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Asymptotic formulae associated with the Wallis power function and digamma function

Let $s, t$ be two given real numbers, $s\not=t$. We determine the coefficients $c_j(s, t)$ such that [\frac{; ; ; ; \Gamma(x+t)}; ; ; ; {; ; ; ; \Gamma(x+s)}; ; ; ; ]^{; ; ; ; 1/(t-s)}; ; ; ; \sim\exp(\psi(x+\sum_{; ; ; ; j=0}; ; ; ; ^{; ; ; ; \infty}; ; ; ; c_j(s, t)x^{; ; ; ; -j}; ; ; ; ))) as $x\to\infty$, where $\psi(x)=\Gamma'(x)/\Gamma(x)$ denotes the digamma function. Also, the analysis of the coefficients in the asymptotic expansion of the composition $\exp(\psi(x+s))$ is given in details.

Gamma function ; digamma (psi) function ; Bernoulli polynomials ; Asymptotic expansions

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