arXiv subjects

R. B. Paris

Publications and source records attributed to R. B. Paris.

At least 37 records · Page 2Linked to original sources

arXiv2015

An asymptotic expansion for the Stieltjes constants

The Stieltjes constants $γ_n$ appear in the coefficients in the Laurent expansion of the Riemann zeta function $ζ(s)$ about the simple pole $s=1$. We present an asymptotic expansion for $γ_n$ as $n\rightarrow \infty$ based on the approach described by Knessel and Coffey [Math. Comput. {\bf 80} (2011) 379--386]. A truncated form of this expansion with explicit coefficients is also given. Numerical results are presented that illustrate the accuracy achievable with our expansion.

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arXiv2015

Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function \[{}_1Ψ_1(ρ,k; ρ,0;x)= \sum_{n=0}^\infty\frac{Γ(k+ρn)}{Γ(ρn)}\,\frac{x^n}{n!}\qquad (|x|<\infty)\] when the parameter $ρ\in (-1,0)\cup (0,\infty)$ and the argument $x$ is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter $k$ is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of $ρ$. The asymptotics of ${}_1Ψ_1(ρ,k;ρ,0;x)$ are obtained under numerous assumptions on the behavior of the arguments $k$ and $x$ when the parameter $ρ$ is both positive and negative. We also provide some integral representations and structural properties involving the `reduced' Wright function ${}_0Ψ_1(-\!\!\!-; ρ,0;x)$ with $ρ\in (-1,0)\cup (0,\infty)$, which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions ${}_0Ψ_1(-\!\!\!-; \pmρ, 0;\cdot)$ and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

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arXiv2015

A note on an integral associated with the Kelvin ship-wave pattern

The velocity potential in the Kelvin ship-wave source can be partly expressed in terms of space derivatives of the single integral \[F(x,ρ,α)=\int_{-\infty}^\infty \exp\,[-\frac{1}{2}ρ\cosh (2u-iα)] \cos (x\cosh u)\,du,\] where $(x, ρ, α)$ are cylindrical polar coordinates with origin based at the source and $-π/2\leqα\leqπ/2$. An asymptotic expansion of $F(x,ρ,α)$ when $x$ and $ρ$ are small, but such that $M\equiv x^2/(4ρ)$ is large, was given using a non-rigorous approach by Bessho in 1964 as a sum involving products of Bessel functions. This expansion, together with an additional integral term, was subsequently proved by Ursell in 1988. Our aim here is to present an alternative asymptotic procedure for the case of large $M$. The resulting expansion consists of three distinct parts: a convergent sum involving the Struve functions, an asymptotic series and an exponentially small saddle-point contribution. Numerical computations are carried out to verify the accuracy of our expansion.

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arXiv2015

A note on a class of exact solutions for a doubly anharmonic oscillator

We examine a class of exact solutions for the eigenvalues and eigenfunctions of a doubly anharmonic oscillator defined by the potential $V(x)=ω^2/2 x^2+λx^4/4+ηx^6/6$, $η>0$. These solutions hold provided certain constraints on the coupling parameters $ω^2$, $λ$ and $η$ are satisfied.

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arXiv2015

On an extension of Watson's lemma due to Ursell

In 1991, Ursell gave a strong form of Watson's lemma for the Laplace integral \[\int_0^\infty e^{-xt}f(t)\,dt\qquad (x\rightarrow+\infty) \] in which the amplitude function $f(t)$ is regular at the origin and possesses a Maclaurin expansion valid in $0\leq t\leq R$. He showed that if the asymptotic series for the integral as $x\rightarrow+\infty$ is truncated after $rx$ terms, where $0<r<R$, then the resulting remainder is exponentially small of order $O(e^{-rx})$. In this note we extend this result to include situations when $f(t)$ has a branch point at $t=0$ and when $x$ is a complex variable satisfying $|\arg\,x|<π/2$.

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arXiv2015

The asymptotic expansion of a generalisation of the Euler-Jacobi series

We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty n^{-w}\e^{-an^p}\] as $a\rightarrow 0$ in $|\arg\,a|<π/2$ for arbitrary finite $p>$ and $w>0$. Our attention is concentrated mainly on the case when $p$ and $w$ are both even integers, where the expansion consists of a {\it finite} algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for $S_p(a;w)$ analogous to the well-known Poisson-Jacobi transformation for the sum with $p=2$ and $w=0$. Numerical results are given to illustrate the accuracy of the expansion obtained.

