Appears in collection : Combinatorics and Arithmetic for Physics

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 (Bessel–type), and is related to the generalized hypergeometric functions of Appell and Kampé de Férie, to the Wigner-Bloch dilogarithm as well as to properties of discriminants and associativity of 2-valued multiplication laws. I will try to tell some stories around of this remarkable function.