1920-21.] Hypergeometric Functions of Two Variables. 
73 
IX. — The Confluent Hypergeometric Functions of Two Variables. 
By Pierre Humbert. Communicated by Professor E. T. Whit- 
taker, F.R.S. 
(MS. received December 6 , 1920. Read January 10 , 1921.) 
Introduction. 
This memoir is devoted to the study of certain new functions, which may 
be regarded as limiting cases of the “ hypergeometric functions of two 
variables ” discovered by Appell.* The relation which the new functions 
bear to Appell’s functions is, in fact, analogous to that which the 
“ confluent hypergeometric functions,” *j* 
and 
z \ , aX , a(a +1) 9 
* (w) =i + _ + _y_^ 
B(y? x ) — 1 + 
7 2 ! 7(7 + 1)" 
x x 5 
+ . 
+ . . . , 
I.7 2 ! 7(7 + 1) 
bear to the ordinary hypergeometric function. 
There are four hypergeometric series of two variables. If we denote 
the product 
X(X + 1) . . . (X + n-1), 
where \ is arbitrary and n a positive integer, by the symbol 
(X, n), 
with the convention (X, 0) = 1, these functions are as follows: — 
F 3 ( a > a '> A P ' ; y;^?) = 22 
F 4 («, P; y,y; x, 
= + 0 O 
(a, m + n)(p, 
n) 
X m i/ n 
Zj 
1=0 
( 7 , m + n) 
m\ n\ 
■ (a, 
m + n)({3, n) 
x m y n 
i 
( 7 , m)( 7 ', n) 
m 
! n ! 
(a, 
m)(a\ n)(f3, m)(/3', 
n) 
x m y n 
1 
( 7 , m + n) 
m\ n\ 
(a, 
m + n)(/3, m + n ) x m y n 
1 
V 
T 
! n 
t ’ 
* J. math, pures appl . , 1882, p. 173 ; 1884, p. 407. 
+ For an account of the confluent hypergeometric functions, see chapter xvi of 
Whittaker and Watson’s Modern Analysis. 
