arXiv subjects

R B Paris

Publications and source records attributed to R B Paris.

59 records · Page 4Linked to original sources

arXiv2014

Comments on "Exactification of Stirling's approximation for the logarithm of the gamma function"

We re-examine the exponentially improved expansion for $\log\,\g(z)$, first considered in Paris and Wood in 1991, to point out that the recent treatment by Kowalenko [Exactification of Stirling's approximation for the logarithm of the gamma function, arXiv:1404.2705] using his procedure of regularisation produces an equivalent result. In addition, we point out an error in his definition of the Stokes multiplier that leads him to make the incorrect statement that the Stokes phenomenon is a jump discontinuity, rather than a smooth transition. We supply a numerical example that clearly demonstrates the smooth transition of the leading subdominant exponential $e^{2πiz}$ across the Stokes line $\arg\,z=\fsπ$.

math.CA↗

arXiv2014

An asymptotic expansion for the generalised quadratic Gauss sum revisited

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,θ)=\sum_{j=1}^{N} \exp (πixj^2+2πijθ),$$ where $x$, $θ$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and $N\rightarrow\infty$ such that $Nx$ is finite. The form of this expansion holds for all values of $Nx+θ$ and, in particular, in the neighbourhood of integer values of $Nx+θ$. A simple bound for the remainder in the expansion is derived. Numerical results are presented to demonstrate the accuracy of the expansion and the sharpness of the bound.

math.CA↗

arXiv2014

Exponential asymptotics of the Voigt functions

We obtain the asymptotic expansion of the Voigt functions $K(x,y)$ and $L(x,y)$ for large (real) values of the variables $x$ and $y$, paying particular attention to the exponentially small contributions. A Stokes phenomenon is encountered as $y\rightarrow+\infty$ with $x>0$ fixed. Numerical examples are presented to demonstrate the accuracy of these new expansions.

math.CA↗

arXiv2014

Evaluations of some terminating hypergeometric ${}_2F_1(2)$ series

Explicit expressions for the hypergeometric series ${}_2F_1(-n, a; 2a\pm j;2)$ and ${}_2F_1(-n, a; -2n\pm j;2)$ for positive integer $n$ and arbitrary integer $j$ are obtained with the help of generalizations of Kummer's second and third summation theorems obtained earlier by Rakha and Rathie. Results for $|j|\leq 5$ derived previously using different methods are also obtained as special cases.

math.CV↗

arXiv2014

Generalization of a quadratic transformation due to Exton

Exton [Ganita 54(2003)13-15] obtained numerous new quadratic transformations involving hypergeometric functions of order two and of higher order by applying various known classical summation theorems to a general transformation formula based on the Bailey transformation. We obtain the generalization of one of the Exton quadratic transformations. The results are derived with the help of a generalization of Dixon's summation theorem for the series ${}_3F_2$ obtained earlier by Lavoie {\em et al}. Several interesting known as well as new special cases and limiting cases are also given.

math.CA↗