This series expansion of a Lerch transcendent seems fixed in V12. However, the following still fails: From the definition of a Lerch transcendent,

Assuming[z > 0, Sum[(-1)^k/(z + k), {k, 0, ∞}]]
(*    LerchPhi[-1, 1, z]    *)

we expect the series-expansion around $z=0$ to give a dominant term of $1/z$ (i.e., the $k=0$ term), but instead it gives the wholly nonsensical

Series[LerchPhi[-1, 1, z], {z, 0, 2}]
(*     -Log[2] + π^2/12*z - 3/4*Zeta[3]*z^2 + O[z]^3    *)

Plot[{LerchPhi[-1, 1, z], -Log[2] + π^2/12*z - 3/4*Zeta[3]*z^2}, {z, 0, 1}]

enter image description here

Is this a bug or am I misunderstanding something?


As J.M. points out, the series expansion of LerchPhi merely omits the singular term, so the given series expansion isn't wholly nonsensical:

Plot[{LerchPhi[-1, 1, z] - 1/z, -Log[2] + π^2/12*z - 3/4*Zeta[3]*z^2}, {z, 0, 1}]

enter image description here

I find this behavior very confusing: the option IncludeSingularTerm seems to default to True when plotting, and defaul to False when series-expanding. Further,

LerchPhi[-1, 1, z, IncludeSingularTerm -> False]

does not exclude the singular term.

A bug?

  • 4
    $\begingroup$ By default, LerchPhi[] has the setting IncludeSingularTerm -> False, so the 1/z term is tacitly excluded for z == 0. $\endgroup$ – J. M. is in limbo Jan 25 at 13:18

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