Dual Hahn polynomials
Encyclopedia
In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials
in the Askey scheme
of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by
for 0≤n≤N
where λ(x)=x(x+γ+δ+1).
give a detailed list of their properties.
Closely related polynomials include the Hahn polynomials
, 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.
Q by switching the roles of x and n: more precisely
Racah polynomials
are a generalization of dual Hahn polynomials
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. They are defined in terms of generalized hypergeometric functions by
for 0≤n≤Nwhere λ(x)=x(x+γ+δ+1).
give a detailed list of their properties.
Closely related polynomials include the Hahn polynomials
Hahn polynomials
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...
, 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.
Relation to other polynomials
Dual Hahn polynomials are related to Hahn polynomialsHahn polynomials
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...
Q by switching the roles of x and n: more precisely
Racah polynomials
Racah polynomials
In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients....
are a generalization of dual Hahn polynomials