Possibly related: I just simplified something to get Sum[((-1)^n*QPolyGamma[0, -Log[2,-n], 1/2])/n, {n, ∞}] == (𝛾-2)*Log@2 (/.𝛾->EulerGamma) In[456]:= N@% Out[456]= True but I already forget how I got the something. Again, QPolyGamma is just logderiv of QGamma, and subsumes these Lamberts: Sum[b^n/(1 - a*b^n), {n, 0,∞}] == (Log[1 - b] + QPolyGamma[0, Log[b,a])/ (a*Log[b]) In the 𝛾 sum, changing the QPolygamma back to a Lambert, then sumswapping: Out[493]= Sum[HurwitzLerchPhi[-1, 1, 1 + 2^n], {n, 0, ∞}] == 2 - EulerGamma - Log[2] In[494]:= N@% Out[494]= False and I am the Virgin Mary. But what the heck is HurwitzLerchPhi[-1, 1, k] ? Out[501]= {Log[2], 1 - Log[2], -(1/2) + Log[2], 5/6 - Log[2], -(7/12) + Log[2], 47/60 - Log[2], -(37/60) + Log[2], 319/420 - Log[2], -(533/840) + Log[2]} In[502]:= % /. Log@2 -> 0 Out[502]= {0, 1, -(1/2), 5/6, -(7/12), 47/60, -(37/60), 319/420, -(533/840)} In[503]:= FindSequenceFunction@% Out[503]= (-1)^#1 ((-1)^#1 LerchPhi[-1, 1, #1] + Log[2]) & GAA! Multiplying by n! Sorry, but 0, 1, 1, 5, 14, 94, 444, 3828, 25584 do not match anything in the OEIS. With and w/o the signs. --rwg On Tue, Jul 18, 2017 at 6:29 AM, Bill Gosper <billgosper@gmail.com> wrote:

but just in case it's still mysterious in the "morning": What is this?

QPolyGamma[0, 1/2, q] + QPolyGamma[0, 1/2 - (I π)/Log[q], q] == -Log[(1 - q)/(1 + q)] + QPolyGamma[0, 1/2, q^2]

q-Euler's constants? In[1199]:= PolyGamma[0, 1/2] // FunctionExpand

Out[1199]= -EulerGamma - Log[4]

Empirically true in the 1st quadrant of the unit disk, and in the whole 4th quadrant. Does it generalize? Does q-digamma have a Jacobi imaginary transformation? zzzzzzzzzz.... --rwg