On two-term hypergeometric recursions with free lower parameters
John Maxwell Campbell, Paul Levrie · The Journal of Difference Equations and Applications · 2024
Let F(n,k) be a hypergeometric function that may be expressed so that n appears within initial arguments of inverted Pochhammer symbols, as in factors of the form 1(n)k. Only in exceptional cases is F(n,k) such that Zeilberger's algorithm produces a two-term recursion for ∑k=0∞F(n,k) obtained via the telescoping of the right-hand side of a difference equation of the form p1(n)F(n+r,k)+p2(n)F(n,k)=G(n,k+1)−G(n,k) for fixed r∈N and polynomials p1 and p2. Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupaş, and Amdeberhan are special cases of hypergeometric transforms introduced in our article.