Romanovski–Bessel polynomials

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1: 18.34 Bessel Polynomials

… ►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 $$\frac{2^{1-a}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{y}_n(x;a)y_m(x;a)x^{a-2}{\mathrm{e}}^{-2x^{-1}}dx=\frac{1-a}{1-a-2n}\frac{n!}{{\left(2-a-n\right)}_n}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil -(1+a)/2\rceil$.

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Symbols:

$y_n(x;a)$: Bessel polynomial, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Lesky (1996, p.183, case a.3))(proved)

Sources:

Romanovski (1929, Type V); Koekoek et al. (2010, (9.13.2))

Referenced by:

(18.34.7_3), §18.34(ii), §18.34(iii), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

… ►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …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)). …

2: 18.39 Applications in the Physical Sciences

… ►The functions ${\varphi }_n$ are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). …

3: 18.3 Definitions

§18.3 Definitions

… ►For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►For $\nu \in ℝ$ and $N>-\frac{1}{2}$ a finite system of Jacobi polynomials $P_n^{(-N-1+\mathrm{i}\nu ,-N-1-\mathrm{i}\nu )}\left(\mathrm{i}x\right)$ (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on $(-\mathrm{\infty },\mathrm{\infty })$ with $w(x)={\left(1+x^2\right)}^{-N-1}{\mathrm{e}}^{2\nu \arctan x}$. … ►

Bessel polynomials

►Bessel polynomials are often included among the classical OP’s. …

4: Bibliography R

… ►

A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Applied Bessel Functions. Dover Publications Inc., New York.

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Notes:

Reprint of original Blackie & Son Limited, London, 1946.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.73(iii).

Encodings:

BibTeX

►

Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

External Links:

ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

… ►

Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

… ►

Sur quelques classes nouvelles de polynômes orthogonaux. C. R. Acad. Sci. Paris 188, pp. 1023–1025.

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External Links:

ZentralBlatt

Referenced by:

(18.34.5_5).

Encodings:

BibTeX

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