Källén function and around. Part 1

There are elementary functions that turn out to be ubiquitous. The Källén function is one of them. Of course, it is incomparably less well-known than the exponential function (although, in a certain sense, it is related to it).
Originally arising in "school-textbooks" mathematics, the Källén function was later "rediscovered" by physicists in the context of scattering amplitude calculations. It is directly connected with the famous combinatorial generating functions (for the Catalan numbers), appears in the study of solutions of classical ODEs, and is related to the generalized hypergeometric functions of Appell and Kampé de Fériet as well as to properties of discriminants.
I will try to tell some stories around of this remarkable function.