Bessel polynomials

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1: 18.34 Bessel Polynomials

§18.34 Bessel Polynomials

… ►See also §10.49(ii). … ►

§18.34(ii) Orthogonality

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

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Symbols:

$y_n(x;a)$: Bessel polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.34.E7

Encodings:

pMML, png, TeX

See also:

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

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2: 18.1 Notation

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Bessel: $y_n(x;a)$.

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3: Bibliography D

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On the zeros of generalized Bessel polynomials. I. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 1–13.

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External Links:

ISSN 0019-3577, Document, MathReview (M. E. Muldoon)

Referenced by:

§10.49(ii), §10.58.

Encodings:

BibTeX

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On the zeros of generalized Bessel polynomials. II. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 14–25.

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External Links:

ISSN 0019-3577, Document, MathReview (M. E. Muldoon)

Referenced by:

§10.49(ii), §10.58.

Encodings:

BibTeX

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Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.

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External Links:

ISSN 0036-1410, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§10.49(ii), §18.34(iii).

Encodings:

BibTeX

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4: 10.49 Explicit Formulas

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10.49.8 $${\mathsf{i}}_n^{(1)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^{n+1}\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.2.9 (modified)

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(ii), §10.49 and Ch.10

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10.49.10 $${\mathsf{i}}_n^{(2)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^n\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${\mathsf{i}}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.2.10 (modified)

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(ii), §10.49 and Ch.10

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10.49.12 $${\mathsf{k}}_n\left(z\right)=\frac{1}{2}\pi {\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathsf{k}}_n\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.2.15

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(ii), §10.49 and Ch.10

… ► ${\sum }_{k=0}^n{a}_k(n+\frac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree $n$. … …

5: Bibliography E

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Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

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External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

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Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.

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External Links:

ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§18.34(ii).

Encodings:

BibTeX

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6: 10.23 Sums

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10.23.8 $$\frac{{𝒞}_{\nu }\left(w\right)}{w^{\nu }}=2^{\nu }\mathrm{\Gamma }\left(\nu \right)\sum _{k=0}^{\mathrm{\infty }}(\nu +k)\frac{{𝒞}_{\nu +k}\left(u\right)}{u^{\nu }}\frac{J_{\nu +k}\left(v\right)}{v^{\nu }}{C}_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.80

Referenced by:

§10.23(ii), §10.23(ii), §10.23(ii), §10.44(ii), §10.60(i), §10.60(ii)

Permalink:

http://dlmf.nist.gov/10.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

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10.23.9 $${\mathrm{e}}^{\mathrm{i}v\cos \alpha }=\frac{\mathrm{\Gamma }\left(\nu \right)}{{(\frac{1}{2}v)}^{\nu }}\sum _{k=0}^{\mathrm{\infty }}(\nu +k){\mathrm{i}}^k{J}_{\nu +k}\left(v\right)C_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.81

Referenced by:

§10.23(ii)

Permalink:

http://dlmf.nist.gov/10.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

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10.23.12 $$\frac{1}{t-z}=J_0\left(z\right)O_0\left(t\right)+2\sum _{k=1}^{\mathrm{\infty }}{J}_k\left(z\right)O_k\left(t\right),$$ .

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Defines:

$O_n\left(x\right)$: Neumann’s polynomial

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $n$: integer, $k$: nonnegative integer, $x$: real variable and $z$: complex variable

A&S Ref:

9.1.84

Permalink:

http://dlmf.nist.gov/10.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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7: Bibliography W

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Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.34(iii).

Encodings:

BibTeX

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8: 18.12 Generating Functions

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18.12.2 $${\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.6 $$\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right){\mathrm{e}}^{z\cos \theta }{(\frac{1}{2}z\sin \theta )}^{\frac{1}{2}-\lambda }{J}_{\lambda -\frac{1}{2}}\left(z\sin \theta \right)=\sum _{n=0}^{\mathrm{\infty }}\frac{C_n^{(\lambda )}\left(\cos \theta \right)}{{\left(2\lambda \right)}_n}{z}^n,$$ $0\le \theta \le \pi$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.12 $${\mathrm{e}}^{xz}{J}_0\left(z\sqrt{1-x^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n\left(x\right)}{n!}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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9: Bibliography G

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Bessel Polynomials. Lecture Notes in Mathematics, Vol. 698, Springer, Berlin-New York.

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External Links:

ISBN 3-540-09104-1, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§10.49(ii).

Encodings:

BibTeX

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10: 3.5 Quadrature

… ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … …