L-orthogonal polynomials associated with related measures (original) (raw)

2010, Applied Numerical Mathematics

A positive measure ψ defined on [a, b] such that its moments μ n = b a t n dψ(t) exist for n = 0, ±1, ±2,. .. , can be called a strong positive measure on [a, b]. When 0 a < b ∞, the sequence of polynomials {Q n } defined by b a t −n+s Q n (t) dψ(t) = 0, s = 0, 1,. .. ,n − 1, exist and they are referred here as L-orthogonal polynomials. We look at the connection between two sequences of L-orthogonal polynomials {Q (1) n } and {Q (0) n } associated with two closely related strong positive measures ψ 1 and ψ 0 defined on [a, b]. To be precise, the measures are related to each other by (t − κ) dψ 1 (t) = γ dψ 0 (t), where (t − κ)/γ is positive when t ∈ (a, b). As applications of our study, numerical generation of new L-orthogonal polynomials and monotonicity properties of the zeros of a certain class of L-orthogonal polynomials are looked at.