1920-21.] Hypergeometric Functions of Two Variables. 83 
so that we can write, for an M function of the aforesaid type, 
x+y 
k- 
ct+&+3 
m av V 2 * e 2 r(fl + })r(& + })22( i) 
„a + i ^ 2(a+m+l)( ^ 
(a + b + 3 \ d“ +1 
x \ 9 -k, m + n) x 
d 6+1 
dy 
rU, 
6+1 ^ «(5+rc+l) 
(>/*)• 
As any differential coefficient of the U polynomials can be expressed in 
terms of the U themselves, we can express any function M where ya and v 
are of the type 9 + \-\-\ i n terms of Hermite’s polynomials. 
£ JL 
Let us take, in particular, a = b= — 1 ; we have at once, with a change 
of variables, 
Mi 
-(j, y f) - V? (-■ x 2 Z ( ■ - 1 r +, *(i - *. « + ") u -(^) u «(^> 
This formula connects the special M function with the parabolic-cylinder 
functions. 
Chapter VI. 
CONNECTION BETWEEN CERTAIN KNOWN FUNCTIONS AND THE 
CONFLUENT HYPERGEOMETRIC FUNCTIONS. 
Several functions of two variables introduced by different authors can 
be connected with some of the seven confluent functions of two variables. 
Of this we shall give three examples. 
1. The Two-variable Polynomials A m>n of Appell . — It is a well-known 
fact that limiting cases of a great number of one-variable polynomials are 
expressible by the W& ; m function or by Bessel functions. For instance, as 
anyone knows, for Legendre functions we have 
J m(x). 
We can establish a similar property for certain two- variable polynomials. 
Let us consider the two-variable polynomials discussed by Appell,* 
and defined by 
A m, J x, y) = X 1 V y (l - x- y) y+y ' 
As shown by Appell himself, they can be written under the form 
A Wt „ = (y, m){ 7, n)( 1 - a - y) m+n F 2 (y + y - 8 ; - m, - n; y, y'; 
x + y - 1’ x + y - 1/ 
Archiv Math. Phys lxvi, 1881, p. 238. 
