18 Orthogonal PolynomialsOther Orthogonal Polynomials18.33 Polynomials Orthogonal on the Unit Circle18.35 Pollaczek Polynomials

For the confluent hypergeometric function ${}_{1}{}^{}F_{1}^{}$ and the generalized hypergeometric function ${}_{2}{}^{}F_{0}^{}$, the Laguerre polynomial ${L}_{n}^{(\alpha )}$ and the Whittaker function ${W}_{\kappa ,\mu}$ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively.

18.34.1 | $$\begin{array}{ll}{y}_{n}(x;a)& ={}_{2}{}^{}F_{0}^{}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_{n}{\left(\frac{x}{2}\right)}^{n}{}_{1}{}^{}F_{1}^{}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x})\\ & =n!{\left(-\frac{1}{2}x\right)}^{n}{L}_{n}^{(1-a-2n)}\left(2{x}^{-1}\right)\\ & ={\left(\frac{1}{2}x\right)}^{1-\frac{1}{2}a}{\mathrm{e}}^{1/x}{W}_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2{x}^{-1}\right).\end{array}$$ | ||

With the notation of Koekoek et al. (2010, (9.13.1)) the left-hand side of (18.34.1) has to be replaced by ${y}_{n}(x;a-2)$. Other notations in use are given by

18.34.2 | ${y}_{n}(x)$ | $={y}_{n}(x;2)=2{\pi}^{-1}{x}^{-1}{\mathrm{e}}^{1/x}{\U0001d5c4}_{n}\left({x}^{-1}\right),$ | ||

${\theta}_{n}(x)$ | $={x}^{n}{y}_{n}({x}^{-1})=2{\pi}^{-1}{x}^{n+1}{\mathrm{e}}^{x}{\U0001d5c4}_{n}\left(x\right),$ | |||

where ${\U0001d5c4}_{n}$ is a modified spherical Bessel function (10.49.9), and

18.34.3 | ${y}_{n}(x;a,b)$ | $={y}_{n}(2x/b;a),$ | ||

${\theta}_{n}(x;a,b)$ | $={x}^{n}{y}_{n}({x}^{-1};a,b).$ | |||

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*.
Sometimes the polynomials ${\theta}_{n}(x;a,b)$ are called
*reverse Bessel polynomials*.
See also §10.49(ii).

18.34.4 | $${y}_{n+1}(x;a)=({A}_{n}x+{B}_{n}){y}_{n}(x;a)-{C}_{n}{y}_{n-1}(x;a),$$ | ||

where

18.34.5 | ${A}_{n}$ | $={\displaystyle \frac{(2n+a)(2n+a-1)}{2(n+a-1)}},$ | ||

${B}_{n}$ | $={\displaystyle \frac{(a-2)(2n+a-1)}{(n+a-1)(2n+a-2)}},$ | |||

${C}_{n}$ | $={\displaystyle \frac{-n(2n+a)}{(n+a-1)(2n+a-2)}}.$ | |||

The product ${A}_{n-1}{A}_{n}{C}_{n}$ of coefficients in (18.34.4)
is positive if and only if $$. Hence the full system
of polynomials ${y}_{n}(x;a)$ cannot be orthogonal on the
line with respect to a positive weight function, but this is possible
for a finite system of such polynomials,
the *Romanovski–Bessel polynomials*, if $$:

18.34.5_5 | $$\frac{{2}^{1-a}}{\mathrm{\Gamma}\left(1-a\right)}{\int}_{0}^{\mathrm{\infty}}{y}_{n}(x;a){y}_{m}(x;a){x}^{a-2}{\mathrm{e}}^{-2{x}^{-1}}dx=\frac{1-a}{1-a-2n}\frac{n!}{{\left(2-a-n\right)}_{n}}{\delta}_{n,m},$$ | ||

$m,n=0,1,\mathrm{\dots},N=\lceil -(1+a)/2\rceil $. | |||

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 ${\mu}_{n}$. Explicit (but complicated) weight functions $w(x)$ taking both positive and negative values have been found such that (18.2.26) holds with $d\mu (x)=w(x)dx$; 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 | $$\frac{1}{2\pi \mathrm{i}}{\int}_{|z|=1}{z}^{a-2}{y}_{n}(z;a){y}_{m}(z;a){\mathrm{e}}^{-2/z}dz=\frac{{(-1)}^{n+a-1}n!{\mathrm{\hspace{0.17em}2}}^{a-1}}{(n+a-2)!(2n+a-1)}{\delta}_{n,m},$$ | ||

$a=1,2,\mathrm{\dots}$, | |||

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 $a$.

18.34.7 | $${x}^{2}{y}_{n}^{\prime \prime}(x;a)+(ax+2){y}_{n}^{\prime}(x;a)-n(n+a-1){y}_{n}(x;a)=0,$$ | ||

where primes denote derivatives with respect to $x$. With functions

18.34.7_1 | $$\begin{array}{ll}{\varphi}_{n}(x;\lambda )& ={\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -\frac{1}{2}}{y}_{n}({\lambda}^{-1}{\mathrm{e}}^{x};2-2\lambda )/n!\\ & ={(-1)}^{n}{\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -n-\frac{1}{2}}{L}_{n}^{(2\lambda -2n-1)}\left(2\lambda {\mathrm{e}}^{-x}\right)\\ & ={(2\lambda )}^{-\frac{1}{2}}{\mathrm{e}}^{x/2}{W}_{\lambda ,n+\frac{1}{2}-\lambda}\left(2\lambda {\mathrm{e}}^{-x}\right)/n!,\end{array}$$ | ||

$n=0,1,\mathrm{\dots},N=\lceil \lambda -\frac{3}{2}\rceil $, $\lambda >\frac{1}{2}$, | |||

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

18.34.7_2 | $$\left(\frac{{d}^{2}}{{dx}^{2}}-{\lambda}^{2}\left({\mathrm{e}}^{-2x}-2{\mathrm{e}}^{-x}\right)-{\left(\lambda -\left(n+\frac{1}{2}\right)\right)}^{2}\right){\varphi}_{n}(x;\lambda )=0.$$ | ||

and

18.34.7_3 | $${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\varphi}_{n}(x;\lambda ){\varphi}_{m}(x;\lambda )dx=\frac{\mathrm{\Gamma}\left(2\lambda -n\right)}{(2\lambda -2n-1)n!}{\delta}_{n,m},$$ | ||

$m,n=0,1,\mathrm{\dots},N=\lceil \lambda -\frac{3}{2}\rceil $. | |||

18.34.8 | $$\underset{\alpha \to \mathrm{\infty}}{lim}\frac{{P}_{n}^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{{P}_{n}^{(\alpha ,a-\alpha -2)}\left(1\right)}={y}_{n}(x;a).$$ | ||

In this limit the finite system of Jacobi polynomials ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty})$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty})$ (see (18.34.5_5)).

For uniform asymptotic expansions of ${y}_{n}(x;a)$ as $n\to \mathrm{\infty}$ 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).