3.3.3 THEORETICAL TESTS 79 Using the well-known formulas x= m(m-1) and 22 = m(m -1Pm -1) c 2 c 6 OOandcmodh=O. (20) Proof. We leave it to the reader to prove that, under these hypotheses, dh,k,c)+dk,h,c)=o(h,k,O)+rr(k,h,O)+~-6 i -3e(h,c)+3. (21) 11 (See exercise 6.) The lemma now must be proved only in the case c = 0. The proof we will give, based on complex roots of unity, is essentially due to L. Carlitz. There is actually a simpler proof that uses only elementary manipulations of sums (see exercise 7)-but the following method reveals more of the mathematical tools that are available for problems of this kind and it is therefore much more instructive. Let f(x) and g(s) be polynomials defined as follows: f(x) = 1+ Z +. . . + xk–l = (xk -1)/(X -1) g(x) = x + 2×2 + . . . + (k -l)xk- = zf (x) (22) = kxk/(x -1) - x(2 -1)/(x -l)!
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