Hypergeometric series \(\phi (a,b;z)\): Further developments (Q1901325)

The function \({}_1 F_1 [n; m+n; z]\), where \(n\in \mathbb{N}\) and \(m\in \mathbb{N}_0\), is shown to be expressible as \(F_1 (z)- e^z F_2 (z)\), where \(F_1\) and \(F_2\) are rational functions. The proof of this result is based upon a special case of Vandermonde's theorem, which is, however, not named as such but rederived over more than five pages.