On the Zeros of a Class of Polynomials Defined by a Three Term Recurrence Relation

Let PN+1(x) be the polynomial which is defined recursively by P0(x) = 0, P1(x) = 1, and αnPn + 1(x) + αn − 1Pn − 1(x) + bnPn(x) = xdnPn(x), n = 1, 2,…, N, where αn, bn, dn are real sequences with αn ≠ 0, for every n = 1, 2, …, N, and k terms of the sequence {dn}∞n = 1, 0 ≤ k < N, are equal to zero. It is proved that if bn > 0 and the sequence {α2n/bnbn + 1}∞n = 1, is a chain sequence then the polynomial PN + 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.