On the Error in the Padé Approximants for a Form of the Incomplete Gamma Function Including the Exponential Function

Abstract

Closed form expressions for all entries of the Padé matrix table and their errors are derived for the incomplete gamma function $H(c,z) = cz^{ - c} e^{ - z} \int_0^z {e^t t^{c - 1} dt} ,\quad R(c) > 0,\quad H(0,z) = e^{ - z} .$ Asymptotic estimates for the error are developed. Let $(\mu ,\nu )$ be a position in the Pade matrix table where $\mu $ and $\nu $ are the degrees of the denominator and numerator polynomials, respectively, which define the Padé approximant. Our error representations hold for c and z fixed, c not a negative integer, with $\mu $ or $\nu $ or both $\mu $ and $\nu $ approaching infinity. Under these conditions the Padé approximants converge along all rows, columns and diagonals. The asymptotic representations are remarkable in that they give very easy to apply and very realistic estimates when the parameters $\mu $ and $\nu $ are rather small. In the case $c = 0$, that is, the exponential function, uniform asymptotic estimates are developed for the main diagonal and first and second subdiagonal entries of the Padé matrix.