14
votes
1answer
322 views

Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...
5
votes
5answers
722 views

Summation of an expression

Hi, Does anyone have an idea about an exact or approximate formulae for the following summation? $$ \sum_{j=1}^n \frac{j^k}{(j-1)!} $$ where k is a positive integer (the denominator of the j^th term ...
0
votes
1answer
779 views

Generalizations of a product formula for the gamma function

Hello and Happy holidays. I am interested in generalizations of the following product formula for the gamma function $\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$: \begin{align} ...
2
votes
0answers
341 views

Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
5
votes
2answers
1k views

Multiplicative integral of $\Gamma(x)$

A recent question on the notion and notation of multiplicative integrals ( What is the standard notation for a multiplicative integral? ) induced me to play with the Riemann products of the Gamma ...