In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series1F1 of one variable and the confluent hypergeometric limit function0F1 of one variable. The first of these double series was introduced by Pierre Humbert (1920).
Definitions[]
The Humbert series Φ1 is defined for |x| < 1 by the double series:
where the Pochhammer symbol (q)n represents the rising factorial:
where the second equality is true for all complex except .
There are four related series of two variables, F1, F2, F3, and F4, which generalize Gauss's hypergeometric series2F1 of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880.
Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "9.26.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc.ISBN978-0-12-384933-5. LCCN2014010276.
Humbert, Pierre (1920). "Sur les fonctions hypercylindriques". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 171: 490–492. JFM47.0348.01.