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R. B. Paris

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

At least 19 recordsLinked to original sources
arXiv2021

Uniform asymptotics of a Gauss hypergeometric function with two large parameters, V

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+ελ,b;c+λ;x),\qquad 0 1$ and the constants $a$, $b$ and $c$ are supposed finite. Use of a standard integral representation shows that the problem reduces to consideration of a simple saddle point near an endpoint of the integration path. A uniform asymptotic expansion is first obtained by employing Bleistein's method. An alternative form of uniform expansion is derived following the approach described in Olver's book [{\it Asymptotics and Special Functions}, p.~346]. This second form has several advantages over the Bleistein form.

math.CA↗
arXiv2020

On some Féjer-type trigonometric sums

We examine the four Féjer-type trigonometric sums of the form \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0 0$ in $0<x<π$. The graph of the sum in this case presents a jump in the neighbourhood of $x=2π/3$. This jump is explained and is quantitatively estimated when $n\to\infty$.

math.NT↗
arXiv2020

A note on an extension of Gelfond's constant

The aim of this note is to provide a natural extension of Gelfond's constant $e^π$ using a hypergeometric function approach. An extension is also found for the square root of this constant. A few interesting special cases are presented.

math.CA↗
arXiv2020

On a new result for the hypergeometric function

The aim of this note is to provide a new identity connected with the Gauss hypergeometric function. This is achieved using results of certain combinatorial identities and a hypergeometric function approach.

math.CA↗
arXiv2020

Extensions of beta and related functions

In this paper, we introduce and investigate a new extension of the beta function by means of an integral operator involving a product of Bessel-Struve kernel functions. We also define a new extension of the well-known beta distribution, the Gauss hypergeometric function and the confluent hypergeometric function in terms of our extended beta function. In addition, some useful properties of these extended functions are also indicated in a systematic way.

math.CA↗
arXiv2020

Generalized Beta-type integral operators

This research note deals with the evaluation of some generalized beta-type integral operators involving the multi-index Mittag-Leffler function $E_{ε_{i}),(ω_{i})}(z)$. Further, we derive a new family of beta-type integrals involving the product of a multi-index Mittag-Leffler function and a generating function of two variables. Some concluding remarks regarding our present investigation are briefly discussed in the last section.

math.CA↗
arXiv2020

Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions

We consider two integrals over $x\in [0,1]$ involving products of the function $ζ_1(a,x)\equiv ζ(a,x)-x^{-a}$, where $ζ(a,x)$ is the Hurwitz zeta function, given by $$\int_0^1ζ_1(a,x)ζ_1(b,x)\,dx\quad\mbox{and}\quad \int_0^1ζ_1(a,x)ζ_1(b,1-x)\,dx$$ when $\Re (a,b)>1$. These integrals have been investigated recently in \cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $ζ(x)$ and two free parameters $a$ and $b$. The limiting forms of these sums when $a+b$ takes on integer values are considered.

math.CA↗
arXiv2020

Refined asymptotics of the Riemann-Siegel theta function

The Riemann-Siegel theta function $\vartheta(t)$ is examined for $t\to+\infty$. Use of the refined asymptotic expansion for $\log\,\g(z)$ shows that the expansion of $\vartheta(t)$ contains an infinite sequence of increasingly subdominant exponential terms, each multiplied by an asymptotic series involving inverse powers of $πt$. Numerical examples are given to detect and confirm the presence of the first three of these exponentials.

math.CA↗
arXiv2019

An asymptotic expansion for a sum of modified Bessel functions with quadratic argument

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \[S_ν(a)=\sum_{n\geq 1} (\frac{1}{2} an^2)^{-ν} K_ν(an^2)\qquad (|\arg\,a|<π/2)\] as the parameter $|a|\to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $a\to 0$ through positive values. Numerical results are presented to support the asymptotic theory.

math.CA↗
arXiv2017

A Feynman integral in Lifshitz-point and Lorentz-violating theories in $R^D\oplus R^m$

We evaluate a one-loop, two-point, massless Feynman integral $I_{D,m}(p,q)$ relevant for perturbative field theoretic calculations in strongly anisotropic $d=D+m$ dimensional spaces given by the direct sum $\mathbb R^D\oplus\mathbb R^m$. Our results are valid in the whole convergence region of the integral for generic (non-integer) co-dimensions $D$ and $m$. We obtain series expansions of $I_{D,m}(p,q)$ in terms of powers of the variable $X:=4p^2/q^4$, where $p=|\bm p|$, $q=|\bm q|$, $\bm p\in\mathbb R^D$, $\bm q\in\mathbb R^m$, and in terms of generalised hypergeometric functions $_3F_2(-X)$, when $X<1$. These are subsequently analytically continued to the complementary region $X\ge 1$. The asymptotic expansion in inverse powers of $X^{1/2}$ is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

hep-th↗
arXiv2016

Asymptotics of a Gauss hypergeometric function with large parameters, III: Application to the Legendre functions of large imaginary order and real degree

