---------- Forwarded message ---------
From: Bill Gosper <billgosper(a)gmail.com>
Date: Fri, Jul 10, 2020 at 7:32 AM
Subject: diGamma sums
To: Wolfram Tech Support
I don't know if there's a general theory, but FindIntegerNullVector seems
to crank these out reliably.
Sum[PolyGamma[0, k]^2/k^2, {k, Infinity}] == (EulerGamma^2*Pi^2)/6 +
(11*Pi^4)/360 - 2*EulerGamma*Zeta[3]
Sum[PolyGamma[0, k]^3/k^2, {k, Infinity}] == (-(1/6))*EulerGamma^3*Pi^2 -
(11*EulerGamma*Pi^4)/120 + 3*EulerGamma^2*Zeta[3] + (1/6)*Pi^2*Zeta[3]
+ (15*Zeta[5])/2
Sum[PolyGamma[0, k]^2/k^3, {k, Infinity}] == -((EulerGamma*Pi^4)/180) +
EulerGamma^2*Zeta[3] + (1/6)*Pi^2*Zeta[3] - (3*Zeta[5])/2
Similarly for HarmonicNumber vs PolyGamma.
Perhaps you should offer such results via the function ConjectureExpand.-)
—Bill
