[102 
1851), pp. 45—47.] 
erties of the function 
be found interesting 
series. 
?r values of r from 
(1) 
:>f n). Hence 
(2) 
it series of fractional 
positive or negative 
in this case being 
a definition of the 
102] ON A DOUBLE INFINITE SERIES. 9 
and also 
S* [n + n'... - If {a (x + x'.. .)}n+«'...-i-k = (7 n+ri , *»<*+*..•>. 
Multiplying the first set of series, and comparing with this last, 
G n+ n'...Sr./... [n - l] r [»' - 1Y ... x n ~^x' n '~^... 
= C n C n '... [n + n' ... — l] fc (x + x'.. .) n+n '- -1 (3) 
(where r, r' denote any positive or negative integer numbers satisfying r + r' + ... = k + 1 — p, 
p being the number of terms in the series n, n', ...). This equation constitutes a 
multinomial theorem of a class analogous to that of the exponential theorem contained 
in the equation (2). 
In particular 
C n+n .... 2 r y... [» - If [n'- If ...= G n G n > (4) 
and if p — 2, writing also m, n for n, n, and k — 1 — r for r', 
G m+7) Xr [w — If [n — l] fc_1_r = G m C n [m + n — l] fc 2 m+n ~ 1 ~ k , (5) 
or putting k = 0 and dividing, 
C m C n + G m+n = 2 ~— x [m - If [n - (6) 
Now the series on the second side of this equation is easily seen to be convergent 
(at least for positive values of m y n). To determine its value write 
F (m, n) = j x m ~ x (1 — x) n ~ l dx; 
J 0 
then 
F (m, n) = I x m ~ l (1 — x) n ~ x dx + f x n ~ x (1 — x) m ~ x dx; 
J 0 Jo 
and by successive integrations by parts, the first of these integrals is reducible to 
om+n-i ->■ [ m ~ l] r [ n — l] -1-r , r extending from — 1 to — oo inclusively, and the second to 
S r [m — l] r [w — l] _1_r , r extending from 0 to oo; hence 
jL 
F (™> n ) = 2 mln-1 [ m — l] r — l]- 1 - r , 
or k> m G n — G m + n — F(w, n), (7) 
2 
C. II.