Polynomials orthogonal on the semicircle. (English) Zbl 0604.42024

The authors study the complex polynomials \(\{\pi_ n\}\) which are orthogonal with respect to the complex-valued inner product \((f,g)=\int^{\pi}_{0}f(e^{i\theta})g(e^{i\theta})d\theta.\) For these polynomials, they obtain a three-term recurrence relation, a linear differential equation of second order and discuss the nature of the zeros. A relationship between these polynomials and the Legendre polynomials is worked out. Finally, they discuss some interesting applications of these polynomials.
Reviewer: A.N.Srivastava


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C55 Spherical harmonics
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)


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