Gosper summability of rational multiples of hypergeometric terms

By the telescoping method, Sun has recently given some hypergeometric series whose sums are related to π. We investigate these from point of view Gosper's algorithm. Given a term tk, we consider Gosper summability r(k)tk for r(k) being rational function k. give an upper bound and lower on degree numerator such that is summable. also show denominator can be read off representation tk+1/tk. Based results, systematic method construct derived known ones. illustrate corresponding super-congruences q-analogue approach.