Humbert series
# Humbert series

In mathematics, Humbert series are a set of seven hypergeometric series $Φ_1,$ $Φ_2,$ $Φ_3,$ $Ψ_1,$ $Ψ_2,$ $Ξ_1,$ $Ξ_2$ of two variables that generalize Kummer's confluent hypergeometric series $_1F_1$ of one variable and the confluent hypergeometric limit function $_0F_1$ of one variable. The first of these double series was introduced by Pierre Humbert (1920).

## Definitions

The Humbert series $Φ_1$ is defined for |x| < 1 by the double series:

    $\Phi _{1}(a,b,c;x,y)=F_{1}(a,b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$

where the Pochhammer symbol $(q)_n$ represents the rising factorial:

    $(q)_{n}=q\,(q+1)\cdots (q+n-1)={\frac {\Gamma (q+n)}{\Gamma (q)}}~,$

where the second equality is true for all complex $q$ except $q=0,-1,-2,\ldots$.

For other values of x the function $Φ_1$ can be defined by analytic continuation.

The Humbert series $Φ_1$ can also be written as a one-dimensional Euler-type integral:

    $\Phi _{1}(a,b,c;x,y)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b}e^{yt}\,\mathrm {d} t,\quad \Re \,c>\Re \,a>0~.$

This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.

Similarly, the function $Φ_2$ is defined for all x, y by the series:

    $\Phi _{2}(b_{1},b_{2},c;x,y)=F_{1}(-,b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$

the function $Φ_3$ for all x, y by the series:

    $\Phi _{3}(b,c;x,y)=\Phi _{2}(b,-,c;x,y)=F_{1}(-,b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$

the function $Ψ_1$ for |x| < 1 by the series:

    $\Psi _{1}(a,b,c_{1},c_{2};x,y)=F_{2}(a,b,-,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}(b)_{m}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,$

the function $Ψ_2$ for all x, y by the series:

    $\Psi _{2}(a,c_{1},c_{2};x,y)=\Psi _{1}(a,-,c_{1},c_{2};x,y)=F_{2}(a,-,-,c_{1},c_{2};x,y)=F_{4}(a,-,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m}y^{n}~,$

the function $Ξ_1$ for |x| < 1 by the series:

    $\Xi _{1}(a_{1},a_{2},b,c;x,y)=F_{3}(a_{1},a_{2},b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a_{1})_{m}(a_{2})_{n}(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~,$

and the function $Ξ_2$ for |x| < 1 by the series:

    $\Xi _{2}(a,b,c;x,y)=\Xi _{1}(a,-,b,c;x,y)=F_{3}(a,-,b,-,c;x,y)=\sum _{m,n=0}^{\infty }{\frac {(a)_{m}(b)_{m}}{(c)_{m+n}\,m!\,n!}}\,x^{m}y^{n}~.$

## Related series

* Main article: Appell series

    There are four related series of two variables, $F_1,$ $F_2,$ $F_3,$ and $F_4,$ which generalize Gauss's hypergeometric series $_2F_1$ of one variable in a similar manner and which were introduced by Paul Émile Appell in 1880.
