Web site templates - 3.3.3 THEORETICAL TESTS 81 which follows from the
3.3.3 THEORETICAL TESTS 81 which follows from the usual method of expanding the right-hand side into partial fractions. Moreover, if o(z) = (X -yl)(z -yz). . . (CC -y,), we have d(Yj) = (Yj -Yl). . . (Y.j -Yj-l)(Yj -Yj+1). . . (Yj -Ym); (28) this identity may often be used to simplify expressions like those in the left-hand side of (27). When h and Ic are relatively prime, the numbers w, w2, . . . , tik–l, ghP1 are all distinct; we can therefore consider formula (27) in the &&a; c ase of the polynomial (z -w). . . (zr - w~- )(x -5). . . (z -ch-l) = (x -l)(zh -1)/(x -1)2, obt aining the following identity in x: This identity has many interesting consequences, and it leads to numerous reci: procity formulas for sums of the type given in Eq. (26). For example, if we differentiate (29) twice with respect to x and let x + 1, we find that Replace j by h -j and by Ic - j in these sums and use (26) to get ~(kh,O)+ 3(h h -1) which is equivalent to the desired result. I Lemma B gives us an explicit function f(h, k, c) such that 4h, k, c) = f(h k c) -o(k h, c) (30) whenever 0 < h 5 k, 0 2 c < k, and h is relatively prime to k. From the definition (16) it is clear that a(k, h, c) = c(k mod h, h, c mod h). (31) Therefore we can use (30) iteratively to evaluate o(h, Ic, c), using a process that reduces the parameters as in Euclid s algorithm. Further simplifications occur when we examine this iterative procedure more closely. Let us set ml = k, m2 = h, cl = c, and form the following tableau: ml = alm2 + m3 cl =hmz+cz m2 = a2m3 + m4 Q = bzm3 + ~3 (32) m3 = a3m4 + m5 ~3 = km4 + ~4 m4 = a4m5 ~4 = km5 + ~5