Not much to add, but here's the expansion with the 6th term, which took ~1 cpu day of pure numerics: QPolyGamma[0,x,q]==PolyGamma[0,x]+(-(3/4)+x/2) (q-1)+1/2 (49/72-(7 x)/12+x^2/12) (q-1)^2+1/6 (-(31/24)+(5 x)/4-x^2/4) (q-1)^3+1/24 (13481/3600-(47 x)/12+(109 x^2)/120+x^3/60-x^4/120) (q-1)^4+1/120 (-(5261/360)+(97 x)/6-(49 x^2)/12-x^3/6+x^4/12) (q-1)^5+1/720 (7588759/105840-(497 x)/6+(5575 x^2)/252+(17 x^3)/12-(44 x^4)/63-x^5/84+x^6/252) (q-1)^6+O[q-1]^7 Where might we meet polynomial sequences where the degrees go 1,2,2,4,4,6,...? --rwg Does anyone know how to use CoefficientList on a Series expanded at other than 0? On Thu, Jul 20, 2017 at 12:28 PM, Bill Gosper <billgosper@gmail.com> wrote:

has polynomials in x as coefficients of (q-1)^n. See gosper.org/psi_q.png Plots are blue digamma overwritten by gold series approximation. For (q-)PolyGamma[n,...], just d^n/dx^n.

QPolyGamma[3, 1/2, q] is phenomenally near π⁴ for .1<q<1:

Table[QPolyGamma[3, 1/2, q] - π⁴, {q, .1`69, .9`69, .1`69}] // N

Out[1343]= {-0.0000557992067391662, -3.46554182307681*10^-8, -8.95673115069408*10^-12, -3.02779751180959*10^-16, -2.86641755545745*10^-22, -4.2550061594941*10^-31, -1.32744226211741*10^-45, -1.41336365314375*10^-65}

Intermediate 69 digit precision because of huge subtractive significance loss, as well as some sub-par Mathematica numerics. --rwg