relation to hypergeometric function

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11: 13.6 Relations to Other Functions

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§13.6(ii) Incomplete Gamma Functions

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§13.6(v) Orthogonal Polynomials

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Laguerre Polynomials

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Charlier Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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12: 15.4 Special Cases

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§15.4(i) Elementary Functions

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§15.4(ii) Argument Unity

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§15.4(iii) Other Arguments

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13: 10.16 Relations to Other Functions

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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14: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

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§14.3(i) Interval

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§14.3(ii) Interval

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§14.3(iii) Alternative Hypergeometric Representations

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14.3.14 $$w_2(\nu ,\mu ,x)=\frac{2^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)}x{\left(1-x^2\right)}^{-\mu /2}𝐅(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu ,\frac{1}{2}\nu -\frac{1}{2}\mu +1;\frac{3}{2};x^2).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order, $\nu$: general degree and $w_2$: function $w_1$,

Referenced by:

§10.59, §14.3(i), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

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15: 18.20 Hahn Class: Explicit Representations

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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16: 8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

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17: 18.5 Explicit Representations

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Laguerre

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Hermite

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18: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

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

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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 hypergeometricfunction.

See also:

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

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19: 33.2 Definitions and Basic Properties

… ►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by … ►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by …

20: 16.16 Transformations of Variables

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§16.16(i) Reduction Formulas

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