L-orthogonal polynomials associated with related measures

摘要

A positive measure ψ defined on [a,b] such that its moments μ_n =∫_a~b t~n dψ(t) exist for n = 0, ±1, ±2,…. Can be called a strong positive measure on [a, b]. When 0 ≤ a ≤ b ≤ ∞,rnthe sequence of polynomials |Q_n| defined by ∫ _a~b t~(-n+s) Q_n(t)dψ(t) = 0, s = 0,1…..n - 1,rnexist and they are referred here as L-orthogonal polynomials. We look at the connection between two sequences of L-orthogonal polynomials {~_n~((1))} and {Q_n~((0))} 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 - k) dψ_1(t) = γ dψ_0(t) . Where (t-k)/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.