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 ).
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
(
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}
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.
References
Appell, Paul ; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13 .Bateman, H. ; Erdélyi, A. (1953). Higher Transcendental Functions, Vol. I (PDF) . New York: McGraw–Hill.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 Academic Press, Inc. ISBN 0-12-384933-0 LCCN 2014010276 . ISBN 978-0-12-384933-5 .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 .
