# Evaluation of infinite series involving special products and their algebraic characterization

In: Mathematica Slovaca, vol. 59, no. 3
Marian Genčev

## Details:

Year, pages: 2009, 365 - 378
Keywords:
summation methods, infinite series, hypergeometric function, transcendence
The aim of the paper is the investigation of special infinite series of the form $∑n=0 ( \frac{\prods=0m1n (s+a ) \prods=0m2n (s+b )} {\prods=0(m1+m2)n+1 (s+a+b )} )c·((P(n)) / (θn))· fn(n)$ where $(a,b,m1,m2,θ ,c,P(n))\in\mathbb{R}4× \mathbb{C}×\{\pm 1\}× \overline{\mathbb{Q}}[n]$ and $\{fn(n)\}n\in\N0$ is a sequence of rational functions. A general summation method for the sum above in the case of the special choice of parameters $a$, $b$ and $fn(n)$ is included. We find the $2m$-tuple of rational numbers $αi$, $βj$ ($1≤ i≤ m$, $1≤ j≤ m$) for which $m+1Fm( . \begin{array}{cc} 1,&α1, … , αm1+1, … , βm+1 \end{array} | x )\notin\overline{\Q}$ iff $m+1Fm( . \begin{array}{cc} 1,&β1, … , βm1, … , αm \end{array} | x )\in\overline{\Q}$ and vice versa.