1920-21.] Hypergeometric Functions of Two Variables. 81 
We shall, in general, denote by W kjfXiV (x, y) a solution of this system 
with the condition that it reduces to W ktfJi (x) for y = 2i/ + l = 0, and to 
W k, v (y) f° r ^ = 2^ + 1 = 0. 
The general solution of the system S is readily found to be of the form 
2/) T y ) + fx f ~ y) “l - ~ fJlj _ v (x, ?/), 
the C’s being arbitrary constants. 
Numerous recurrence formulas may be written for the M function. We 
shall only give the following as an example: 
M 
& — v ' 
dx~~ K ' tx ' v V ' * 2/ ‘ 2^+1 
The expansions for ”T 2 furnish analogous results. W 7 e thus obtain, 
bearing in mind the definition of the one- variable M function and its 
relation with <E>, 
and 
M lx v) f-;ya!" Wi ('‘ + l '" i;+1 ’ m > M /.a 
V (2/t+l, m)m! •’fe) 
M (a v ) - v m+ " +i (_ t + r ~ k + 1 , m) 
2d (2v + 1, m) m ! 
By transforming the formula 
^ 2 ( a ; y>y*> v) = ( - 1) a X2- a * ? + T^ ( y, *)$(-», y , 2 /) 
m ! n ! 
we obtain 
M, „ m y)={- 
7« 71 • '' J • 
Let us consider the one-variable M functions which occur under the 
symbol of summation. The expansion of the first one is 
M 
( -• m, p) 
pto(2/x-+L^! 
but, as m is an integer, the product ( — m, p) vanishes whenever p is 
greater than m, so that the sum represents not an infinite series, but a 
polynomial of degree m in x, 
The question is now, what is this polynomial ? Let us write 
= ( - rn t ™ - 2) 
^(2/x+l, m-q)(m-q) ! 
q=m X m ~ q 
= r(2/x + t)m ! — l) w_? — 77 , ; lVO , r 
v A ’ f-fp ’ q \ (m - q) \ T(2 /jl + m — q) 
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