as Bessel polynomials

SearchAdvancedHelp

(0.005 seconds)

11—20 of 82 matching pages

11: Bibliography W

… ►

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

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.34(iii).

Encodings:

BibTeX

…

12: 10 Bessel Functions

Chapter 10 Bessel Functions

…

13: 18.15 Asymptotic Approximations

… ►

18.15.6 $${(\sin \frac{1}{2}\theta )}^{\alpha +\frac{1}{2}}{(\cos \frac{1}{2}\theta )}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)=\frac{\mathrm{\Gamma }\left(n+\alpha +1\right)}{2^{\frac{1}{2}}{\rho }^{\alpha }n!}\left({\theta }^{\frac{1}{2}}{J}_{\alpha }\left(\rho \theta \right)\sum _{m=0}^M\frac{A_m(\theta )}{{\rho }^{2m}}+{\theta }^{\frac{3}{2}}{J}_{\alpha +1}\left(\rho \theta \right)\sum _{m=0}^{M-1}\frac{B_m(\theta )}{{\rho }^{2m+1}}+{\epsilon }_M(\rho ,\theta )\right),$$

ⓘ

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, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ , ${\epsilon }_M(\rho ,\theta )$ , $A_m(\theta )$: coefficient and $B_m(\theta )$: coefficient

Permalink:

http://dlmf.nist.gov/18.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►

18.15.19 $$L_n^{(\alpha )}\left(\nu x\right)=\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha }{x}^{\frac{1}{2}\alpha +\frac{1}{4}}{(1-x)}^{\frac{1}{4}}}\left({\xi }^{\frac{1}{2}}{J}_{\alpha }\left(\nu \xi \right)\sum _{m=0}^{M-1}\frac{A_m(\xi )}{{\nu }^{2m}}+{\xi }^{-\frac{1}{2}}{J}_{\alpha +1}\left(\nu \xi \right)\sum _{m=0}^{M-1}\frac{B_m(\xi )}{{\nu }^{2m+1}}+{\xi }^{\frac{1}{2}}\mathrm{env}{J}_{\alpha }\left(\nu \xi \right)O\left(\frac{1}{{\nu }^{2M-1}}\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\xi$ , $A_m(\xi )$: coefficient, $B_m(\xi )$: coefficient and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

… ►Here $J_{\nu }\left(z\right)$ denotes the Bessel function (§10.2(ii)), $\mathrm{env}{J}_{\nu }\left(z\right)$ denotes its envelope (§2.8(iv)), and $\delta$ is again an arbitrary small positive constant. …

14: Bibliography G

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.49(ii), (18.34.1), (18.34.2), (18.34.6).

Encodings:

BibTeX

…

15: Errata

… ►

18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n$$

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${}_0{}^{}𝐅_1^{}$ functions.

… ►

18.34.1 $$y_n(x;a)={}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_n{\left(\frac{x}{2}\right)}^n{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x})=n!{\left(-\frac{1}{2}x\right)}^n{L}_n^{(1-a-2n)}\left(2x^{-1}\right)={\left(\frac{1}{2}x\right)}^{1-\frac{1}{2}a}{\mathrm{e}}^{1/x}{W}_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right)$$

This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.

►

18.34.2

$y_n(x)=y_n(x;2)=2{\pi }^{-1}{x}^{-1}{\mathrm{e}}^{1/x}{𝗄}_n\left(x^{-1}\right),$

${\theta }_n(x)=x^n{y}_n(x^{-1})=2{\pi }^{-1}{x}^{n+1}{\mathrm{e}}^x{𝗄}_n\left(x\right)$

This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind.

… ►

The Legendre polynomial $P_n$ was mistakenly identified as the associated Legendre function $P_n$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

►

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

…

16: 3.5 Quadrature

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

17: 10.60 Sums

… ►

10.60.2 $$\frac{\sin w}{w}=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗃}_n\left(u\right)P_n\left(\cos \alpha \right).$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

10.1.45

Referenced by:

§10.60(iii)

Permalink:

http://dlmf.nist.gov/10.60.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(i), §10.60 and Ch.10

… ►

10.60.7 $${\mathrm{e}}^{\mathrm{i}z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathrm{i}}^n{𝗃}_n\left(z\right)P_n\left(\cos \alpha \right),$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.47

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

►

10.60.8 $${\mathrm{e}}^{z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right),$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.36

Permalink:

http://dlmf.nist.gov/10.60.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

►

10.60.9 $${\mathrm{e}}^{-z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right).$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.37

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

►

10.60.10 $$J_0\left(z\sin \alpha \right)=\sum _{n=0}^{\mathrm{\infty }}(4n+1)\frac{(2n)!}{2^{2n}{(n!)}^2}{𝗃}_{2n}\left(z\right)P_{2n}\left(\cos \alpha \right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.48

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

…

18: 18.41 Tables

… ►

§18.41(i) Polynomials

►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

19: 18.11 Relations to Other Functions

… ►

Jacobi

►

18.11.5 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(1-\frac{z^2}{2n^2}\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{P}_n^{(\alpha ,\beta )}\left(\cos \frac{z}{n}\right)=\frac{2^{\alpha }}{z^{\alpha }}{J}_{\alpha }\left(z\right).$$

ⓘ

Symbols:

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

A&S Ref:

22.15.1

Referenced by:

§18.11(ii), §18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

►

Laguerre

►

18.11.6 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{L}_n^{(\alpha )}\left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha }}{J}_{\alpha }\left(2z^{\frac{1}{2}}\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.2

Referenced by:

§18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

…

20: Bibliography H

… ►

Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

ⓘ

External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

…