310 History of the Theory of Numbers. [Chap, x 
Glaisher"" considered the set Gn[\l/(d), x(^>- •• ) of the values of \l/(d), 
x(cO, • • • when d ranges over all the di\isors of n, and wrote —G{\J/, x,-) 
for G{—\f/, — X, • •)• By use of the ^-function (12), he proved (p. 377) that 
the numbers given by 
GM-Gn-i(d, d=^l)+Gn-z{d, d^l, d=^2)-G,.M d^l, ^±2, ^±3) + . . . 
all cancel if n is not a triangular number, but reduce to one 1, two 2's, 
three 3's, . . ., g g's, each taken with the sign ( — )*~\ if n is the ^th tri- 
angular number ^(gr+l)/2. For example, if n = 6, whence g = S, 
{1, 2, 3, 6) - {1, 5; 2, 6; 0, 4} + {l, 3; 2, 4; 0, 2; 3, 5; -1, 1} 
= {1,2,2,3,3,3). 
Let \l/{d) be an odd function, so that \f/{ — d)=—\l/{d), and let 2r/(c?) 
denote the sum of the values of f(d) when d ranges over the divisors of r. 
Then the above theorem implies that 
2„iA(d) -2„-i iHd) +4^{d=^ 1) ) +2„_i_2 [rPid) +rl^{d^ 1) -{-^p{d^ 2) ) 
-2„_i_2-3 {rpid) +yp{d^ 1) +iA(d± 2) +^(d± 3) 1 + . . . 
= 5( - 1)''-^ {,^(1) +2^(2) +3V^(3) + . . . +g^l^{g) } , 
where 5 = or 1 according as n is not or is of the form ^(^+l)/2, and where 
\l/{d^i) is to be replaced by \}/{d+i)-{-yl/{d — i). Taking \l/{d) =d"', where m 
is odd, we obtain Glaisher's^°^ recursion formula for (Tm{n), other forms of 
which are derived. For the function^^ ^3, we derive 
Un)+Un-1)+Un-S) + . . .+Q{nn-l)-{l'-2'mn-3) 
+ (l2-22+3^)r(n-6)-...) 
= (-l)''-^(l^-2H3'- . . . +(-l)'-y) or 0, 
according as n is of the form ^(^+l)/2 or not. 
Next he proved a companion theorem to the first : 
2d-\-7 \, 
-[2d-7];+ 
^/ 2d+l \ ^ / 2d+3 V^ / 2d+5 \ r f 
all cancel if n is not a triangular number, but reduce to 1, 3, 5,. .., 2^ — 1, 
each taken with the sign ( — )", together with ( — l)''+^(2^+l) taken g 
times, if n is the gth triangular number ^(^+l)/2. For example, if n = 6, 
{-tt J-;?}-{-t"M-3;-"}={-'-'-- ^- ^- ^}- 
Hence if x(c^) be any even function, so that x(—d) = x{d), 
2„{x(2d+l)-x(2d-l)l-2„_i{x(2ci+3)-x(2d-3)l+S„_3-.... 
= 5(-l)''-M^x(2^ + l)-x(l)-x(3)- . . . -x(2^-l)l. 
Taking ^(A;) = fc'""*"\ where k and m are odd, we get Glaisher's^°^ formula. 
"oProc. London Math. Soc, 22, 1890-1, 359-410. Results stated in London, Edinb., Dublin 
Phil. Mag., (5), 33, 1892, 54-61. 
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