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- W3148095003 abstract "We investigate Binet's series, convergent for $operatorname{Re}left( zright) >0$, [ muleft( zright) =sum_{m=1}^{infty}frac{c_{m}}{prod_{k=0}^{m-1}(z+k)}% ] for the Binet function [ muleft( zright) =logGammaleft( zright) -left( z-frac{1}% {2}right) log z+z-frac{1}{2}logleft( 2piright) ] and contribute to the classical theory of the Gamma function $Gammaleft( zright) $ by correcting an unfortunate error in Binet's original computation. After a brief review of the Binet function $muleft( zright) $, several different expressions for the rational coefficients $c_{m}$ in Binet's convergent expansion as well as two integral representations for $c_{m}$ are presented. In addition, we demonstrate the important property that all but the first two coefficients are negative, i.e. $c_{m} 2$, while $c_{1}=frac{1}{12}$ and $c_{2}=0$. We compare the corrected Binet series with Stirling's emph{asymptotic} expansion and discuss the advantage of both series. Finally, we compute the corresponding factorial series for the derivatives of the Binet function and apply those series to the digamma and polygamma functions." @default.
- W3148095003 created "2021-04-13" @default.
- W3148095003 creator A5001877366 @default.
- W3148095003 date "2021-02-09" @default.
- W3148095003 modified "2023-09-27" @default.
- W3148095003 title "Convergent Binet series in the theory of the Gamma function" @default.
- W3148095003 hasPublicationYear "2021" @default.
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