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 1F1 of one variable and the confluent hypergeometric limit function 0F1 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:
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:
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