288 History of the Theory of Numbers. [Chap, x 
and proved the second formula in Liouville's first article and the first two 
summation formulas of Liouville.-^ He proved 
i.(2.-l)=2.-l+z[^-^], .= [^], 
where 77 = 1 or according as 2o-— 1 is divisible by 3 or not. The last two 
were later generaUzed by Gegenbauer.^^ 
E. Lionnet^ proved the first two formulas of Liouville.^^ 
J. Liou^ille^ noted that, if q is divisible by the prime a, 
(r,(a5)+a''a-M^|j = (a'' + l)(r,(g). 
C. Sardi^^ denoted by A„ the coefficient of x" in Jacobi's^'' series for s^, 
so that An = unless n is a triangular number. From that series he got 
S(-l)P(2p+l)(7{7i-p(p + l)/2)=(-l)^'+^'/W3orO {t = Vl+8n), 
p 
according as n is or is not a triangular number, and 
|.4„+A„_,cr(l)+...+Ai(r(n-l)+Ao(r(n)=0. 
This recursion formula determines A„ in terms of the c's, or (T{n) in terms of 
the A's. In each case the values are expressed by means of determinants of 
order n. 
IM. A. Andreievsk}^^ wrote N^h^^i for the number of the divisors of the 
form 4/i± 1 of n = a'^h^ . . . , where a, b,. . . are distinct primes. We have 
where d ranges over all the di\'isors of n and the symbols are Legendre's. 
Evidently „ /-i\a' 
S (— ^) = a + l if a = 4Z + l, 
a'=o\ a } 
= or 1 if a = 4Z-l, 
according as a is odd or even. Hence, if any prime factor 4Z — 1 of n occurs 
to an odd power, we have iV4A+i=iV4A_i. Next, let *| 
where each p, is a prime of the form 4Z + 1 , each g, of the form 4? — 1 . Then 
iV4A+i-iV4.-i = (ai + l)(ao + l). . . =r(^), D = q,\^\ . .. 
^'Souv. Ann. Math., (2), 7, 1868, 68-72. 
"Jour, de math., (2), 14, 1869, 263-4. 
"Giomale di Mat., 7, 1869, 112-5. 
3%Iat. Sbomik (Math. Soc. Moscow), 6, 1872-3, 97-106 (Russian). 
