Dr. Glaisher on Arithmetical Recurring Formula. 6 1 



together. Some of these theorems are given in the paper 

 just referred to, and others in a continuation which will 

 shortly be communicated to the same Society. In this note I 

 have only mentioned the two theorems which give rise to 

 cr-formulae by simple addition, and to the corresponding 

 ^-formulae by addition after changing the signs of the even 

 numbers, my object being merely to supply an arithmetical 

 explanation of the general resemblance, and differences in 

 detail, which are observable in corresponding 'pairs of a - - and 

 f-formulse. 



§ 18. The recurring formulae which are deducible from the 

 general theorems by adding the cubes and higher uneven 

 powers of the numbers in each group, and adding the same 

 powers after changing the signs of the even numbers, neces- 

 sarily exhibit the same kinds of resemblance and discrepancy 

 as the cr- and ^-formulas of §§ 1, 2, and 11. 



§ 19. Thus, for example, by adding the cubes of the numbers 

 in the general theorem of § 4 we find 



a s (n) — 3<r B {n — 1) +5cr 3 (n — 3) — 7a d (n — 6) + 9<7 3 (?2 — 10) — &c. 



-6{o-(7z-l)-(l 2 + 2 2 ) ( 7(n-3) + (l 2 + 2 2 + 3 2 )a(n-6)-&c.} 



= [(-l>- 1 {l 4 + 2 4 + 3 4 + ...+/}] ; 



where (r s (n) denotes the sum of the cubes of all the divisors 

 of n, and the term enclosed in square brackets only occurs 

 when n is a triangular number, \g{cj + 1) . When n is not a 

 triangular number the right-hand member of the equation is 

 zero. 



By adding the cubes of the numbers, after changing the 

 signs of the even numbers, we find 



W») +Ss(n-1) + ?,(n-3) + f,(n-6) +&(n-10) 



+ 6{£(«-l)-(l 2 -2 2 )f(«-3)+(l 2 -2 2 + 3 2 X(«-6)-&c.} 



= [(-l)»-i{l«-2« + 3*-... + (-l)»-yi], 

 where 5 3 (n) denotes the excess of the sum of the cubes of the 

 uneven divisors of n above the sum of the cubes of the even 

 divisors, and the square brackets have the same meaning as 

 in the case of the cr-formula. 



Reducing the expressions on the left-hand side of these 

 equations, they become respectively 



b- 3 (w)-3o- 3 (w--l)+5(73(n--3)--7cr 3 (n--6) + 9o-3(w--10)-&c. 



2{3(7(n-l)-5.3o-rn-3) + 7,6o-(w-6)-9.1Qo-(w-10)+&c] 



and 



? s (») + &(«-l) + ?3{«-3) + ? 8 (n-6)+? 3 (»-10) +&e. 



f6{?(?i-l) + 3?(n-3) + 6?(n-6) + 10?(n-10)+&c.j. 



