In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1904) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Whittaker's equation is
It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functionsMκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functionsM and U by
Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models.
Jacquet, Hervé (1966), Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B262: A943––A945, ISSN0151-0509, MR0200390