INTERLACING PROPERTIES OF ZEROS OF ASSOCIATED POLYNOMIALS

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:59

Article

PEHERSTORFER, F ; SCHMUCKENSCHLAGER, M

关键词: ZEROS;    INTERLACING PROPERTY;    ORTHOGONAL POLYNOMIALS;    FUNCTIONS OF THE 2ND KIND;    ASSOCIATED POLYNOMIALS;    JACOBI WEIGHTS;    JACOBI POLYNOMIALS;    ULTRASPHERICAL POLYNOMIALS;   

DOI  :  10.1016/0377-0427(94)00014-R

来源: Elsevier

【 摘 要 】

Let w be a nonnegative integrable weight function on [-1, 1] and let p(n+1)(x)=x(n+1) +... be the polynomial of degree n+1 orthogonal with respect to w. Furthermore, let p(n)((1))(x)=x(n) +... denote the polynomials associated with P-n+1 and p(n)((1-x2)) (X)=X(n) +.... the polynomials orthogonal with respect to the weight function (1-x(2))w(x). In this paper we give necessary and sufficient conditions such that the zeros of p(n)((1)) and p(n)((1-x2)) strictly interlace on [-1, 1] for large n. In particular this problem is studied for the Jacobi weights w(alpha beta)(x)=(1-x)(alpha)(1+x)(beta), alpha,beta epsilon(-1, infinity). In this case p(n)((1-x2)) = p'(n+1)/(n+1). For a large class of parameters, including, e.g. the ultraspherical case alpha=beta, it is shown that the interlacing property holds for each n is an element of N. Also a fairly complete description of the parameters for which the interlacing property does not hold is given.

【 授权许可】

Free