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1: 10.25 Definitions

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§10.25(ii) Standard Solutions

… ►The defining property of the second standard solution $K_{\nu }\left(z\right)$ of (10.25.1) is …

2: 14.21 Definitions and Basic Properties

… ►Standard solutions: the associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $P_{\nu }^{-\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{-\nu -1}^{\mu }\left(z\right)$. … …

3: 10.2 Definitions

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§10.2(ii) Standard Solutions

… ►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

4: 14.2 Differential Equations

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§14.2(i) Legendre’s Equation

… ►Standard solutions: ${𝖯}_{\nu }\left(\pm x\right)$, ${𝖰}_{\nu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}\left(\pm x\right)$, $P_{\nu }\left(\pm x\right)$, $Q_{\nu }\left(\pm x\right)$, $Q_{-\nu -1}\left(\pm x\right)$. … ►

§14.2(ii) Associated Legendre Equation

… ►Standard solutions: ${𝖯}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(\pm x\right)$, $P_{\nu }^{\mu }\left(\pm x\right)$, $P_{\nu }^{-\mu }\left(\pm x\right)$, ${𝑸}_{\nu }^{\mu }\left(\pm x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(\pm x\right)$. …

5: 13.2 Definitions and Basic Properties

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Standard Solutions

►The first two standard solutions are: … ►Another standard solution of (13.2.1) is $U(a,b,z)$, which is determined uniquely by the property … ►

13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.7)

Permalink:

http://dlmf.nist.gov/13.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)$ has been added to the original equation to express an explicit connection between the two standardsolutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$.

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.8 $$U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)=z^{-a}\sum _{s=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{s}\right){\left(a\right)}_s{z}^{-s}.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.8)

Permalink:

http://dlmf.nist.gov/13.2.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)$ has been added to the original equation to express an explicit connection between the two standardsolutions of Kummer’s equation.

Suggested 2014-02-10 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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6: 10.47 Definitions and Basic Properties

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§10.47(ii) Standard Solutions

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7: 12.2 Differential Equations

§12.2 Differential Equations

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where …

8: 13.14 Definitions and Basic Properties

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Standard Solutions

►Standard solutions are: …

9: 14.3 Definitions and Hypergeometric Representations

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14.3.8

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $m$: nonnegative integer, $\nu$: general degree and : real variable

Referenced by:

§14.3(ii)

Permalink:

http://dlmf.nist.gov/14.3.E8

Encodings:

pMML, png, TeX

See also:

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

►As standard solutions of (14.2.2) we take the pair and , where …

10: 9.2 Differential Equation

… ►Standard solutions are: …