Orthogonal Polynomials for the Oscillatory-gegenbauer Weight

Gradimir V. Milovanović, Aleksandar S. Cvetković, Zvezdan M. Marjanović

This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional $\mathcal{L}:\mathcal{P}\to\mathbb{C}$, where $\mathcal{L}=\int_{-1}^1 p(x)\,d\mu(x)$, $d\mu(x)=(1-x^2)^{\lambda-1/2} \exp(i\zeta x)\,dx$, and $\mathcal{P}$ is a linear space of all algebraic polynomials. Here, we prove an extension of our previous existence theorem for rational $\lambda\in(-1/2,0]$, give some hypothesis on three-term recurrence coefficients, and derive some differential relations for our orthogonal polynomials, including the second order differential equation.