relations to other functions

SearchAdvancedHelp

(0.029 seconds)

11—20 of 162 matching pages

11: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

►

§19.10(i) Theta and Elliptic Functions

…

12: 7.1 Special Notation

… ►

13: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

…

14: 16.24 Physical Applications

… ►

§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

…

15: 25.13 Periodic Zeta Function

… ►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: … ►Also, …

16: 15.17 Mathematical Applications

… ►This topic is treated in §§15.10 and 15.11. …

17: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

►

§15.9(i) Orthogonal Polynomials

… ►

Jacobi

… ►

§15.9(ii) Jacobi Function

… ►

§15.9(iv) Associated Legendre Functions; Ferrers Functions

…

18: 7.5 Interrelations

§7.5 Interrelations

… ► …

19: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

►

§14.3(i) Interval

… ►

§14.3(ii) Interval

… ►

§14.3(iii) Alternative Hypergeometric Representations

… ►

§14.3(iv) Relations to Other Functions

…

20: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

… ►

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

ⓘ

Defines:

$y_n(x;a)$: Bessel polynomial

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: 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:

pMML, png, TeX

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

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … ►

18.34.8 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{P_n^{(\alpha ,a-\alpha -2)}\left(1\right)}=y_n(x;a).$$

ⓘ

Symbols:

$y_n(x;a)$: Bessel polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.34(iii)

Permalink:

http://dlmf.nist.gov/18.34.E8

Encodings:

pMML, png, TeX

See also:

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

…