In mathematics, the continuous dual Hahn polynomials are a family of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...

in the

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 , ...

of hypergeometric orthogonal polynomials. They are defined in terms of

generalized hypergeometric function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, whic ...

s by

S_n(x^2;a,b,c)= _3F_2(-n,a+ix,a-ix;a+b,a+c;1).\

Complex plot of Continuous Duall Hahn Polynomials

give a detailed list of their properties. Closely related polynomials include the dual Hahn polynomials ''R''_''n''(''x'';γ,δ,''N''), the continuous Hahn polynomials ''p''_''n''(''x'',''a'',''b'', , ), and the

Hahn polynomials In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn . The Hahn class is a name for spe ...

. These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the

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 The polynomials are given in terms of basic hypergeometric ...

''Q''_''n''(''x'';α,β, ''N'';''q''), and so on.

Relation to other polynomials

* Wilson polynomials are a generalization of continuous dual Hahn polynomials

References

* * *{{dlmf, id=18.19, title=Hahn Class: Definitions, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Special hypergeometric functions Orthogonal polynomials