# §18.34 Bessel Polynomials

## §18.34(i) Definitions and Recurrence Relation

For the confluent hypergeometric function ${{}_{1}F_{1}}$ and the generalized hypergeometric function ${{}_{2}F_{0}}$, the Laguerre polynomial $L^{(\alpha)}_{n}$ 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 $y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right)=n!\left(-\tfrac{1}{2}x\right)^{n}L^{(1-a-2n)}% _{n}\left(2x^{-1}\right)=\left(\tfrac{1}{2}x\right)^{1-\frac{1}{2}a}{\mathrm{e% }}^{1/x}W_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right).$ ⓘ Defines: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Bessel polynomial Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$: alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$: $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Proved: Grosswald (1978, p.38, Theorem 1); Combine (18.5.12) and items (i), (v), (vii) of the mentioned theorem for $b=2$(proved) Referenced by: (18.34.2), (18.34.7_1), §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.1) Permalink: http://dlmf.nist.gov/18.34.E1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function. See also: Annotations for §18.34(i), §18.34 and Ch.18

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

 18.34.2 $\displaystyle y_{n}(x)$ $\displaystyle=y_{n}\left(x;2\right)=2{\pi}^{-1}x^{-1}{\mathrm{e}}^{1/x}\mathsf% {k}_{n}\left(x^{-1}\right),$ $\displaystyle\theta_{n}(x)$ $\displaystyle=x^{n}y_{n}(x^{-1})=2{\pi}^{-1}x^{n+1}{\mathrm{e}}^{x}\mathsf{k}_% {n}\left(x\right),$ ⓘ Symbols: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Bessel polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$: modified spherical Bessel function, $n$: nonnegative integer and $x$: real variable Source: Grosswald (1978, p.7) Proof sketch: Combine (18.34.1), (13.18.9) and (10.47.9). Referenced by: §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.2) Permalink: http://dlmf.nist.gov/18.34.E2 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind. See also: Annotations for §18.34(i), §18.34 and Ch.18

where $\mathsf{k}_{n}$ is a modified spherical Bessel function (10.49.9), 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. Sometimes the polynomials $\theta_{n}(x;a,b)$ are called reverse 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

The product $A_{n-1}A_{n}C_{n}$ of coefficients in (18.34.4) is positive if and only if $n<\tfrac{1}{2}(1-a)$. Hence the full system of polynomials $y_{n}\left(x;a\right)$ 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 $\frac{2^{1-a}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}y_{n}\left(x;a\right)y_% {m}\left(x;a\right)x^{a-2}{\mathrm{e}}^{-2x^{-1}}\,\mathrm{d}x=\frac{1-a}{1-a-% 2n}\,\frac{n!}{{\left(2-a-n\right)_{n}}}\,\delta_{n,m},$ $m,n=0,1,\dots,N=\lceil-\ifrac{(1+a)}{2}\rceil$.

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

the integration path being taken in the positive rotational sense. 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 $x^{2}y_{n}''\left(x;a\right)+(ax+2)y_{n}'\left(x;a\right)-n(n+a-1)y_{n}\left(x% ;a\right)=0,$ ⓘ Symbols: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Bessel polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: (18.34.7_2) 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 $x$. With functions

 18.34.7_1 $\phi_{n}(x;\lambda)={\mathrm{e}}^{-\lambda\,{\mathrm{e}}^{-x}}\left(2\lambda\,% {\mathrm{e}}^{-x}\right)^{\lambda-\frac{1}{2}}\ifrac{y_{n}\left(\lambda^{-1}{% \mathrm{e}}^{x};2-2\lambda\right)}{n!}=(-1)^{n}{\mathrm{e}}^{-\lambda\,{% \mathrm{e}}^{-x}}\left(2\lambda\,{\mathrm{e}}^{-x}\right)^{\lambda-n-\frac{1}{% 2}}L^{(2\lambda-2n-1)}_{n}\left(2\lambda\,{\mathrm{e}}^{-x}\right)=(2\lambda)^% {-\frac{1}{2}}\,{\mathrm{e}}^{x/2}\,W_{\lambda,n+\frac{1}{2}-\lambda}\left(2% \lambda{\mathrm{e}}^{-x}\right)/n!\,,$ $n=0,1,\dots,N=\left\lceil\lambda-\frac{3}{2}\right\rceil$, $\lambda>\frac{1}{2}$, ⓘ

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

 18.34.7_2 $\left(\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}-\lambda^{2}\left({\mathrm{e}}% ^{-2x}-2{\mathrm{e}}^{-x}\right)-\left(\lambda-\left(n+\tfrac{1}{2}\right)% \right)^{2}\right)\phi_{n}(x;\lambda)=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $x$: real variable Proof sketch: This follows by straightforward computation from (18.34.7) and (18.34.7_1). Referenced by: (18.39.17), (18.39.18) Permalink: http://dlmf.nist.gov/18.34.E7_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.34(iii), §18.34 and Ch.18

and

 18.34.7_3 $\int_{-\infty}^{\infty}\phi_{n}(x;\lambda)\phi_{m}(x;\lambda)\,\mathrm{d}x=% \frac{\Gamma\left(2\lambda-n\right)}{(2\lambda-2n-1)\,n!}\,\delta_{n,m},$ $m,n=0,1,\dots,N=\left\lceil\lambda-\frac{3}{2}\right\rceil$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $\left\lceil\NVar{x}\right\rceil$: ceiling of $x$, $\,\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Proof sketch: This follows by straightforward computation from (18.34.5_5) and (18.34.7_1). Referenced by: §18.34(iii), (18.39.17), §18.39(i), Erratum (V1.2.0) §18.34 Permalink: http://dlmf.nist.gov/18.34.E7_3 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.34(iii), §18.34 and Ch.18
 18.34.8 $\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)% }{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right).$ ⓘ

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

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).