ON THE ZEROS OF A CLASS OF POLYNOMIALS DEFINED BY A 3 TERM RECURRENCE RELATION

[摘要] Let P(N+1)(x) be the polynomial which is defined recursively by P0(x)=0, P1(x)=1, and alpha(n)P(n+1)(x)+alpha(n-1)P(n-1)(x)+b(n)P(n)(x)=xd(n)P(n)(x), n=1, 2,...,N, where alpha(n), b(n), d(n) are real sequences with alpha(n) not-equal 0, for every n = 1, 2, ..., N, and k terms of the sequence {d(n)}n=1 infinity, 0 less-than-or-equal-to k < N, are equal to zero. It is proved that if b(n) > 0 and the sequence {alpha(n)2/b(n)b(n+1)}n=1 infinity, is a chain sequence then the polynomial P(N+1)(x) is of degree N - k and has real and simple zeros different from zero. This result generalizes and simplifies previously known results with respect to the class of polynomials whose zeros are real and simple. (C) 1994 Academic Press, Inc.

[发布日期] 1994-03-01