On polynomials orthogonal on a circular arc

Abstract

Polynomials {π R k } orthogonal on a circular arc with respect to the complex inner product ( f,g) = ∫ π-ϕ ϕ f 1(θ) · g 1(θ) w 1(θ) dθ, where ϕ ϵ (0, 1 2 π), and for f( z) the function f 1(θ) is defined by f 1(θ) = f(−i R+e iθ( R 2 + 1) 1 2 ), R = tan ϕ, have been introduced by de Bruin (1990). In this paper the functions of the second kind, as well as the corresponding associated polynomials, are introduced. Some recurrence relations and identities of Christoffel-Darboux type are proved. Also, the corresponding Stieltjes' polynomials which are orthogonal to all lower-degree polynomials with respect to a complex measure on Γ R = { z ∈ C : z = −i R + e iθ ( R 2 + 1) 1 2 , ϕ ⩽ θ ⩽ π - ϕ, tan ϕ = R} are investigated. A class of polynomials orthogonal on a symmetrical circular arc in the down half plane is also introduced. Finally, in the Jacobi case w( z) = (1 − z) α (1 + z) β, α,β > −1, a linear second-order differential equation for π R n ( z) is obtained.