228 TAUNO METSÄNKYLÄ

Let us keep n and к fixai. First assume that /[/e. Then /a^=/0 =d or dp with d > 1. Since the order of ^^ is a p-power, ф{а)^1 (mod p) unless a is divisible by /. Thus we find, because Ö, Ф ф~^, that Щ){а)^в(а)ф{а) = 0(ö) (mod p) for all a e Z. Consequently, by (7), с^(#) = с^^(о) (mod p) which proves the assertion.

Secondly let {IJ^) = 1. Now /0^ = M or Idp with (/,d) = 1, and by using the fact that a^/ = (1 + bp)*"^^ (modp""^ ^) we can write

Си^Лвф ) = - ^ E '1;'^'0(5„(№0 + Ф"^^)^(5„(№0 + Ф"^^).

To get rid of ф, observe that among the numbers s„(riaj)-bip"'^^ (/ — 0,...,W~1) those divisible by / are precisely s„{riaJ)-h{i^-{-jl)p"^^ with j = 0,..., d — 1, where f^ is defined by

s „ { riaj ) + цр"^' = /5„(№), 0 ^ i, ^ / -1.

It follows that

ас^АОф ) ^ - I Y 10(5„(№0 + Ф"^^)

+ I Z' %'^jl)e{K{rioik)+i¥^') (modp). To reformulate this congruence, note that

= ÏjO(^+jp«^^) (modp)

for all I? e Z. If d > 1, we therefore infer, on recalling (9), that dc^^M) = dcj,UQ) - вШсМ (mod p).

Thus the proposition is established in this case. Now let d == 1. Then в ^ of and our congruence reduces to

с , + Д# ) s 0(0 E i,0(s„(№)) = I f,r (mod p)

or

( 11 ) Cik^-^e»**^) s - -^p^ E (««(^öCikO - ЫпчМ (mod p).