Binet's factorial series and extensions to Laplace transforms

Abstract

We investigate a generalization of Binet's factorial series in the parameter $\alpha$ \[ μ\left( z\right) =\sum_{m=1}^{\infty}\frac{b_{m}\left( α\right) }{\prod_{k=0}^{m-1}(z+α+k)}% \] due to Gilbert, for the Binet function \[ μ\left( z\right) =\logΓ\left( z\right) -\left( z-\frac{1} {2}\right) \log z+z-\frac{1}{2}\log\left( 2π\right) \] After a review of the Binet function $\mu \left(z\right)$ and Gilbert's investigations of $\mu \left(z\right)$, several properties of the Binet polynomials $b_m\left(\alpha \right)$ are presented. We compare Gilbert's generalized factorial series with Stirling's asymptotic expansion and demonstrate by a numerical example that, with a same number of terms evaluated, the Gilbert 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.

We have integrated Gilbert's investigations of 1876

References in corpus (1)

• The Mittag-Leffler function