Multibasic and Mixed Hypergeometric Gosper-Type Algorithms

Gosper’s summation algorithm finds a hypergeometric closed form of an indefinite sum of hypergeometric terms, if such a closed form exists. We extend his algorithm to the case when the terms are simultaneously hypergeometric and multibasic hypergeometric. We also provide algorithms for finding polynomial as well as hypergeometric solutions of recurrences in the mixed case. We do not require the bases to be transcendental, but only that q1 1 · · · qm m 6= 1 unless k1 = · · · = km = 0. Finally, we generalize the concept of greatest factorial factorization to the mixed hypergeometric case.