20 N. F. DUPUIS 



and the law of the coefiicieuts is established. 



C08 8= 1 — — H + 4- 



2 ! 4 ! (2k) ! 



IV. 



Expression of the General Beenoullian Number as a Combinational 



Determinant. 



Wheu the fuuctiou is expanded in the form 



e" — 1 



1 h_5, B.,+ (-) 



11+1 ^ 



2 2! 4! (2n)l 



the quantities denoted by Bi, B,, B„ are called the first, second, m"' BernouUian 



numbers. 



This definition gives directly 



x = .t- a + - + -+■•—•■ • f]l + -A-+ •••(-) 5"... [; 



(_ 2!3! n! )( 22! (2n)I ) 



whence equating coefficients of ay" in the expanded product gives — 



^^ ^ l(2n)! 3! (2n — 2)!^5 1 (2n — 4)T ^ 2(2n) ! (2w+ 1) ! J 



or multiplying through by {2n) ! 



/ ^..^ n (2n)! 5 I (2n) ! ^ _ ., (2n)\ _ (2w) I ]_ 



^~^ 1 " 3TX2n— 2)! "-'^ 5! (2n— 4)! '"- ^ ^ 2(i;n) ! (2k+1)!J 



But 



(2n) ! (2n)|_ _ 2w— 1 



2(2n)! (2n+l)! 2(2n+l). 



and 



(2w) ! _ (2n)l _ _g« 



(2?- + 1) ! (2» — 2r) ! (2r+.l)(2r) ! (2n— 2r) ! 2r+i' 



,-.(-)" |5,._icr5„-i+icrB„.,-+ +2^ 



2n — 1 



0; 



2(2n+l) ! 



whence 



2n— 1 



5,. = icr5,..i-iCf5„.,+ (_)"_!_ q;;.,B, + (-)'.+i 



2(2w + 1) 



