Complex numbers and polynomials - University College Cork

and, for k ≥ 2,c k =Again,( mk)b 0 +(( ) ( ) ( ) ( ) ( )m m m m mb 1 + b 2 = c + b + a.k − 1 k − 2 k k − 1 k − 2m∑i=0( mi)x i )(n∑j=0and comparing coeffs of powers of x,( ) m + n= ∑ ( mkiIn particular, if k = m = nk=0i=0i+j=k( nj)x j ) = (1 + x) m (1 + x) n( ) 2n=nj=0= (1 + x) m+nm+n∑( ) m + n=x k ,kk=0)( nj), k = 0, 1, 2 . . . , m + n.n∑( ) 2 n.kSince (1 − x 2 ) n = (1 + x) n (1 − x) n , we see in the same way thatn∑( ) nn∑( ) nn∑( ) n2n∑(−1) k x 2k = x i (−1) j x j = x ∑ ( ) ( )n n k (−1) j ,ki ji jwhenceIn particular,and2m+1∑i=02m∑i=0∑i+j=k( ) ( ) {n n(−1) j =i j( 2(2m + 1)( 4mi7.3 Interpolationik=0) ( )2(2m + 1)(−1) i 2m + 1 − i) ( ) ( 4m4m(−1) i = (−1) m2m − imk=0i+j=k0, if k is odd,(−1) k/2( nk/2), if k is even.= 0, m = 0, 1, 2, . . .), m = 0, 1, 2, . . .As we’ve mentioned, if we know the roots of a poly and their multiplicities, we candetermine the poly to within a constant multiple. What if we know the values thata poly takes on a prescribed set? Can we determine it? In geometric terms, can wealways find a poly of smallest degree to pass through a finite set of points in the(x, y)-plane? Since a poly is a function, the x-coordinates of the points better bedistinct, but the y-coordinates don’t have to be.21