150 History of the Theory of Numbers. [Chap, v 
where i ranges over the integers ^n which are prime to n, while pi, P2> • • • 
denote the distinct prime factors of n. If f{t) = l, then \l/{n, d)=n/d 
and (13) becomes 
^in)=n-i:^+X-^-...=n(l-^)(l-^).... 
Pi P1P2 \ Pi/ V P2/ 
Next, take f(t) = ao+ait-\-a2f+ .... Using hyperboUc functions, 
S/(0 = Jcoth(./2)=l+^-4+..., 
provided Z be replaced by nJXn)J_r{n), where 
/i(n) =/'(n) -a„ Mn) =f{n) -2a2, . . ., /_i(n) = ff{n)dn. 
Hence, since Ji(n) =(j){n), 
2/(0 =^/-i(^) +^^-i(n)/i(n) -^V_3(n)/3(n) + . . . . 
In particular, for f{t)=t'', we get ^^-(n). In Prouhet's^^ first formula, 5 
may be replaced by the g. c. d. A,,, b of a and h. The generalization 
J,{ah)=Ma)J,{h)j^^ 
is proved. From (12) we get by addition* 
(14) i\-]j,{j) = l'+2' 
y=iLjJ 
+ . . . +n*. 
Taking n = l, 2,..., n, we obtain equations whose solution gives Jk(n) 
expressed as a determinant of order n in which the elements of the last 
coluimi are 1, 1+2*, 1+2^+3*, . . ., while for s<n the sth column consists 
of s — 1 zeros followed by s units, then s twos, etc. For s>0, the element 
in the (s + l)th row and rth column in Glaisher's^"^ first determinant is 
1 or according as r/s is integral or fractional. 
J. Valyi^°^ used J2{n) ■^({>{n) in his enumeration of the n-fold perspective 
polygons of n sides inscribed in a cubic curve. 
H. Weber^os proved (10) for k = 2. 
L.Carlini209 gave without references (10), (11), (12), with</)(^) for J„(A;). 
E. Cesaro^io noted that (12) implies (10). For, if 2/(d) =F(n), we have 
by inversion (Ch. XIX), /(n) =Xii{d)F(n/d). The case f=Ji gives 
Jijn) _^IJLid) 
The latter is a case of G{n) ='2g{d) and hence, with (12) and 
W)QQ^2,Wf(^). 
♦This work, Mess. Math., 20, 1890-1, p. 161, for k = l, is really due to Dirichlet." Formula 
(14) is the case p = 1 of Gegenbauer's, p. 217. 
"^Math. Nat. Berichte aus Ungarn, 9, 1890, 148; 10, 1891, 171. 
"'EUiptische Functioncn, 1891, 225; ed. 2, 1908 (Algebra III), 215. 
"»Periodico di Mat., 6, 1891, 119-122. 
"o/fcid., 7, 1892, 1-6. 
