounded complex domains, with error bounds provided. Coupled with…

We express the $q$-Pochhammer symbol $(z;q)_\infty$ as an infinite product of gamma functions, analogously to how Narukawa expressed the elliptic gamma function as an infinite product of hyperbolic gamma functions. This identity is used to…

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and…

A generalised Struve function has recently been introduced by Ali, Mondal and Nisar [J. Korean Math. Soc. {\bf 54} (2017) 575--598] as \[(\frac{1}{2} z)^{\nu+1}\sum_{n=0}^\infty\frac{(\frac{1}{2} z)^{2n}}{\Gamma(n+\frac{3}{2})…

Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic…

We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…