A note on the moment generating function for the reciprocal gamma distribution

Abstract

In this note we consider the function φ ( t ) = ∫ 0 ∞ e − t x / Γ ( x ) d x \varphi (t) = \smallint _0^\infty {e^{ - tx}}/\Gamma (x)\, dx and use the Euler-Maclaurin expansion with the step-length h = 1 / 4 h = 1/4 to obtain some useful (from a numerical point of view) formulae. Numerical values of φ ( t ) \varphi (t) correct to 11D are given for t = 0.0 ( 0.1 ) 5.0 t = 0.0(0.1)5.0 .