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arXiv2015

The asymptotics of a generalised Beta function

We consider the generalised Beta function introduced by Chaudhry {\it et al.\/} [J. Comp. Appl. Math. {\bf 78} (1997) 19--32] defined by \[B(x,y;p)=\int_0^1 t^{x-1} (1-t)^{y-1} \exp \left[\frac{-p}{4t(1-t)}\right]\,dt,\] where $\Re (p)>0$ and the parameters $x$ and $y$ are arbitrary complex numbers. The asymptotic behaviour of $B(x,y;p)$ is obtained when (i) $p$ large, with $x$ and $y$ fixed, (ii) $x$ and $p$ large, (iii) $x$, $y$ and $p$ large and (iv) either $x$ or $y$ large, with $p$ finite. Numerical results are given to illustrate the accuracy of the formulas obtained.

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arXiv2015

On an extension of extended Beta and hypergeometric functions

Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically investigate several properties of each of these extended functions, namely their various integral representations, Mellin transforms, derivatives, transformations, summation formulas and asymptotics. Relevant connections of certain special cases of the main results presented here are also pointed out.

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arXiv2015

On the asymptotics of a sequence of lacunary binomial-type polynomials

We examine the asymptotics of a sequence of lacunary binomial-type polynomials $\wp_n(z)$ as $n\rightarrow\infty$ that have arisen in the problem of the expected number of independent sets of vertices of finite simple graphs. We extend the recent analysis of Gawronski and Neuschel by employing the method of steepest descents applied to an integral representation. The case of complex $z$ with $|z|<1$ is also considered. Numerical results are presented to illustrate the accuracy of the resulting expansions.

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arXiv2015

A derivation of two transformation formulas contiguous to that of Kummer's second theorem via a differential equation approach

The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummer's second transformation for the confluent hypergeometric function ${}_1F_1$ using a differential equation approach.

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arXiv2015

A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer $m$

We obtain an asymptotic expansion for the sum \[S(a;w)=\sum_{n=1}^\infty \frac{e^{-an^2}}{n^{w}}\] as $a\rightarrow 0$ in $|\arg\,a|<π/2$ for arbitrary finite $w>0$. The result when $w=2m$, where $m$ is a positive integer, is the analogue of the well-known Poisson-Jacobi transformation for the sum with $m=0$. Numerical results are given to illustrate the accuracy of the expansion.

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arXiv2014

A note on a hypergeometric transformation formula due to Slater with an application

In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$ series with one pair of parameters differing by unity expressed as a linear combination of two ${}_3F_2(1)$ series.

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arXiv2014

The expansion of a finite number of terms of the Gauss hypergeometric function of unit argument and the Landau constants

We obtain convergent inverse factorial expansions for the sum $S_n(a,b;c)$ of the first $n$ terms of the Gauss hypergeometric function of unit argument valid for $n\geq 1$. The form of these expansions depends on the location of the parametric excess $s:=c-a-b$ in the complex $s$-plane. The leading behaviour as $n\rightarrow\infty$ agrees with previous results in the literature. The case $a=b=1/2$, $c=1$ corresponds to the Landau contants.

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arXiv2014

On the asymptotic expansion of $Γ(x)$, Lagrange's inversion theorem and the Stirling coefficients

We show how the asymptotic expansion for the gamma function $Γ(x)$, 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 (possibly) new closed-form representation for the Stirling coefficients is given.

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arXiv2014

On extensions of two results due to Ramanujan

The aim in this note is to provide a generalization of an interesting entry in Ramanujan's Notebooks that relate sums involving the derivatives of a function Phi(t) evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating 3F2 hypergeometric functions of argument equal to 2, recently obtained in Kim et al. [Two results for the terminating 3F2(2) with applications, Bull. Korean Math. Soc. 49 (2012) pp. 621{633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences.

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arXiv2013

An alternative proof of the extended Saalschutz summation theorem for the r+3Fr+2(1) series with applications

A simple proof of a new summation formula for a terminating r+3Fr+2(1) hypergeometric series, representing an extension of Saalschutz's formula for a 3F2(1) series, is given for the case of r pairs of numeratorial and denominatorial parameters differing by positive integers. Two applications of this extended summation theorem are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde-Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kamp?e de F?eriet function of two variables given by Exton and Cvijovi?c and Miller.

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arXiv2013

A note on a series containing the Laguerre polynomials

Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently obtained in the literature.

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arXiv2013

On two Thomae-type transformations for hypergeometric series with integral parameter differences

We obtain two new Thomae-type transformations for hypergeometric series with r pairs of numeratorial and denominatorial parameters differing by positive integers. This is achieved by application of the so-called Beta integral method developed by Krattenthaler and Rao [Symposium on Symmetries in Science (ed. B. Gruber), Kluwer (2004)] to two recently obtained Euler-type transformations. Some special cases are given.

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