1920-21.] Hypergeometric Functions of Two Variables. 
79 
known relations between contiguous hypergeometric functions of one 
variable; thus 
£?*!(<*+ 1; P+ 1 ; y+l; x,y)+ ^(a + 1; /?; y+1; x, y) 
7 7 
= + 1 ; /? ; y ; y) - ^i(<* ; P ; y ; ^ y) 
$i(a ; /3 + 1 ; y ; x, y) = ^(a '> Pi 7 i y) + — ' $i( a + 1 ; £ + 1 i 7 + 1 i ^ 2/)- 
7 
The relation 
iU-i(a; P; y , x, y) = “£*>,(<* + 1 ; 0 + 1; y+1; *, ?/) 
shows that the derivates of the < h 1 function are expressible in terms of the 
function itself. 
Similar formulae may be obtained for d? 2 . 
<3^ may be expressed by a simple definite integral 
*i(«J Pi 7 ; y) - r(a)r ( (y - «) f ~ “) T ' a ~ 1 ( 1 - 
while <f> 2 may be expressed by the double integral 
ri * »> - rmnml-f -<?> // *‘~ v " (1 - - 
(u^O, v^O, l - u - v^O). 
Formulae of the same type may be obtained for the H functions : thus 
we have 
ft ’ “> ^ *’ y > = r(.)r(,Wy- a -,-) // 
the field being the same as above. 
\y-a — a'-l 
dudv . 
Chapter V. 
THE FUNCTION AND ITS TRANSFORMATIONS. 
The function Tg proves to be the most interesting of the seven, as its 
properties afford a very direct generalisation of the one- variable confluent 
hypergeometric function. To render this fact more conspicuous, we shall 
substitute for A r 2 a new function, just as Whittaker * studied, instead of T>, 
his functions M or W. 
We therefore make the following change of parameters: 
a = /x -f v — /*' -f 1 
7= 2/x-f 1 
y =2v+ 1, 
* Bull. Amer. Math. Noc., iv, p. 125. 
