Wilson polynomials

2020 Mathematics Subject Classification: Primary: 33A65, Secondary: 33A3081C40 [MSN][ZBL]

$ \def\iy{\infty} \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\la{\lambda} \def\Ga{\Gamma} $

Orthogonal polynomials defined in terms of generalized hypergeometric series by

$${W_n(x^2;a,b,c,d)\over (a+b)_n(a+c)_n(a+d)_n}$$

$$={}_4F_3\left( {-n,n+a+b+c+d,a+ix,a-ix\atop a+b,a+c,a+d};1\right), $$ where $(a)_n=\Ga(a+n)/\Ga(a)=a(a+1)\ldots(a+n-1)$ is the Pochhammer symbol. They satisfy the orthogonality relations

$$\int_0^\iy W_n(x^2)W_m(x^2)w(x)\,dx=0,\quad n\ne m,$$ where

$$w(x)=\left| {\Ga(a+ix)\Ga(b+ix)\Ga(c+ix)\Ga(d+ix)\over\Ga(2ix)}\right|^2$$ and $\Re(a,b,c,d)>0$, with complex parameters appearing in conjugate pairs. See J.A. Wilson [Wi] for the more general orthogonality when one parameter is negative and finitely many discrete mass points occur.

Wilson polynomials are closely related to classical orthogonal polynomials, since they are eigenfunctions of a second-order difference operator:

$$A(x)W_n((x-i)^2)+B(x)W_n(x^2)+C(x)W_n((x+i)^2)$$

$$=\la_n W_n(x^2)$$ for certain functions $A,B,C$ not depending on $n$ and for eigenvalues $\la_n$. There are $q$-analogues of Wilson polynomials, known as Askey–Wilson polynomials (cf. [AsWi]), which contain Wilson polynomials as limit cases. Askey–Wilson polynomials are also orthogonal polynomial eigenfunctions of a second-order difference operator and they are believed to be the most general orthogonal polynomials with this property, in the sense that all other classes with this property can be obtained from them by specialization of parameters or as limit cases.

There is an important variant of the Wilson polynomials called Racah polynomials, defined by

$$R_n(\la(x);\al,\be,\ga,\de)$$

$$={}_4F_3\left({ -n,n+\al+\be+1,-x,x+\ga+\de+1 \atop \al+1,\be+\de+1,\ga+1};1\right)$$ where $\la(x)=x(x+\ga+\de+1)$, $\be+\de+1=-N$ and $n=0,1,\ldots,N$. These satisfy orthogonality relations of the form

$$\sum_{x=0}^N R_n(\la(x))R_m(\la(x))w(x)=0,\quad n\ne m,$$ for certain explicit weights $w(x)$. They have an interpretation as Racah coefficients for tensor products of irreducible representations of the group $\def\SU{\textrm{SU}}\SU(2)$.

The complete set of limit cases of Wilson and Racah polynomials is often written as a directed graph which is known as the Askey tableau, see the Appendix to [AsWi] as well as the references given there. Here the four-parameter families of Wilson and Racah polynomials are in the top level, while there are lower levels with families depending on $3$, $2$, $1$, or $0$ parameters. In general, one parameter is lost with each limit transition. The $3$-parameter level contains continuous Hahn polynomials (cf. [As]) and continuous dual Hahn polynomials (continuous weight functions) and Hahn polynomials and dual Hahn polynomials (discrete weights). The $2$-parameter level contains Meixner–Pollaczek polynomials and Jacobi polynomials (continuous weight functions) and the (discrete) Krawtchouk and Meixner polynomials. The $1$-parameter level contains (continuous) Laguerre polynomials and (discrete) Charlier polynomials. The $0$-parameter bottom level contains just the Hermite polynomials, which are limit cases of all other classes.

References

[As]	R. Askey, "Continuous Hahn polynomials" J. Phys. A: Math. Gen., 18 (1985) pp. L1017-L1019 MR0812420 Zbl 0582.33007
[AsWi]	R. Askey, J. Wilson, "Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials" Mem. Amer. Math. Soc., 319 (1985) MR0783216 Zbl 0572.33012
[AtSu]	N.M. Atakishiyev, S.K. Suslov, "On the Askey–Wilson polynomials" Constr. Approx., 8 (1992) pp. 363–369 MR1164075
[Ko]	T.H. Koornwinder, "Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials" M. Alfaro (ed.) J.S. Dehesa (ed.) F.J. Marcellna (ed.) J.L. Rubio de Francia (ed.), Orthogonal Polynomials and Their Applications, Lect. notes in math., 1329, Springer (1988) pp. 46–72 MR0740416
[NiSuUv]	A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, "Classical orthogonal polynomials of a discrete variable", Springer (1991) (Translated from Russian) MR1149380 Zbl 0743.33001
[Wi]	J.A. Wilson, "Some hypergeometric orthogonal polynomials" SIAM J. Math. Anal., 11 (1980) pp. 690–701 MR0579561 Zbl 0454.33007