Three Finite Classes of Hypergeometric Orthogonal Polynomials and Their Application in Functions Approximation

Abstract

Classical orthogonal polynomials of Jacobi, Laguerre and Hermite are characterized as the infinite sequences of orthogonal polynomials so that they satisfy a second order differential equation of the form <$$>(ax^{2} + bx + c) y_{n}^{\\prime \\prime} (x) + (\\hbox{d}x + e) y_{n}^{\\prime} (x) - n ((n - 1) a + {d}) y_{n} (x) = 0; \\quad n \\in Z^{+}</$$> in which a,b,c,d,e are parameters independent of n. In this work, we present three other sequences of hypergeometric polynomials which are special solutions of above equation and are finitely orthogonal with respect to three particular weighting functions on infinite intervals. These classes have respectively relation with the Jacobi, Ultraspherical and Laguerre polynomials, in particular, third class is directly related to generalized Bessel polynomials and has also relation with the Laguerre polynomials. General properties of these sequences such as Orthogonality relation, Rodrigues type formula, Recurrence relations, Shift operators and Generating function are indicated. Under the Dirichlet conditions the function f ( x ) is approximatable in terms of finite sum of each of these classes so that one can consider any arbitrary precision degree n = N for foresaid approximations. Finally, estimating three kinds of definite integrals using Gauss integration theory and polynomials weight functions are introduced.