ON A SYSTEM OF FUNCTIONS GENERATING CLASSICAL ORTHOGONAL SYSTEMS OF ALGEBRAIC POLYNOMIALS
In contrast to the two well-known methods for introducing classical orthogonal systems of algebraic polynomials (see [1–5], respectively), this paper proposes a completely different method for introducing these same polynomials. At the same time, integral representations for them that are different from the known ones are also given. The latter, in comparison with the Rodrigues formulas known in the literature, are simple and, what is more important, the integrands are not multivalued, but single-valued. In this paper it is proved that the system of functions I^(α,β)_n,n_1,a=æ_n,n_1,a 2π_i_C(1-xa)^n_1+ζ(xa)^-ζ Γ(n_1+ζ+α+1)Γ(-ζ+β+1)·dζ^m_v=0(ζ+v), where n_1≥ n≥ 0 are integers, α>-1, β>-1, a>0 are arbitrary numbers, x(0,a), the simple contour C encloses neighborhoods of points 0,-1,-2,...,-n, æn,n_1,a=Γ(n_1+n+α+β=2), generates the above-mentioned orthogonal polynomials, namely: for n_1=n