Binet's factorial series and extensions to Laplace transforms

TL;DR

This paper explores a generalized factorial series for the Binet function, compares it with Stirling's expansion, and extends the method to Laplace transforms, demonstrating improved accuracy with optimized parameters.

Contribution

It introduces a generalized factorial series for the Binet function, analyzes its properties, compares it with Stirling's expansion, and extends the approach to Laplace transforms.

Findings

Gilbert's generalized factorial series can outperform Stirling's expansion with optimal parameters.

Numerical examples show improved accuracy of the generalized series.

Extension of Binet's method to factorial series of Laplace transforms.

Abstract

We investigate a generalization of Binet's factorial series in the parameter $\alpha$ \[ \mu\left( z\right) =\sum_{m=1}^{\infty}\frac{b_{m}\left( \alpha\right) }{\prod_{k=0}^{m-1}(z+\alpha+k)}% \] due to Gilbert, 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 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.