Humbert series

The Humbert series Φ₁ is defined for |x| < 1 by the double series:

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

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

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

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

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:

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

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:

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

the function Φ₃ for all x, y by the series:

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

the function Ψ₁ for |x| < 1 by the series:

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

the function Ψ₂ for all x, y by the series:

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

the function Ξ₁ for |x| < 1 by the series:

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

and the function Ξ₂ for |x| < 1 by the series:

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

• Main article: Appell series

There are four related series of two variables, F₁, F₂, F₃, and F₄, which generalize Gauss's hypergeometric series ₂F₁ 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.