We obtain the asymptotic expansion for the Gauss hypergeometric function \[F(a-λ,b+λ;c+iαλ;z)\] for $λ\rightarrow+\infty$ with $a$, $b$ and $c$ finite parameters by application of the method of steepest descents. The quantity $α$ is real, so that the denominatorial parameter is complex and $z$ is a finite complex variable restricted to lie in the sector $|\arg (1-z)|<π$. We concentrate on the particular case $a=0$, $b=c=1$, which is associated with the Legendre functions of real degree and imaginary order. The resulting expansions are of Poincaré type and hold in restricted domains of the $z$-plane. An expansion is given at the coalescence of two saddle points. Numerical results illustrating the accuracy of the different expansions are given.

math.CA↗
arXiv2016

Integrals of products of Hurwitz zeta functions and the Casimir effect in $ϕ^4$ field theories

We evaluate two integrals over $x\in [0,1]$ involving products of the function $ζ_1(a,x)\equiv ζ(a,x)-x^{-a}$ for $\Re (a)>1$, where $ζ(a,x)$ is the Hurwitz zeta function. The evaluation of these integrals for the particular case of integer $a\geq 2$ is also presented. As an application we calculate the $O(g)$ weak-coupling expansion coefficient $c_{1}(\varepsilon)$ of the Casimir energy for a film with Dirichlet-Dirichlet boundary conditions, first stated by Symanzik [Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1-44] in the framework of $gϕ^4_{4-\varepsilon}$ theory.

math.CA↗
arXiv2016

Some results associated with Bernoulli and Euler numbers with applications

In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\mbox{sech} t$ and $\coth t$. For example, we prove that for $t > 0$ and $N\in\mathbb{N}:=\{1, 2, \ldots\}$, \[\mbox{sech}\, t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t) \] with \[ R_N(t)=\frac{(-1)^{N}2t^{2N}}{π^{2N-1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+\frac{1}{2})^{2N-1}\Big(t^2+π^2(k+\frac{1}{2})^2\Big)}, \] and \[\mbox{sech}\, t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+Θ(t, N)\frac{E_{2N}}{(2N)!}t^{2N} \] with a suitable $0 < Θ(t, N) < 1$. Here $E_n$ are the Euler numbers. By using the obtained results, we deduce some inequalities and completely monotonic functions associated with the ratio of gamma functions. Furthermore, we give a (presumably new) quadratic recurrence relation for the Bernoulli numbers.

math.CA↗
arXiv2015

On the asymptotic expansions of products related to the Wallis, Weierstrass and Wilf formulas

For all integers $n\geq1$, let \begin{align*} W_n(p,q)=\prod_{j=1}^{n}\left\{e^{-p/j}\left(1+\frac{p}{j}+\frac{q}{j^2}\right)\right\} \end{align*} and \begin{align*} R_n(p, q)=\prod_{j=1}^{n}\left\{e^{-p/(2j-1)}\left(1+\frac{p}{2j-1}+\frac{q}{(2j-1)^2}\right)\right\}, \end{align*} where $p$, $q$ are complex parameters. The infinite product $W_{\infty}(p,q)$ includes the Wallis and Wilf formulas, and also the infinite product definition of Weierstrass for the gamma function, as special cases. In this paper, we present asymptotic expansions of $W_n(p,q)$ and $R_n(p, q)$ as $n\to\infty$. In addition, we also establish asymptotic expansions for the Wallis sequence.

math.CA↗
arXiv2015

The asymptotics of the Struve function ${\bf H}_ν(z)$ for large complex order and argument

We re-examine the asymptotic expansion of the Struve function ${\bf H}_ν(z)$ for large complex values of $ν$ and $z$ satisfying $|\arg\,ν|\leqπ/2$ and $|\arg\,z|<π/2$. Watson's analysis covers only the case of $ν$ and $z$ of the same phase with $ν/z$ in the intervals $(0,1)$ and $(1,\infty)$. The domains in the complex $ν/z$-plane where the expansion takes on different forms are obtained.

math.CA↗
arXiv2015

A note on a modified Bessel function integral

We investigate the integral \[\int_0^\infty \cosh^μ\!t\,K_ν(z\cosh t)\,dt \qquad \Re(z)>0,\] where $K$ denotes the modified Bessel function, for non-negative integer values of the parameters $μ$ and $ν$. When the integers are of different parity, closed-form expressions are obtained in terms of $z^{-1}e^{-z}$ multiplied by a polynomial in $z^{-1}$ of degree dependent on the sign of $μ-ν$.

math.CA↗