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 series 1 F 1 of one variable and the confluent hypergeometric limit function 0 F 1 of one variable. The first of these double series was introduced by Pierre Humbert (1920 ).
The Humbert series Φ1 is defined for |x | < 1 by the double series:
Φ 1 ( a , b , c ; x , y ) = F 1 ( a , b , − , c ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b ) m ( c ) m + n m ! n ! x m y n , {\displaystyle \Phi _{1}(a,b,c;x,y)=F_{1}(a,b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} where the Pochhammer symbol (q )n represents the rising factorial:
( q ) n = q ( q + 1 ) ⋯ ( q + n − 1 ) = Γ ( q + n ) Γ ( q ) , {\displaystyle (q)_{n}=q\,(q+1)\cdots (q+n-1)={\frac {\Gamma (q+n)}{\Gamma (q)}}~,} where the second equality is true for all complex q {\displaystyle q} except q = 0 , − 1 , − 2 , … {\displaystyle q=0,-1,-2,\ldots } .
For other values of x the function Φ1 can be defined by analytic continuation .
The Humbert series Φ1 can also be written as a one-dimensional Euler -type integral :
Φ 1 ( a , b , c ; x , y ) = Γ ( c ) Γ ( a ) Γ ( c − a ) ∫ 0 1 t a − 1 ( 1 − t ) c − a − 1 ( 1 − x t ) − b e y t d t , ℜ c > ℜ a > 0 . {\displaystyle \Phi _{1}(a,b,c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b}e^{yt}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.} This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.
Similarly, the function Φ2 is defined for all x , y by the series:
Φ 2 ( b 1 , b 2 , c ; x , y ) = F 1 ( − , b 1 , b 2 , c ; x , y ) = ∑ m , n = 0 ∞ ( b 1 ) m ( b 2 ) n ( c ) m + n m ! n ! x m y n , {\displaystyle \Phi _{2}(b_{1},b_{2},c;x,y)=F_{1}(-,b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} the function Φ3 for all x , y by the series:
Φ 3 ( b , c ; x , y ) = Φ 2 ( b , − , c ; x , y ) = F 1 ( − , b , − , c ; x , y ) = ∑ m , n = 0 ∞ ( b ) m ( c ) m + n m ! n ! x m y n , {\displaystyle \Phi _{3}(b,c;x,y)=\Phi _{2}(b,-,c;x,y)=F_{1}(-,b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} the function Ψ1 for |x | < 1 by the series:
Ψ 1 ( a , b , c 1 , c 2 ; x , y ) = F 2 ( a , b , − , c 1 , c 2 ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( b ) m ( c 1 ) m ( c 2 ) n m ! n ! x m y n , {\displaystyle \Psi _{1}(a,b,c_{1},c_{2};x,y)=F_{2}(a,b,-,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,} the function Ψ2 for all x , y by the series:
Ψ 2 ( a , c 1 , c 2 ; x , y ) = Ψ 1 ( a , − , c 1 , c 2 ; x , y ) = F 2 ( a , − , − , c 1 , c 2 ; x , y ) = F 4 ( a , − , c 1 , c 2 ; x , y ) = ∑ m , n = 0 ∞ ( a ) m + n ( c 1 ) m ( c 2 ) n m ! n ! x m y n , {\displaystyle \Psi _{2}(a,c_{1},c_{2};x,y)=\Psi _{1}(a,-,c_{1},c_{2};x,y)=F_{2}(a,-,-,c_{1},c_{2};x,y)=F_{4}(a,-,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,} the function Ξ1 for |x | < 1 by the series:
Ξ 1 ( a 1 , a 2 , b , c ; x , y ) = F 3 ( a 1 , a 2 , b , − , c ; x , y ) = ∑ m , n = 0 ∞ ( a 1 ) m ( a 2 ) n ( b ) m ( c ) m + n m ! n ! x m y n , {\displaystyle \Xi _{1}(a_{1},a_{2},b,c;x,y)=F_{3}(a_{1},a_{2},b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,} and the function Ξ2 for |x | < 1 by the series:
Ξ 2 ( a , b , c ; x , y ) = Ξ 1 ( a , − , b , c ; x , y ) = F 3 ( a , − , b , − , c ; x , y ) = ∑ m , n = 0 ∞ ( a ) m ( b ) m ( c ) m + n m ! n ! x m y n . {\displaystyle \Xi _{2}(a,b,c;x,y)=\Xi _{1}(a,-,b,c;x,y)=F_{3}(a,-,b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m}(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~.} There are four related series of two variables, F 1 , F 2 , F 3 , and F 4 , which generalize Gauss's hypergeometric series 2 F 1 of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880. Appell, Paul ; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13 . (see p. 126) Bateman, H. ; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF) . New York: McGraw–Hill. (see p. 225) 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. ISBN 978-0-12-384933-5 . LCCN 2014010276 . Humbert, Pierre (1920). "Sur les fonctions hypercylindriques". Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French). 171 : 490–492. JFM 47.0348.01 .