New posts in polygamma

Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(2n+1)}=2-2\ln(2)$

integration sequences-and-series digamma-function polygamma

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

calculus special-functions gamma-function conjectures polygamma

The meaning and definition of $\psi^{(-2)}(x)$, and the convergence of some related series involving the Möbius function

derivatives convergence-divergence mobius-function polygamma digamma-function

Show some properties of the Digamma Function

complex-analysis special-functions gamma-function polygamma

Great difficulty in finding the residues of $\frac{\mathrm{Log}\Gamma\left(\frac{z+ai}{2\pi i}\right)}{\cosh(z)+1}$

complex-analysis limits gamma-function residue-calculus polygamma

In an attempt to find $I = \int_0^\infty \frac{t}{e^t-1}dt$

real-analysis calculus laplace-transform riemann-zeta polygamma

Generalized FoxTrot Series $F(a,b,q,x) = \sum_{k=q}^{\infty} \dfrac {(-1)^{k+1} k^a}{k^b+x}$

calculus sequences-and-series closed-form polygamma

What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

combinatorics summation gamma-function polygamma combinatorial-number-theory

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

integration definite-integrals closed-form conjectures polygamma

Integrating $\int_0^1\frac{\ln^2x\ln(1+x)}{1+x^2} dx$ using real methods

real-analysis calculus integration harmonic-numbers polygamma

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

definite-integrals logarithms closed-form polylogarithm polygamma

The most complete reference for identities and special values for polylogarithm and polygamma functions

reference-request special-functions gamma-function polylogarithm polygamma

Evaluate $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_n}{(2n+1)^3}$

real-analysis calculus integration harmonic-numbers polygamma

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

closed-form conjectures polylogarithm experimental-mathematics polygamma

Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$ [duplicate]

calculus sequences-and-series special-functions polygamma

Closed form for $\sum_{n=0}^\infty\frac{\Gamma\left(n+\tfrac14\right)}{2^n\,(4n+1)^2\,n!}$

sequences-and-series hypergeometric-function polylogarithm experimental-mathematics polygamma

Evaluating $\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dx$

integration definite-integrals logarithms closed-form polygamma

Re-Expressing the Digamma

calculus special-functions gamma-function harmonic-numbers polygamma

Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$

definite-integrals closed-form hypergeometric-function polylogarithm polygamma

A closed form of $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$?

calculus integration sequences-and-series closed-form polygamma