Classical Orthogonal Polynomials of a Discrete Variable

Abstract

The basic properties of the polynomials p _n (x) that satisfy the orthogonality relations

$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$

((2.0.1))

hold also for the polynomials that satisfy the orthogonality relations of a more general form, which can be expressed in terms of Stielties integrals

$$ \int_a^b {{p_n}(x)} {p_m}(x)dw(x) = 0\quad (m \ne n), $$

((2.0.2))

where w(x) is a monotonic nondecreasing function (usually called the distribution function). The orthogonality relation (2.0.2) is reduced to (2.0.1) in the case when the function w(x) has a derivative on (a, b) and w′(x) = ϱ(x). For solving many problems orthogonal polynomials are used that satisfy the orthogonality relations (2.0.2) in the case when w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ _i at the points x = x _i . In this case the orthogonality relation (2.0.2) can be rewritten in the form

$$ \sum\limits_i {{p_n}({x_i})pm} ({x_i}){\rho_i} = 0\quad (m \ne n). $$

((2.0.3))