If I write
f[i_] := Expand[FunctionExpand[x^(i/5) LerchPhi[x, 1, i/5]]]
i < 5, the result is a linear combination (over
k from 1 to 5) of logarithms where only appears
x^(i/5). The coefficients are complex inside and outside the logarithms. I try to find the recurrence relations for all these coefficients (they are fractional powers of -1). I am not focusing on fractions of 5, but I search for the general answer ... if any !