75 
1920-21.] Hypergeometric Functions of Two Variables. 
both /3 and ft tend to infinity. These two functions will be denoted by 
the symbol 'F : their expressions are 
*i(«; fi;y,yi *, 2/) = ZlVfn"’ V”. 
(y> m )(y j w) m ! n ! 
^o( a ; y» 7 ; x > 2/) = 2Z 
From the function 
F 3 (a, a, ft ft; y ; i/)=22 
(a, m + n) x m y n 
(y, m)(y', 71) m \ n\ 
(a, m)( a', rc)(ft m)(ft, ») 
(y, m + n) 
! *, 1 
may be obtained in like manner a new function by dividing y by ft, and 
making ft-> oo . This is the function * 
H,(«, a'-, P;y,x, 2 /)=22 ( “’ " )(A ^ — 
(y, m + n) m \ n ! 
Similarly, replacing y by y/a'ft, and making a and ft infinite, we 
obtain a function 
b( a > P ; y ; ^ y) = 2 2 
(a, m)(ft ra) ic w 2/ r 
(y, m -j- w) m ! n ! 
Chapter II. 
VARIOUS EXPANSIONS FOR THE FUNCTIONS; RELATIONS BETWEEN THEM. 
The seven confluent functions which we have introduced, and defined by 
double power-series, may also be represented by simple power-series in x, 
or in y, by performing the process of confluence on the similar expressions 
given by Appell for the four F functions. We thus find 
, , 0 N xWa, m)(6, m) ,, , x x m 
$ i( a ; P m > y> x, y) = Zj , , — -$(a,+ ?», y+m, y) 
(y, rn) ml 
_ ^ (a, m) 
3T 
2 7 — — ( F(a + m, ft y + m: x)~\ 
,S(y , w) x r > / 
and similar formulae for the other confluent functions. 
We shall next consider formulae derived from the definite-integral 
values of the F functions, such as 
F 0 = 
r(y)r(y') 
- ux - vy)~*dudv. 
r(f3)T(B‘)T(y- mv - n 
We have 
(1 - ux-vy)~ a =( - l)" a [l - (1 - ux) - (1 - vy)]~ a 
= ( - ir“2 - «*)"( i - ^)- 
m ! n ! 
