classification of parameters

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1: 31.2 Differential Equations

… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). …

2: 31.14 General Fuchsian Equation

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$\alpha \beta ={\displaystyle \sum _{j=1}^N}{a}_j{q}_j.$

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3: 2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

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4: 18.27 $q$-Hahn Class

… ►A (nonexhaustive) classification of such systems of OP’s was made by Hahn (1949). …These families depend on further parameters, in addition to $q$. The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. … ►Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. …

5: 16.21 Differential Equation

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16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

… ►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\mathrm{\infty }$. … ►

16.21.2 $$G_{p,q}^{1,p}(z{\mathrm{e}}^{(p-m-n-1)\pi \mathrm{i}};\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_j,b_1,\mathrm{\dots },b_{j-1},b_{j+1},\mathrm{\dots },b_q}),$$ $j=1,\mathrm{\dots },q$.

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Symbols:

$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

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6: 16.4 Argument Unity

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§16.4(i) Classification

… ►The last condition is equivalent to the sum of the top parameters plus $2$ equals the sum of the bottom parameters, that is, the series is 2-balanced. … ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. … ►These series contain $\mathit{6}j$ symbols as special cases when the parameters are integers; compare §34.4. … ►Contiguous balanced series have parameters shifted by an integer but still balanced. …

7: 31.12 Confluent Forms of Heun’s Equation

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31.12.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+ϵ\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.2 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\delta }{z^2}+\frac{\gamma }{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^2}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.4 $$\frac{d^{2}w}{{dz}^2}+\left(\gamma +z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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8: 1.13 Differential Equations

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§1.13(ii) Equations with a Parameter

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Variation of Parameters

… ►For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. … ►A standard form for second order ordinary differential equations with $x\in ℝ$, and with a real parameter $\lambda$, and real valued functions $p(x),q(x),$ and $\rho (x)$, with $p(x)$ and $\rho (x)$ positive, is ►

1.13.26 $${\left(p(x)u^{\prime }(x)\right)}^{\prime }+\left(\lambda \rho (x)-q(x)\right)u(x)=0.$$

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Symbols:

$p(x)$: function, $q(x)$: function, $\rho (x)$: function, $u(x)$: solution, $\lambda$: real parameter and $x$: real variable

Referenced by:

§1.13(viii), §1.13(viii), Erratum (V1.2.0) §1.13

Permalink:

http://dlmf.nist.gov/1.13.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(viii), §1.13 and Ch.1

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9: 16.8 Differential Equations

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§16.8(i) Classification of Singularities

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16.8.3 $$\left(\mathit{\vartheta }(\mathit{\vartheta }+b_1-1)\mathrm{\cdots }(\mathit{\vartheta }+b_q-1)-z(\mathit{\vartheta }+a_1)\mathrm{\cdots }(\mathit{\vartheta }+a_p)\right)w=0.$$

ⓘ

Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\mathit{\vartheta }$: differential operator and $w$: solution

Referenced by:

§16.8(ii), §16.8(ii), §16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.7 $${\tilde{w}}_j(z)={(-z)}^{-a_j}{}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_j,1-b_1+a_j,\mathrm{\dots },1-b_q+a_j}{1-a_1+a_j,\mathrm{\dots }\ast \mathrm{\dots },1-a_{q+1}+a_j};\frac{1}{z}),$$ $j=1,\mathrm{\dots },q+1$,

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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, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: solution

Permalink:

http://dlmf.nist.gov/16.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.10 $$\underset{|\alpha |\to \mathrm{\infty }}{lim}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\alpha }{b_1,\mathrm{\dots },b_q};\frac{z}{\alpha })={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z).$$

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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, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(iii), §16.8 and Ch.16

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16.8.11 $$\underset{|\beta |\to \mathrm{\infty }}{lim}{}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q,\beta };\beta z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z),$$

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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, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(iii), §16.8 and Ch.16

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