The built-in function FindSequenceFunction
is quite good at recognizing hypergeometric terms, i.e. terms $c_k$ for which $c_{k+1}/c_k$ is a rational function of $k$.
FindSequenceFunction[CoefficientList[Series[
Hypergeometric2F1[1/3(2-I Sqrt[2]), 1/3(2 + I Sqrt[2]), 1/3, x],
{x, 0, 10}] // Normal, x] // Simplify]
evaluates nicely to:
(*
(Pochhammer[2/3 - (I Sqrt[2])/3, -1 + #1] Pochhammer[2/3 + (I Sqrt[2])/3, -1 + #1]) /
(Pochhammer[1/3, -1 + #1] Pochhammer[1, -1 + #1]) &
*)
Nonetheless, I haven't been able to get Mathematica to recognize the sum of two hypergeometric terms. For example:
FindSequenceFunction[CoefficientList[Series[
Hypergeometric2F1[1/3 (2 - I Sqrt[2]), 1/3 (2 + I Sqrt[2]), 1/3, x] +
Hypergeometric2F1[1/3 (4 - I Sqrt[2]), 1/3 (4 + I Sqrt[2]), 5/3, x],
{x, 0, 10}] // Normal, x] // Simplify]
returns:
(*
FindSequenceFunction[{2, 16/5, 141/40, 5653/1540, 46239/12320, 217707/57200, 11190509/2912000, 17236629861/4453904000, 277267543443/71262464000, 2883838646953/738075520000, 22686466063740149/5786512076800000}]
*)
I appreciate that identifying sums of hypergeometric terms might be more difficult than recognizing one such term because the ratio of successive terms is then no longer a rational function of the index.
Question: Is it possible to trick Mathematica into trying a little (or a lot) harder to recognize the sum of two hypergeometric terms?