Chap. Ill] SYMMETRIC FUNCTIONS MODULO p. 99 
It is shown that (1^^^)" equals the number of combinations of n-\-p — l 
things p — 1 at a time. Various algebraic relations between binomial 
coefficients are derived. 
L. Gegenbauer^^^ considered the polynomial 
p-2+k 
f{x)= S hix' {l-p<k^p-l) 
i=0 
and proved that 
V/(X)/X^-2= -h,.2 (mod p), k<p-l, 
X=l 
'xf{X)/\^-'= -6p_2-fc2p-3 (mod p), k = p-\, 
X=l 
and deduced the theorem on the divisibility of s„ by p. 
E. Lucas^^^ proved the theorem on the divisibility of s„ by p by use of the 
symbolic expression (s+l)"— s" for x" — 1. 
N. Nielsen^^^" proved that if p is an odd prime and if k is odd and 
\<k<p — \, the sum of the products of 1, . . ., p — 1 taken A; at a time is 
divisible by p^. For k=p—2 this result is due to Wolstenholme.^^^ 
N. M. Ferrers^^^ proved that, if 2n+l is a prime, the sum of the products 
of 1, 2, . . . , 2n taken r at a time is divisible by 2n+l if r<2n [Lagrange^^], 
while the sum of the products of the squares of 1, . . . , n taken r at a time is 
divisible by 2n+l if r<n. [Other proofs by Glaisher.^^^] 
J. Perott^^^ gave a new proof that s^ is divisible by p if n<p — l. 
R. Rawson^^^ proved the second theorem of Ferrers. 
G. Osborn^^° proved for r<p — l that s^. is divisible by p if r is even, by 
p^ if r is odd; while the sum of the products of 1, . . ., p — l taken r at a 
time is divisible by p^ if r is odd and l<r<p. 
J. W. L. Glaisher^^^ stated theorems on the sum Sriai,..., a,) of the 
products of fli, . . . , rti taken r at a time. If r is odd, Sr{l, . . . , n) is divisible 
by n+1 (special case n+1 a prime proved by Lagrange and Ferrers). If r 
is odd and > 1, and if n+1 is a prime> 3, Sr{l, . . . , n) is divisible by {n-{-iy 
[Nielsen^^^'']. If r is odd and >1, and if w is a prime >2, Sril,. . ., n) is 
divisible by n^. If n+1 is a prime, Sr{l^,. • ■, n^) is divisible by n+1 for 
r = l,...,n — 1, except for r = n/2, when it is congruent to ( — 1)1+"/^ j^odulo 
n+1. If p is a prime ^n, and k is the quotient obtained on dividing n+1 
by p, then aSp_i(1,..., n)=— A; (mod p); the case n = p — 1 is Wilson's 
theorem. 
"^Sitzungsber. Ak. Wiss. Wien (Math.), 95 II, 1887, 616-7. 
"«Th6orie des nombres, 1891, 437. 
286aNyt Tidsskrift for Mat., 4, B, 1893, 1-10. 
"'Messenger Math., 23, 1893-4, 56-58. 
288BuU. des sc. math., 18, I, 1894, 64. Other proofs. Math. Quest. Educ. Times, 58, 1893, 109; 
4, 1903, 42. 
"•Messenger Math., 24, 1894-5, 68-69. 
""Ibid., 25, 1895-6, 68-69. 
"'Ibid., 28, 1898-9, 184-6. Proofs"*. 
