§18.34 Bessel Polynomials

§18.34(i) Definitions and Recurrence Relation

For the confluent hypergeometric function ${{}_{1}F_{1}}$ and the generalized hypergeometric function ${{}_{2}F_{0}}$ see §16.2(ii) and §16.2(iv).

 18.34.1 $y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right).$

Other notations in use are given by

 18.34.2 $\displaystyle y_{n}(x)$ $\displaystyle=y_{n}\left(x;2\right),$ $\displaystyle\theta_{n}(x)$ $\displaystyle=x^{n}y_{n}(x^{-1}),$ ⓘ Symbols: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Bessel polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.34(i) Permalink: http://dlmf.nist.gov/18.34.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.34(i), §18.34 and Ch.18

and

 18.34.3 $\displaystyle y_{n}(x;a,b)$ $\displaystyle=y_{n}\left(2x/b;a\right),$ $\displaystyle\theta_{n}(x;a,b)$ $\displaystyle=x^{n}y_{n}(x^{-1};a,b).$ ⓘ Symbols: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Bessel polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.34(i) Permalink: http://dlmf.nist.gov/18.34.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.34(i), §18.34 and Ch.18

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 $y_{n+1}\left(x;a\right)=(A_{n}x+B_{n})y_{n}\left(x;a\right)-C_{n}y_{n-1}\left(% x;a\right),$

where

 18.34.5 $\displaystyle A_{n}$ $\displaystyle=\frac{(2n+a)(2n+a-1)}{2(n+a-1)},$ $\displaystyle B_{n}$ $\displaystyle=\frac{(a-2)(2n+a-1)}{(n+a-1)(2n+a-2)},$ $\displaystyle C_{n}$ $\displaystyle=\frac{-n(2n+a)}{(n+a-1)(2n+a-2)}.$ ⓘ Defines: $A_{n}$: coefficient (locally), $B_{n}$: coefficient (locally) and $C_{n}$: coefficient (locally) Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.34.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.34(i), §18.34 and Ch.18

§18.34(ii) Orthogonality

Because the coefficients $C_{n}$ in (18.34.4) are not all positive, the polynomials $y_{n}\left(x;a\right)$ cannot be orthogonal on the line with respect to a positive weight function. There is orthogonality on the unit circle, however:

 18.34.6 $\frac{1}{2\pi\mathrm{i}}\int_{|z|=1}z^{a-2}y_{n}\left(z;a\right)y_{m}\left(z;a% \right)e^{-2/z}\mathrm{d}z=\frac{(-1)^{n+a-1}n!\,2^{a-1}}{(n+a-2)!(2n+a-1)}% \delta_{n,m},$ $a=1,2,\dots$,

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 $x^{2}y_{n}''\left(x;a\right)+(ax+2)y_{n}'\left(x;a\right)-n(n+a-1)y_{n}\left(x% ;a\right)=0,$ ⓘ Symbols: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Bessel polynomial, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.34.E7 Encodings: TeX, pMML, png See also: Annotations for §18.34(iii), §18.34 and Ch.18

where primes denote derivatives with respect to $x$.

 18.34.8 $\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)% }{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right).$

For uniform asymptotic expansions of $y_{n}\left(x;a\right)$ as $n\to\infty$ 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).