101 



sum of all the terms arising from giving to m every integral value from 

 n to r, zero being included when n is negative. We immediately see 

 that 



— n, r n r-t- 1 



mm m —Hi m m— 1 



— », CO X 



and the generating function oi u = S . u t . 



J X 



Art. 18. This most important Theorem may be proved more simply 

 as follows : 



CO CO 00 CO 



SSa =SSa 



m [n m, n n m m,n 



00 05 CO 00 



= Sa + Sa + Sa + &c. + Sa + &c. 



m m, 1 m m,2 m m,3 n m,n 



= a + a + a + a + &c. + a + &c. 



1, 1 2, 1 3, 1 4, 1 m, 1 



+ a + a + a + &c. + a + &c. 



1, 2 2, 2 3, 2 m— 1, 2 



+ a + a + &c. + a + &c. 



1, 3 2, 3 m— 2, 3 



+ a + &c. + a + &c. ' 



1. 4 m— 3, 4 



+ &c. + &c. + &c. 

 + &c. + a + &c. 



^ m—n+l, n 



+ &c. + &c. + &c. 

 + a + &c. 



1, m 



+ &C. 



12 3 4 m 



= Sa + Sa + Sa + Sa + &c. + Sa + &c. 



n 3— n, 1 n :>—«»» n 4— n, n n 6— n, n n m+l— «, » 



CD m 



= S.Sa 



m n m— n + l,fi ^ 



