Digital Library of Mathematical Functions
About the Project
NIST
18 Orthogonal PolynomialsOther Orthogonal Polynomials

§18.34 Bessel Polynomials

Contents

§18.34(i) Definitions and Recurrence Relation

For the confluent hypergeometric function F11 and the generalized hypergeometric function F02 see §16.2(ii) and §16.2(iv).

18.34.1 yn(x;a)=F02(-n,n+a-1-;-x2)=(n+a-1)n(x2)nF11(-n-2n-a+2;2x).

Other notations in use are given by

18.34.2 yn(x) =yn(x;2),
θn(x) =xnyn(x-1),

and

18.34.3 yn(x;a,b) =yn(2x/b;a),
θn(x;a,b) =xnyn(x-1;a,b).

Often only the polynomials (18.34.2) are called Bessel polynomials, while the polynomials (18.34.1) and (18.34.3) are called generalized Bessel polynomials. See also §10.49(ii).

18.34.4 yn+1(x;a)=(Anx+Bn)yn(x;a)-Cnyn-1(x;a),

where

18.34.5 An =(2n+a)(2n+a-1)2(n+a-1),
Bn =(a-2)(2n+a-1)(n+a-1)(2n+a-2),
Cn =-n(2n+a)(n+a-1)(2n+a-2).

§18.34(ii) Orthogonality

Because the coefficients Cn in (18.34.4) are not all positive, the polynomials yn(x;a) cannot be orthogonal on the line with respect to a positive weight function. There is orthogonality on the unit circle, however:

18.34.6 12π|z|=1za-2yn(z;a)ym(z;a)-2/zz=(-1)n+a-1n! 2a-1(n+a-2)!(2n+a-1)δn,m,
a=1,2,,

the integration path being taken in the positive rotational sense.

Orthogonality can also be expressed in terms of moment functionals; see Durán (1993), Evans et al. (1993), and Maroni (1995).

§18.34(iii) Other Properties

18.34.7 x2yn′′(x;a)+(ax+2)yn(x;a)-n(n+a-1)yn(x;a)=0,

where primes denote derivatives with respect to x.

18.34.8 limαPn(α,a-α-2)(1+αx)Pn(α,a-α-2)(1)=yn(x;a).

For uniform asymptotic expansions of yn(x;a) as n in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). For uniform asymptotic expansions in terms of Hermite polynomials see López and Temme (1999b).

For further information on Bessel polynomials see §10.49(ii).