Let me give an example. Recently I derived the following:
Sqrt[Pi] = (2^(5+3 j) (Gamma[5+2 j] Gamma[8+3 j] HypergeometricPFQ[{1,5/2+j,3+j},{3+(3 j)/2,7/2+(3 j)/2},-1]+2 Gamma[7+2 j] Gamma[6+3 j] HypergeometricPFQ[{1,7/2+j,4+j},{4+(3 j)/2,9/2+(3 j)/2},-1]))/(Gamma[5+2 j] Gamma[6+3 j] Gamma[8+3 j] (HypergeometricPFQRegularized[{1,5/2+j,3+j},{3+(3 j)/2,7/2+(3 j)/2},-1]+(15+11 j+2 j^2) HypergeometricPFQRegularized[{1,7/2+j,4+j},{4+(3 j)/2,(3 (3+j))/2},-1])) (*)
Since above (*) holds for an arbitrary integer "j" (as confirmed in both Mathematica and Maple), I call it identity...
I would like to find out whether (*) is well known and/or represents some kind of corollary or does it represnt the new previously unknown independent result?
Is there some functionality (and/or feature) available in Mathematica or in app (such as WolframAlpha} to give me the answer on my above stated question?
Alexander R. Povolotsky
NB Above identity (*) is the subcase of more general, based on four parameters {j,k,l,m}, derived by me identity, which is true for
arbitrary j,k,l,m.
Sqrt[Pi] = (1/(2^j)*((k Gamma[5 + 2 j] Gamma[1 + l] HypergeometricPFQ[{1, 5/2 + j, 3 + j}, {3 + j + l/2,7/2 + j + l/2}, -1])/Gamma[6 + 2 j + l] + ((k + m) Gamma[7 + 2 j] Gamma[1 + l] HypergeometricPFQ[{1, 7/2 + j, 4 + j}, {4 + j + l/2,9/2 + j + l/2}, -1])/Gamma[8 + 2 j + l]))/(2^(-5 - 3 j -l) Gamma[5 + 2 j] Gamma[1 + l] (k HypergeometricPFQRegularized[{1, 5/2 + j,3 + j}, {3 + j + l/2, 7/2 + j + l/2}, -1] +1/2 (3 + j) (5 + 2 j) (k + m) HypergeometricPFQRegularized[{1,7/2 + j, 4 + j}, {4 + j + l/2, 9/2 + j + l/2}, -1])) (**)
