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By W. W. Sawyer

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1.

Is given explicitly by \) (k-1) (r,n) p 2v ~ (-1)~ (k + 2v - ~ - 2) (k+v- 2)! (2\1)! ~! 1)! ~=O and is, up to a change of notation and normalization, the so-called Gegenbauer or "ultraspherical" polynomial, studied in any text on orthowe have chosen the normalization so as to make gonal polynomials; a polynomial with integral coefficients in k,r,n (actually, - . 1 a function of r of the polynomial where and (k-1) Pzv n in a minimal way would still have integral coefficients as for fixed (k-1) Pzv p~~-l) kE lN).

Cjl qni;r 2 ,a lj! j X= lj! in 1jJ Li E - JN and on (19) one checks that r E Z:::/2mNZ::: ) , qni;r 4nm- r N = (a, Q,) in 1jJ i. e. ) r Letting (so that C(r,L'l) depends we easily deduce that the coefficient L 1 N c*(r,L'l) X E NZ::: x Z, for depends only on C(r,r 2 -4nm) denote the coefficient of only on 2 (A. (T, z+h) A. (mod N) is any integer such that that the coefficient of m e C(R,L'l) R (mod 2Nm) R= r(mod 2m) Li (which depends, as it should, on1y on this to •1• - and on r (mod 2m)). we find that the coefficient of " 'V-'i'z,a qni;r in App1ying "j T 0 'i'"" is given by (20) I: c*(n,r) ak- 1 c* (r, r a 2 - 4nm) ad=2 2 wi th c* a re1a ted as in (19) to the coefficient q(R 2 -L'l)/4m~;R in " From eq.

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A Concrete Approach to Abstract Algebra by W. W. Sawyer


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