CHAP. X] SUM AND NUMBER OF DlVISORS. 295
in which the differences between the arguments of <r in the successive terms are 2, 4, 6, 8, . . . , and those between the coefficients are 5, 10, 15, . . . , while <r(0) =0. Finally, there is a similar recursion formula for A(n).
Glaisher64 proved his43 recursion formula for 0(n), gave a more complicated one and the following for a(ri) :
where s = r unless r<r(l) is the last term of a group, in which case, s=r-fl. He proved Jacobi's11 statement and concluded from the same proof that E(n) =TLE(ni) if n-Hrii, the n's being relatively prime. It is evident that E(pr)*=r+l if p is a prime 4ra-fl, while E(pr)*=l or 0 if p is a prime 4m +3, according as r is even or odd. Also i?(2r) = l. Hence we can at once evaluate E(ri) . He gave a table of the values of E(ri) , n = 1, . . . , 1000. By use of elliptic functions he found the recursion formulae
... =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-8)+E(n-10)~. . . = 0 or (-l)»{(-i)«-«/«i_i}/4,
according as n is not or is a triangular number 1, 3, 6, 10, . . .. He gave recursion formulae for
S(2n) = E(2) +#(4) + . . . +E(2n),
The functions E, SjO, cr are expressed as determinants. J. P. Gram640 deduced results of Berger51 and Cesaro.54 Ch. Hermite66 expressed <r(l)+<r(3)+. . .+<r(2n-l), <r(3)+<r(7)+ . . .
+<r(4n-l) and o-(l)+<r(5)+ . . . +(r(4n + l) as sums of functions
Chr. Zeller66 gave the final formula of Catalan.42
J. W. L. Glaisher67 noted that, if in Halphen's40 formula, n is a triangular number, <r(n—ri) is to be given the value n/3; if, however, we suppress the undefined term <r(0), the formula is
cr(n)-3<r(n-l)+5<r(n-3)-- . . . =0 or (-l)f-1(l2+22+ . . .+r2),
according as n is not a triangular number or is the triangular number r(r+l)/2. He reproduced two of his63'64'76 own recursion formulas for <r(ri) (with \l/ for cr in two) and added
- {cr(n-15) + . ..} + ... =A-B,
"Proc. London Math. Soc., 15, 1883-4, 104-122.
•^Det K. Danske Vidensk. Selekabs Skrifter, (6), 2 1881-6 (1884), 215-220 296.
68Amer. Jour. Math., 6, 1884, 173-4.
"Acta Math., 4, 1884, 415-6.
"Proc. Cambr. Phil. Soc., 5, 1884, 108-120.