Psi function or digamma function?

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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
 

Answers and Replies

  • #2
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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
 

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