# Classical Orthogonal Polynomials of a Discrete Variable

Chapter

## Abstract

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

*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)

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

(2.0.2)

*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)

## Keywords

Difference Equation Orthogonal Polynomial Recursion Relation Discrete Variable Hermite Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1991