On an application of Binet’s second formula

Zhang, Ruiming

doi:10.1090/proc/13711

On an application of Binet’s second formula
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by Ruiming Zhang PDF

Proc. Amer. Math. Soc. 145 (2017), 5267-5272 Request permission

Abstract:

In this work we apply the second Binet formula for Euler’s gamma function $\Gamma (x)$ and a Laplace transform formula to derive an infinite series expansion for the auxiliary function $f(x)$ in the computations of sine integral and cosine integral functions in terms of $\log \Gamma (x)$ and the Möbius function. Then we apply Möbius inversion to obtain a Kummer type series expansion for $\log \Gamma (x)$. Unlike the original Kummer formula, our formula is not a Fourier series anymore. By differentiating the series expansion for $f(x)$ we obtain an infinite series expansion for the auxiliary function $g(x)$ associated with sine integral and cosine integral functions as well.

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