Identities of Rothe-Abel- Schläfli-Hurwitz-type

Abstract

Several convolution identities, containing many free parameters, are shown to follow in a very simple way from a combinatorial construction. By specialization of the parameters one can find many of the known generalizations or variations of Abel's generalization of the binomial theorem, including those obtained by Rothe, Schläfli, and Hurwitz. A convolution identity related to Mellin's expansion of algebraic functions, proposed recently by Louck (but contained in equivalent form in earlier work by Raney and Mohanty), and a counting formula for labelled trees by rising edges, due to Kreweras, are also shown to follow from the general approach.