Chap. X] SuM AND NuMBER OF DiVISORS. 295 
in which the differences between the arguments of a in the successive terms 
are 2, 4, 6, 8, ... , and those between the coefficients are 5, 10, 15, ... , while 
o-(O) =0. Finally, there is a similar recursion formula for A(n). 
Glaisher^^ proved his^^ recursion formula for Q{n), gave a more compli- 
cated one and the following for (j{n) : 
o-(n)-2{o-(n-l)+o-(n-2)) +3{(7(n-3)+(r(n-4)+(7(n-5)! - . . . 
+ (-irV{...+(r(l)) = (-l)V-s)/6, 
where s = r unless ra-(l) is the last term of a group, in which case, s = r+l. 
He proved Jacobi's^^ statement and concluded from the same proof that 
E{n) =JlE{ni) if n=nn„ the n's being relatively prime. It is evident that 
E{p')=r-\-l if p is a prime 4m+l, while £'(pO = l or if p is a prime 
4m+3, according as r is even or odd. Also £'(2'') = 1. Hence we can at 
once evaluate E{n). He gave a table of the values oi E{n),n = \,. . ., 1000. 
By use of elliptic functions he found the recursion formulae 
E{n)-2E{n-^)+2E{n-\io)-2E{n-m)+ . . . =0 or (-l)'^-^^/^^, 
for n odd, according as n is not or is a square; for any n. 
E{n)-E{n-l)-E{n-S)+E{n-6)-{-E{n-10)- . . . 
= or (-ir{(-l)('-i)/2^-l}/4, ^^ Vs^^fl, 
according as n is not or is a triangular number 1, 3, 6, 10, . . .. He gave 
recursion formulae for 
S{2n) =E(2)+E(4)+ . . . +E{2n), 
S{2n-l)=E{l)-\-E{S)+ . . .+Ei2n-1). 
The functions E, S, 6, a are expressed as determinants. 
J. P. Gram^^" deduced results of Berger^^ and Cesaro.^'* 
Ch. Hermite^^ expressed (T(l)+<r (3) + . • - +o-(2n-l), (t(3) +o-(7) + . . . 
+o-(4n — 1) and o-(1)+<j(5)+ . . . +(7(4n+l) as sums of functions 
E,{x)=^{[xf-^[x\]/2. 
Chr. Zeller^^ gave the final formula of Catalan.^^ 
J. W. L. Glaisher®^ noted that, if in Halphen's^" formula, n is a triangular 
number, (T{n—n) is to be given the value n/3; if, however, we suppress the 
undefined term (7(0), the formula is 
(T(n)-3(j(n-l)+5(7(n-3)- . . . =0 or {-lY-\l''+2''+ . . .+r''), 
according as n is not a triangular number or is the triangular number 
r(r+ 1)/2. He reproduced two of his^^'^"*'^^ own recursion formulas for 
<T{n) (with yp for <j in two) and added 
o-(n)-{(7(n-2)+o-(n-3)+(r(n-4)j + !(7(n-7)-f(r(n-8)+(7(n-9) 
+o-(n-10)+(7(n-ll)[-{(T(n-15)+...) + ...=A-B, 
«^Proc. London Math. Soc, 15, 1883-4, 104-122. 
"°Det K. Danske Vidensk. Selskabs Skrifter, (6), 2 1881-6 (1884), 215-220 296. 
"^Amer. Jour. Math., 6, 1884, 173-4. 
6«Acta Math., 4, 1884, 415-6. 
6Troc. Cambr. Phil. Soc, 5, 1884, 108-120. 
