By Shalosh B. Ekhad
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Written: Nov. 6, 2019
Soon after this short note was posted in the arxiv, Michael Schlosser asked us whether a similar proof, using the Almkvist-Zeilberger algorithm, could be given to two curious integral identities discovered and proved by him and George Gasper in this interesting paper that appeared in the Ramanujan Journal v. 13 (2007), 229-242. It sure can! All you need is to run the following input file, once you have made sure to read the Maple package EKAHD.txt into the Maple session.
But the Gasper-Schlosser identities are much deeper! Even though the recurrence satisfied by the sequence that is identically 1 is FIRST-ORDER: f(b+1)-f(b)=0, the Almkvist-Zeilberger algorithm produced THIRD-ORDER recurrences, that are also satisfied by the constant sequence 1. It remained to check these identities for the base cases b=0, b=1, b=2, but that is purely routine calculus 1.
Finally, Shalosh only proved it for b NON-NEGATIVE integers, but the usual "nonsense" extends it to all b (β in their paper), where the integrals make sense.
Another indication for the depth of the Gasper-Schlosser identities is time. It took 6 seconds for Shalosh to prove them (3 seconds for each), while the identity in the above-mentioned Ekhad-Zeilberger-Zudilin note took 0.1 seconds.