Binet's factorial series and extensions to Laplace transforms.

We investigate a generalization of Binet's factorial series in the parameter $\alpha$ \[ \mu\left( z\right) =\sum_{m=1}^{\infty}\frac{b_{m}\left( \alpha\right) }{\prod_{k=0}^{m-1}(z+\alpha+k)}% \] for the Binet function \[ \mu\left( z\right) =\log\Gamma\left( z\right) -\left( z-\frac{1}% {2}\right) \log z+z-\frac{1}{2}\log\left( 2\pi\right) \] After a brief review of the Binet function $\mu\left( z\right) $, several properties of the Binet polynomials $b_{m}\left( \alpha\right) $ are presented. We compute the corresponding factorial series for the derivatives of the Binet function and apply those series to the digamma and polygamma functions. We compare Binet's generalized factorial series with Stirling's \emph{asymptotic} expansion and demonstrate by a numerical example that, with a same number of terms evaluated, the Binet generalized factorial series with an optimized value of $\alpha$ can beat the best possible accuracy of Stirling's expansion. Finally, we extend Binet's method to factorial series of Laplace transforms.