HypergeometricSE[exp, nu] expresses Hypergeometric functions by their series expansion in terms of a sum (the Sum is omitted and nu, running from 0 to \infty, is the summation index).
HypergeometricSE[Hypergeometric2F1[a, b, c, z], \[Nu]]\frac{\Gamma (c) z^{\nu } \Gamma (a+\nu ) \Gamma (b+\nu )}{\Gamma (a) \Gamma (b) \Gamma (\nu +1) \Gamma (c+\nu )}
HypergeometricSE[HypergeometricPFQ[{a, b, c}, {d, e}, z], \[Nu]]\frac{\Gamma (d) \Gamma (e) z^{\nu } \Gamma (a+\nu ) \Gamma (b+\nu ) \Gamma (c+\nu )}{\Gamma (a) \Gamma (b) \Gamma (c) \Gamma (\nu +1) \Gamma (d+\nu ) \Gamma (e+\nu )}