A summation formula for a ${}_3F_2(1)$ hypergeometric series
A summation formula is derived for the hypergeometric series of unit argument ${}_3F_2(1,1,c;d,n+2;1)$, where $n=0, 1, 2, \ldots$ and $\Re (d-c+n)>0$.
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arXiv subjects
R B Paris
Publications and source records attributed to R B Paris.
On the properties of the $(p,ν)$-extension of the Whittaker function $M_{κ,μ}(z)$
In this paper, we obtain a $(p,ν)$-extension of the Whittaker function $M_{κ,μ}(z)$ by using the extended confluent hypergeometric function of the first kind $Φ_{p,ν}(b;c;z)$ introduced in Parmar et al. [J. Classical Anal. 11 (2017) 81--106]. Also, we derive some of the main properties of this function, namely several integral representations, a summation formula, the analogue of Kummer's transformation formula, an asymptotic representation, the Mellin transform, a differential formula and some inequalities.
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The expansion of the confluent hypergeometric function on the positive real axis
The asymptotic expansion of the Kummer function ${}_1F_1(a; b; z)$ is examined as $z\to+\infty$ on the Stokes line $\arg\,z=0$. The correct form of the subdominant algebraic contribution is obtained for non-integer $a$. Numerical results demonstrating the accuracy of the expansion are given.
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The asymptotics of a generalised Struve function
A generalised Struve function has recently been introduced by Ali, Mondal and Nisar [J. Korean Math. Soc. {\bf 54} (2017) 575--598] as \[(\frac{1}{2} z)^{ν+1}\sum_{n=0}^\infty\frac{(\frac{1}{2} z)^{2n}}{Γ(n+\frac{3}{2}) Γ(an+ν+\frac{3}{2})},\] where $a$ is a positive integer. In this paper, we obtain the asymptotic expansions of this function for large complex $z$ when $a$ is a real parameter satisfying $a>-1$. Some numerical examples are presented to confirm the accuracy of the expansions.
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A $(p,ν)$-extension of the Appell function $F_1(\cdot)$ and its properties
In this paper, we obtain a $(p,v)$-extension of the Appell hypergeometric function $ F_{1}(\cdot)$, together with its integral representation, by using the extended Beta function $B_{p,v}(x,y)$ introduced in arXiv:1502.06200. Also, we give some of its main properties, namely the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, some new integral representations of the extended Appell function$ F_{1,p,v}(\cdot)$ involving Meijer's $G$-function are obtained.
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A $(p,ν)$-extension of Srivastava's triple hypergeometric function and its properties
In this paper, we obtain a $(p,ν)$-extension of Srivastava's triple hypergeometric function $H_B(\cdot)$, together by using the extended Beta function $B_{p,ν}(x,y)$ introduced in arXiv:1502.06200. We give some of the main properties of this extended function, which include several integral representations involving Exton's hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.
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A note on the asymptotics of the modified Bessel functions on the Stokes lines
We employ the exponentially improved asymptotic expansions of the confluent hypergeometric functions on the Stokes lines discussed by the author [Appl. Math. Sci. {\bf 7} (2013) 6601--6609] to give the analogous expansions of the modified Bessel functions $I_ν(z)$ and $K_ν(z)$ for large $z$ and finite $ν$ on $\arg\,z=\pmπ$ (and, in the case of $I_ν(z)$, also on $\arg\,z=0$). Numerical results are presented to illustrate the accuracy of these expansions.
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Some remarks on the theorems of Wright and Braaksma on the Wright function ${}_pΨ_q(z)$
We carry out a numerical investigation of the asymptotic expansion of the so-called Wright function ${}_pΨ_q(z)$ (a generalised hypergeometric function) in the case when exponentially small terms are present. This situation is covered by two theorems of Wright and Braaksma. We demonstrate that a more precise understanding of the behaviour of ${}_pΨ_q(z)$ is obtained by taking into account the Stokes phenomenon.
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A uniform asymptotic expansion for the incomplete gamma functions revisited
A new uniform asymptotic expansion for the incomplete gamma function $Γ(a,z)$ valid for large values of $z$ was given by the author in {\it J. Comput. Appl. Math.} {\bf 148} (2002) 323--339. This expansion contains a complementary error function of an argument measuring transition across the point $z=a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $Γ(a,a)$.
