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If I write

f[i_] := Expand[FunctionExpand[x^(i/5) LerchPhi[x, 1, i/5]]]

for any 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 !

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I don't see a k anywhere in your expression with the Lerch transcendent... –  J. M. Jun 14 '13 at 14:10
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