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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 theBinet 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 areview of the Binet function $\mu\left( z\right) $ and Gilbert's investigationsof $\mu\left( z\right) $, several properties of the Binet polynomials$b_{m}\left( \alpha\right) $ are presented. We compare Gilbert's generalizedfactorial series with Stirling's asymptotic expansion and demonstrate by anumerical example that, with a same number of terms evaluated, the Gilbertgeneralized factorial series with an optimized value of $\alpha$ can beat thebest possible accuracy of Stirling's expansion. Finally, we extend Binet'smethod to factorial series of Laplace transforms.

Binet's factorial series and extensions to Laplace transforms