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§18.34 Bessel Polynomials

  1. §18.34(i) Definitions and Recurrence Relation
  2. §18.34(ii) Orthogonality
  3. §18.34(iii) Other Properties

§18.34(i) Definitions and Recurrence Relation

For the confluent hypergeometric function F11 and the generalized hypergeometric function F02, the Laguerre polynomial Ln(α) and the Whittaker function Wκ,μ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively.

18.34.1 yn(x;a)=F02(n,n+a1;x2)=(n+a1)n(x2)nF11(n2na+2;2x)=n!(12x)nLn(1a2n)(2x1)=(12x)112ae1/xW112a,12(a1)+n(2x1).

With the notation of Koekoek et al. (2010, (9.13.1)) the left-hand side of (18.34.1) has to be replaced by yn(x;a2). Other notations in use are given by

18.34.2 yn(x) =yn(x;2)=2π1x1e1/x𝗄n(x1),
θn(x) =xnyn(x1)=2π1xn+1ex𝗄n(x),

where 𝗄n is a modified spherical Bessel function (10.49.9), and

18.34.3 yn(x;a,b) =yn(2x/b;a),
θn(x;a,b) =xnyn(x1;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. Sometimes the polynomials θn(x;a,b) are called reverse Bessel polynomials. See also §10.49(ii).

18.34.4 yn+1(x;a)=(Anx+Bn)yn(x;a)Cnyn1(x;a),


18.34.5 An =(2n+a)(2n+a1)2(n+a1),
Bn =(a2)(2n+a1)(n+a1)(2n+a2),
Cn =n(2n+a)(n+a1)(2n+a2).

§18.34(ii) Orthogonality

The product An1AnCn of coefficients in (18.34.4) is positive if and only if n<12(1a). Hence the full system of polynomials yn(x;a) cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if a<1:

18.34.5_5 21aΓ(1a)0yn(x;a)ym(x;a)xa2e2x1dx=1a1a2nn!(2an)nδn,m,

The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments μn. Explicit (but complicated) weight functions w(x) taking both positive and negative values have been found such that (18.2.26) holds with dμ(x)=w(x)dx; see Durán (1993), Evans et al. (1993), and Maroni (1995).

Orthogonality of the full system on the unit circle can be given with a much simpler weight function:

18.34.6 12πi|z|=1za2yn(z;a)ym(z;a)e2/zdz=(1)n+a1n! 2a1(n+a2)!(2n+a1)δn,m,

the integration path being taken in the positive rotational sense. See Ismail (2009, (4.10.9)) for orthogonality on the unit circle for general values of a.

§18.34(iii) Other Properties

18.34.7 x2yn′′(x;a)+(ax+2)yn(x;a)n(n+a1)yn(x;a)=0,

where primes denote derivatives with respect to x. With functions

18.34.7_1 ϕn(x;λ)=eλex(2λex)λ12yn(λ1ex;22λ)/n!=(1)neλex(2λex)λn12Ln(2λ2n1)(2λex)=(2λ)12ex/2Wλ,n+12λ(2λex)/n!,
n=0,1,,N=λ32, λ>12,

expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have

18.34.7_2 (d2dx2λ2(e2x2ex)(λ(n+12))2)ϕn(x;λ)=0.


18.34.7_3 ϕn(x;λ)ϕm(x;λ)dx=Γ(2λn)(2λ2n1)n!δn,m,
18.34.8 limαPn(α,aα2)(1+αx)Pn(α,aα2)(1)=yn(x;a).

In this limit the finite system of Jacobi polynomials Pn(α,β)(x) which is orthogonal on (1,) (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on (0,) (see (18.34.5_5)).

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).