bb[j_, q_, k_] := (-1)^j*
Sum[(Binomial[q, 1 + i + k - q] Binomial[q,
    1 - i + j + k - q] Gamma[1 + i + k] Gamma[
    1 - i + j + k])/(Gamma[1 + i] Gamma[1 - i + j]), {i, 0, j}];
GG2[t_, q_, k_] :=
Sum[bb[j, q, k]*(1/(j + 1))*t^(j + 1), {j, 2*(q - k - 1),
2*(2 q - k - 1)}];
s[q_, k_] :=
2^k * ((2 q - 2)! (2 q)!)/(4 q -
  1)! Product[((4 q - 2 i + 1) (q - i) (2 i - 1))/(
2 q - 2 i - 1), {i, 1, k}];
GG2[1, 5, 2]
s[5, 2]
GG2[1, q, k] - s[q, k]

8/5005

8/5005

-(((-4)^k (2 q)! (-2 + 2 q)! Gamma[1/2 + k] Pochhammer[1/2 - 2 q,
k] Pochhammer[1 - q, k])/(
Sqrt[[Pi]] (-1 + 4 q)! Pochhammer[3/2 - q, k])) + !(
*UnderoverscriptBox[([Sum]), (j =
 2 (((-1) - k + q))), (2 (((-1) - k + 2 q)))](
 *FractionBox[(1), (((1 + j)) Gamma[1 + j])](
*SuperscriptBox[(((-1))), (j)] Binomial[q,
1 + k - q] Binomial[q, 1 + j + k - q] Gamma[1 + k] Gamma[
1 + j + k] HypergeometricPFQ[{(-j), 1 + k,
 1 + k - 2 q, (-1) - j - k + q}, {(-j) - k,
 2 + k - q, (-j) - k + 2 q}, 1])))