Graduate Psi function or digamma function?

[Belgium 12]

Hi everyone,
concerning serie representation of psi function.
In te solution of bessel function of the second kind we have the following expressions for the psi functions

psi(m+1) and psi(n+m+1) then they give the series for the two psi functions ie(or digamma function)

sum
k from 1 to m of 1/k and ( for the first)
sum k from 1 to m+n of 1/k (for the second)

Is it possible to explain where this serie come from maybe using defintion of the psi function.I can't write latex sorry.

Tank you veru much

 

[fresh_42]

The LaTeX part is easier than your actual question:
https://www.physicsforums.com/help/latexhelp/

We have $\psi(x)=\dfrac{d}{dx}\log(\Gamma(x))$ and are interested in $\psi(n)=H_{n-1} -\gamma=H_{n-1}-\lim_{n \to \infty}\left(H_n-\log(n)\right)$.
So we have heuristically a difference quotient of
$$
\log(\Gamma(n))=\log \left(\prod_k k \right)= \sum_k \log k = \sum_k \int_1^k \dfrac{1}{x}\,dx
$$
on the left hand side and the same difference on the right hand side.

An exact proof is a bit more to do. See this nice collection of formulas around $\Gamma(x)$ and $\psi(x)$:
http://fractional-calculus.com/gamma_digamma.pdf