Krawtchouk polynomials

Krawtchouk polynomials

[¦kräv‚chək ‚päl·ə′nō·mē·əlz]
(mathematics)
Families of polynomials which are orthogonal with respect to binomial distributions.
References in periodicals archive ?
(x - n + 1) [sigma] (X) = 1 translated Charlier polynomials [sigma] (x) = x falling factorials, Charlier, Meixner, Krawtchouk polynomials deg([sigma](x), x) = 2 Hahn polynomials
show that for [alpha] = 1/4 translated Laguerre polynomials and for [alpha] < 1/4 Meixner and Krawtchouk polynomials are solutions of (6.1).
The polynomials we consider here can be treated as two-dimensional q-analogs of Krawtchouk polynomials. Some properties of these polynomials are investigated: the difference equation of the Sturm-Liouville type, the weight function, the corresponding eigenvalues including the explicit description of their multiplicities.
In the case of the group SO(5), as shown in [1], investigation of the eigenfunctions of generators of representation lead to a class of orthogonal polynomials that may be considered as two-dimensional analogs of classical Krawtchouk polynomials [3].
As it is shown in [1], equation (3.7) can be considered as two-dimensional analog of the equation for Krawtchouk polynomials. Equation (3.6) can be treated as twodimensional q-analog of the equation for Krawtchouk polynomials.