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The asymptotics of the Touchard polynomials: a uniform approximation
The asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$, where $|z|$ may be finite or allowed to be large like $O(n)$, has been recently considered in \cite{P1}. When $z=-x$ is negative, it is found that there is a coalesence of two contributory saddle points when $n/x=1/e$. Here we determine the expansion when $n$ and $x$ satisfy this condition and also a uniform two-term approximation involving the Airy function in the neighbourhood of this value. Numerical results are given to illustrate the accuracy of the asymptotic approximations obtained.
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The asymptotics of the Touchard polynomials
We examine the asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$. In our treatment $|z|$ may be finite or allowed to be large like $O(n)$. We employ the method of steepest descents to a suitable integral representation of $T_n(z)$ and find that the number of saddle points that contribute to the expansion depends on the values of $n$ and $z$. Numerical results are given to illustrate the accuracy of the various expansions.
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On the Wilker and Huygens-type inequalities
Chen and Cheung [C.-P. Chen, W.-S. Cheung, Sharpness of Wilker and Huygens type inequalities, J. Inequal. Appl. 2012 (2012) 72, \url{http://dx.doi.org/10.1186/1029-242X-2012-72}] established sharp Wilker and Huygens-type inequalities. These authors also proposed three conjectures on Wilker and Huygens-type inequalities. In this paper, we consider these conjectures. We also present sharp Wilker and Huygens-type inequalities.
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The Stokes phenomenon and the Lerch zeta function
We examine the exponentially improved asymptotic expansion of the Lerch zeta function $L(λ,a,s)=\sum_{n=1}^\infty \exp (2πniλ)/(n+a)^s$ for large complex values of $a$, with $λ$ and $s$ regarded as parameters. It is shown that an infinite number of subdominant exponential terms switch on across the Stokes lines $\arg\,a=\pmπ/2$. In addition, it is found that the transition across the upper and lower imaginary $a$-axes is associated, in general, with unequal scales. Numerical calculations are presented to confirm the theoretical predictions.
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Exponentially small expansions associated with a generalised Mathieu series
We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^γ}{(n^λ+a^λ)^μ}\qquad (μ>0)\] when the parameters $λ$ ($>0$) and $γ$ are even integers for large complex $a$ in the sector $|\arg\,a|<π/λ$. The asymptotics in this case consist of a {\it finite} algebraic expansion together with an infinite sequence of increasingly subdominant exponentially small expansions. When $μ$ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of the expansion obtained.
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Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities
We obtain an asymptotic evaluation of the integral \[\int_{\sqrt{2n+1}}^\infty e^{-x^2} H_n^2(x)\,dx\] for $n\rightarrow\infty$, where $H_n(x)$ is the Hermite polynomial. This integral is used to determine the probability for the quantum harmonic oscillator in the $n$th energy eigenstate to tunnel into the classically forbidden region. Numerical results are given to illustrate the accuracy of the expansion.
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An extension of a series containing Laguerre polynomials
Expressions for the summation of the series involving the Laguerre polynomials \[S_m(\pmν, \pm p)\equiv e^{-x}\sum_{n=0}^\infty \frac{x^n\,L_n^{(ν)}(x)}{(1\pm ν\pm p)_n}\frac{(f+m)_n}{(f)_n}\] for any non-negative integers $m$ and $p$ are obtained in terms of generalized hypergeometric functions. These results extend previous work in the literature.
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A derivation of two quadratic transformations contiguous to that of Gauss via a differential equation approach
The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.
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The Stokes phenomenon associated with the periodic zeta function $F(a,s)$
The exponentially improved large-$a$ expansion for the Hurwitz zeta function $ζ(s,a)$ is exploited to examine the expansion of the periodic zeta function $F(a,s)$ in the upper half-plane of the variable $a$. It is shown that a double Stokes phenomenon takes place in the vicinity of the positive imaginary $a$-axis as $|a|\ra\infty$. This is a consequence of the fact that constituent parts of $F(a,s)$ involve two Hurwitz zeta functions resulting in two parallel Stokes lines at unit distance apart. Numerical calculations confirm the theoretical predictions.
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