arXiv subjects

Mykola A. Shpot

Publications and source records attributed to Mykola A. Shpot.

2 recordsLinked to original sources

arXiv2024

Two closed-form evaluations for the generalized hypergeometric function ${}_4F_3(\frac1{16})$

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function $_3F_2$ into even and odd components, together with the use of two known results for $_3F_2(\pm\frac14)$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $π^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds.

math.CA↗

arXiv2016

Feynman integral in $\mathbb R^1\oplus\mathbb R^m$ and complex expansion of $_2F_1$

Closed form expressions are proposed for the Feynman integral $$ I_{D, m}(p,q) = \int\frac{d^my}{(2π)^m}\int\frac{d^Dx}{(2π)^D} \frac1{(x-p/2)^2+(y-q/2)^4} \frac1{(x+p/2)^2+(y+q/2)^4} $$ over $d=D+m$ dimensional space with $(x,y),\,(p,q)\in \mathbb R^D \oplus \mathbb R^m$, in the special case $D=1$. We show that $I_{1,m}(p,q)$ can be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function $_2F_1$, as well as generalized hypergeometric $_2F_2$ and $_3F_2$, Horn $H_4$ and Appell $F_2$ functions. Several interesting relations are derived between the real and imaginary parts of $_2F_1$ and the function $H_4$.

math.CA↗