Some Properties of Wigner Polynomials

Such well-known scientists as Legendre, Gegenbauer, Jacobi, Lager and others were engaged in the study of various properties of orthogonal polynomials. They introduced the concept of various polynomials and determined their properties. With these polynomials the series in orthogonal polynomials are composed and the issue of representing a function with these series is studied; the convergence and summability of these series are also studied by various methods. In the presented paper, the so-called Wigner polynomials are considered. These polynomials are involved in the definition of generalized spherical functions. Therefore, determining the properties of these polynomials would be useful for studying the convergence and summability of Fourier series with respect to generalized spherical functions. In this paper, some basic properties of Wigner polynomials are studied. In particular, asymptotic formulas and some estimates for these polynomials are determined.