R. Paris

On the $ν$ -zeros of the Bessel functions of purely imaginary order

R B Paris

The $ν$ -zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x > 0$ . In the case of the modified Bessel function of the second kind $K_{i ν} (x)$ , i…

An asymptotic approximation for the Riemann zeta function revisited

R B Paris

We revisit a representation for the Riemann zeta function $ζ (s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal.…

The asymptotic expansion of the Humbert hyper-Bessel function

R B Paris

We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a $_{0} F_{2}$ hypergeometric function by \[J_{m,n}(x)=\frac{(x/3)^{m+n}}{m! n!}\,{}_0F…

An extension of an asymptotic result of Tricomi concerning a definite integral

R B Paris

We consider the expansion of an integral considered by F.G. Tricomi given by \[\int_{-\infty}^\infty x e^{-x^2}(\frac{1}{2}+\frac{1}{2}\mbox{erf}\,x)^{m} dx\] as $m \to \infty$ . The…

The asymptotic expansion of Kratzel's integral and an integral related to an extension of the Whittaker function

R B Paris

We consider the asymptotic expansion of Krätzel's integral \[F_{p,ν}(x)=\int_0^\infty t^{ν-1} e^{-t^p-x/t}\,dt\qquad (|\arg\,x|<π/2),\] for $p > 0$ as $∣ x ∣ \to \infty$ in the sector $…

The asymptotic expansion of a Mathieu-exponential series

R B Paris

We consider the asymptotic expansion of the functional series \[S_μ^\pm(a;λ)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-λn}}{(n^2+a^2)^μ}\] for $λ > 0$ and $μ \geq 0$ as $∣ a ∣ \to \infty$ in…

Asymptotic expansion of the Wright function for large variable and parameter

R B Paris

We consider the asymptotic expansion of the Wright function \[W_{λ,μ}(z)=\sum_{n=0}^\infty\frac{z^n}{n! Γ(λn+μ)}\qquad (λ>-1)\] for large (positive and negative) variable and large…

The asymptotic expansion of the Bateman and Havelock functions of large order and argument

R B Paris

Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals \[\frac{2}π\int_0^{π/2} \!\!\!\begin{array}{c} \cos\\\sin\end{array}\!(x\tan u-νu…

math.CA2021★ 1 cited

On the $ν$ -zeros of the modified Bessel function $K_{i ν} (x)$ of positive argument

R B Paris

The modified Bessel function of the second kind $K_{i ν} (x)$ of imaginary order for fixed $x > 0$ possesses a countably infinite sequence of real zeros. Recently it has been shown tha…

The numerical evaluation of the Riesz function

R B Paris

The behaviour of the generalised Riesz function defined by \[S_{m,p}(x)=\sum_{k=0}^\infty \frac{(-)^{k-1}x^k}{k! ζ(mk+p)}\qquad (m\geq 1,\ p\geq 1)\] is considered for large positi…

The asymptotic expansion of a sum appearing in an approximate functional equation for the riemann zeta function

R B Paris

A representation for the Riemann zeta function valid for arbitrary complex $s = σ + i t$ is $ζ (s) = \sum_{n = 0}^{\infty} A (n, s)$ , where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left…

The asymptotic expansion of the Bernoulli polynomials of the second kind

R B Paris

We consider the Bernoulli polynomials of the second kind, which can be related to the generalised Bernoulli polynomials $B_{n}^{(n)} (z)$ . The asymptotic expansions of the scaled poly…

The asymptotic expansion of a function due to L.L. Karasheva

R B Paris

We consider the asymptotic expansion for $x \to \pm \infty$ of the entire function \[F_{n,σ}(x;μ)=\sum_{k=0}^\infty \frac{\sin\,(nγ_k)}{\sin γ_k}\,\frac{x^k}{k! Γ(μ-σk)},\quad γ_k=\fr…

math.CA2021★ 7 cited

On the asymptotic of Wright functions of the second kind

Richard Paris, Armando Consiglio, Francesco Mainardi

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book {\it Fractional Calculus and Waves in Linear Viscoelasticit…

Asymptotics of some integrals involving modified Bessel and hyper-Bessel functions

R B Paris

We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{Γ(1+ν)|J_ν(x)|}{(x/2)^ν}\right)^{\!n}dx\] for large $n$ in which the B…

math.CA2021★ 1 cited

Asymptotics of a Mathieu-Gaussian series

R B Paris

We consider the asymptotic expansion of the functional series \[S_{μ,γ}(a;λ)=\sum_{n=1}^\infty \frac{n^γe^{-λn^2/a^2}}{(n^2+a^2)^μ}\] for real values of the parameters $γ$ , $λ > 0$ a…

math.CA2020★ 2 cited

Asymptotics of some generalised sine-integrals

R B Paris

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.…

A note on an extension of Gelfond's constant

A. K. Rathie, R. B. Paris

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…

A note on a generalization of two well-known Cominatorial identities via a Hypergeometric series approach

Arjun K. Rathie, Insuk Kim, Richard B. Paris

In this note, we aim to provide generalizations of (i) Knuth's old sum (or Reed Dawson identity) and (ii) Riordan's identity using a hypergeometric series approach.

On a new result for the hypergeometric function

A. K. Rathie, R. B. Paris

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 hyperge…

Extensions of beta and related functions

M. Ghayasuddin, M. Ali, R. B. Paris

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 def…

Generalization of Szász operators involving multiple Sheffer polynomials

Mahvish Ali, Richard B. Paris

The present work deals with the mathematical investigation of some generalizations of the Szász operators. In this work, the multiple Sheffer polynomials are introduced. The genera…