Hahn polynomials
Encyclopedia
In mathematics, the Hahn polynomials are a family of orthogonal polynomials
in the Askey scheme
of hypergeometric orthogonal polynomials, introduced by Chebyshev in 1875 and rediscovered by . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials
. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.
Hahn polynomials are defined in terms of generalized hypergeometric functions by
give a detailed list of their properties.
Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the continuous Hahn polynomials
pn(x,a,b, ), and the continuous dual Hahn polynomials
Sn(x;a,b,c). These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials
Qn(x;α,β, N;q), and so on.
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
in the Askey scheme
Askey scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , and has since been extended by and to cover basic...
of hypergeometric orthogonal polynomials, introduced by Chebyshev in 1875 and rediscovered by . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials
Charlier polynomials
In mathematics, Charlier polynomials are a family of orthogonal polynomials introduced by Carl Charlier....
. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.
Hahn polynomials are defined in terms of generalized hypergeometric functions by
give a detailed list of their properties.
Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the continuous Hahn polynomials
Continuous Hahn polynomials
In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials...
pn(x,a,b, ), and the continuous dual Hahn polynomials
Continuous dual Hahn polynomials
In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials...
Sn(x;a,b,c). These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials
Q-Hahn polynomials
In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...
Qn(x;α,β, N;q), and so on.