9 citations · 76 across the 82 of their papers we have counts for
14 papers · 1 filter
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 as 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 and as in…
Wright functions of the second kind and Whittaker functions
Francesco Mainardi, Richard B. Paris, Armando Consiglio
In the framework of higher transcendental functions the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory a…
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…
On the -zeros of the modified Bessel function of positive argument
R B Paris
The modified Bessel function of the second kind of imaginary order for fixed 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 is , 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 . 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 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…
A note on some infinite sums of Hurwitz zeta functions
R B Paris
We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where is…
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 in which the B…
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 , a…