If there are any reference about the expansion of $$\Gamma\left(\frac{p-x}{q}\right),$$ where $p,q$ are integers with $1\leq p\leq q$?
2026-03-26 07:38:35.1774510715
Find reference about expansion of gamma function
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Around $x=0$, $$\log \left(\Gamma \left(\frac{p-x}{q}\right)\right)=\sum_{n=0}^\infty (-1)^n \frac{ \psi ^{(n-1)}\left(\frac{p}{q}\right)}{q^n\,n!}\,x^n$$ Around $x=p$, $$\log \left(\Gamma \left(\frac{p-x}{q}\right)\right)=-\log \left(\frac{p-x}{q}\right)-\gamma \,\left(\frac{p-x}{q}\right)+\sum_{n=2}^\infty (-1)^n \frac{\zeta(n)}n \left(\frac{p-x}{q}\right)^n$$