§18.34 Bessel Polynomials

Keywords:: Bessel polynomials
Referenced by:: §10.49(ii), §3.5(viii)
Permalink:: http://dlmf.nist.gov/18.34

§18.34(i) Definitions and Recurrence Relation
§18.34(ii) Orthogonality
§18.34(iii) Other Properties

§18.34(i) Definitions and Recurrence Relation

Notes:: See Ismail (2005, (4.10.4)–(4.10.6), (4.10.13)).
Keywords:: Bessel polynomials, confluent hypergeometric functions, generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/18.34.i

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

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

Defines:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : Bessel polynomial
Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.34(i)
Permalink:: http://dlmf.nist.gov/18.34.E1
Encodings:: TeX, pMML, png

Other notations in use are given by

18.34.2

$y_{n}(x)=\mathop{y_{{n}}\/}\nolimits\!\left(x;2\right),$

$\theta _{n}(x)=x^{n}y_{n}(x^{{-1}}),$

Symbols:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : 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

and

18.34.3

$y_{n}(x;a,b)=\mathop{y_{{n}}\/}\nolimits\!\left(2x/b;a\right),$

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

Symbols:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : 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

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

Symbols:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : Bessel polynomial, $n$ : nonnegative integer, $x$ : real variable, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient
Referenced by:: §18.34(ii)
Permalink:: http://dlmf.nist.gov/18.34.E4
Encodings:: TeX, pMML, png

where

18.34.5

$A_{n}=\frac{(2n+a)(2n+a-1)}{2(n+a-1)},$

$B_{n}=\frac{(a-2)(2n+a-1)}{(n+a-1)(2n+a-2)},$

$C_{n}=\frac{-n(2n+a)}{(n+a-1)(2n+a-2)}.$

Symbols:: $n$ : nonnegative integer, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/18.34.E5
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

§18.34(ii) Orthogonality

Notes:: See Ismail (2005, (4.10.9)).
Keywords:: Bessel polynomials, moment functionals
Permalink:: http://dlmf.nist.gov/18.34.ii

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

18.34.6 $\frac{1}{2\pi i}\int _{{|z|=1}}z^{{a-2}}\mathop{y_{{n}}\/}\nolimits\!\left(z;a\right)\mathop{y_{{m}}\/}\nolimits\!\left(z;a\right)e^{{-2/z}}dz=\frac{(-1)^{{n+a-1}}n!\, 2^{{a-1}}}{(n+a-2)!(2n+a-1)}\delta _{{n,m}},$ $a=1,2,\dots$ ,

Symbols:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : Bessel polynomial, $\delta _{{j,k}}$ : Kronecker delta, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.34.E6
Encodings:: TeX, pMML, png

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

Notes:: See Ismail (2005, (4.10.7), (4.10.10)).
Keywords:: Bessel polynomials, Jacobi polynomials, differential equations
Permalink:: http://dlmf.nist.gov/18.34.iii

18.34.7 $x^{2}{\mathop{y_{{n}}\/}\nolimits^{{\prime\prime}}}\!\left(x;a\right)+(ax+2){\mathop{y_{{n}}\/}\nolimits^{{\prime}}}\!\left(x;a\right)-n(n+a-1)\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)=0,$

Symbols:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : Bessel polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.34.E7
Encodings:: TeX, pMML, png

where primes denote derivatives with respect to $x$ .

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

Symbols:: $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ : Bessel polynomial, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.34.E8
Encodings:: TeX, pMML, png

For uniform asymptotic expansions of $\mathop{y_{{n}}\/}\nolimits\!\left(x;a\right)$ 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).

§18.34 Bessel Polynomials

Contents

§18.34(i) Definitions and Recurrence Relation

§18.34(ii) Orthogonality

§18.34(iii) Other Properties