Clear[x, m, n, z, c, c0, fi];
c := 1/4
c0 := 1/8
fi := 5/8
n := 3/4
m := n + 1
z := c/c0
x := 1/Phi*
Sqrt[m/2/c0^(m - 2)]*2/(2 - m)*(z^(1 - m/2)*
Hypergeometric2F1[1/2 - 1/m, 1/2, 3/2 - 1/m, z^-m] -
Hypergeometric2F1[1/2 - 1/m, 1/2, 3/2 - 1/m, 1])
Series[Hypergeometric2F1[c, c0, 8, x], {x, 0, 1}]
