Series expansion of the Gamma function and its reciprocal

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.