In mathematics, the 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 on a non-uniform lattice

x(s)=s(s+1)

and are defined as :

w_n^ (s,a,b)=\frac _3F_2(-n,a-s,a+s+1;a-b+a,a+c+1;1)

for

n=0,1,...,N-1

and the parameters

a,b,c

are restricted to

-\frac .

Note that (u)_k is the

rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \ ...

, otherwise known as the Pochhammer symbol, and

_3F_2(\cdot)

is the

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 give a detailed list of their properties.

Orthogonality

The dual Hahn polynomials have the orthogonality condition :

\delta_d_n^2

for

n,m=0,1,...,N-1

. Where

\Delta x(s)=x(s+1)-x(s)

, :

\rho(s)=\frac

and :

d_n^2=\frac.

Numerical instability

As the value of

n

increases, the values that the discrete polynomials obtain also increases. As a result, to obtain numerical stability in calculating the polynomials you would use the renormalized dual Hahn polynomial as defined as :

\hat w_n^(s,a,b)=w_n^(s,a,b)\sqrt

for

n=0,1,...,N-1

. Then the orthogonality condition becomes :

\sum^_\hat w_n^(s,a,b)\hat w_m^(s,a,b)=\delta_

for

n,m=0,1,...,N-1

Relation to other polynomials

The Hahn polynomials,

h_n(x,N;\alpha,\beta)

, is defined on the uniform lattice

x(s)=s

, and the parameters

a,b,c

are defined as

a=(\alpha+\beta)/2,b=a+N,c=(\beta-\alpha)/2

. Then setting

\alpha=\beta=0

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 ...

become the

Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebys ...

. Note that the dual Hahn polynomials have a ''q''-analog with an extra parameter ''q'' known as the

dual q-Hahn polynomials In mathematics, the dual ''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 hypergeomet ...

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. The Racah polynomials were first defined by and are giv ...

are a generalization of 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