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1: 31.3 Basic Solutions

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31.3.2

a\gamma c_{1}-qc_{0}=0,

ⓘ

Symbols:: $\gamma$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter and $c_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/31.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(i), §31.3 and Ch.31

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31.3.5

z^{1-\gamma}\mathit{H\!\ell}\left(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-% \gamma,\beta+1-\gamma,2-\gamma,\delta;z\right).

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Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.3(iii), §31.7(ii)
Permalink:: http://dlmf.nist.gov/31.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(i), §31.3 and Ch.31

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31.3.6

\mathit{H\!\ell}\left(1-a,\alpha\beta-q;\alpha,\beta,\delta,\gamma;1-z\right),

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Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

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31.3.7

(1-z)^{1-\delta}\*\mathit{H\!\ell}\left(1-a,((1-a)\gamma+\epsilon)(1-\delta)+% \alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-z\right).

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

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31.3.8

\mathit{H\!\ell}\left(\frac{a}{a-1},\frac{\alpha\beta a-q}{a-1};\alpha,\beta,% \epsilon,\delta;\frac{a-z}{a-1}\right),

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Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

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2: 31.1 Special Notation

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$x$ , $y$	real variables.
…
$a$	complex parameter, $\|a\|\geq 1,a\neq 1$ .
$q,\alpha,\beta,\gamma,\delta,\epsilon,\nu$	complex parameters.

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3: 15.7 Continued Fractions

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15.7.1

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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15.7.3

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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15.7.5

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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4: 31.12 Confluent Forms of Heun’s Equation

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31.12.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{% z(z-1)}w=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : 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{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{% \gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\gamma+z\right)z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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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5: 31.14 General Fuchsian Equation

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31.14.1

{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},

\sum_{j=1}^{N}q_{j}=0

.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14 and Ch.31

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31.14.3

w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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31.14.4

\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{% \gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,

\sum_{j=1}^{N}\tilde{q}_{j}=0

,

ⓘ

Defines:: $W(z)$ : solution (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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6: 15.5 Derivatives and Contiguous Functions

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15.5.1

\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1% ;c+1;z\right),

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(i), §15.5 and Ch.15

►

15.5.2

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a% \right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z% \right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(i), §15.5 and Ch.15

►

15.5.3

\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §18.17(iv)
Permalink:: http://dlmf.nist.gov/15.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(i), §15.5 and Ch.15

►

15.5.4

\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)% \right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §18.17(iv)
Permalink:: http://dlmf.nist.gov/15.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(i), §15.5 and Ch.15

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15.5.5

\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}F% \left(a,b;c;z\right)\right)={\left(c-a\right)_{n}}z^{c-a+n-1}(1-z)^{a-n+b-c}\*% F\left(a-n,b;c;z\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §18.17(iv)
Permalink:: http://dlmf.nist.gov/15.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.5(i), §15.5 and Ch.15

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7: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

8: 31.4 Solutions Analytic at Two Singularities: Heun Functions

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31.4.1

(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),

m=0,1,2,\dots

.

ⓘ

Symbols:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.6
Permalink:: http://dlmf.nist.gov/31.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.4 and Ch.31

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31.4.2

q=\cfrac{a\gamma P_{1}}{Q_{1}+q-\cfrac{R_{1}P_{2}}{Q_{2}+q-\cfrac{R_{2}P_{3}}{% Q_{3}+q-\cdots}}},

ⓘ

Symbols:: $\gamma$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Referenced by:: §31.18
Permalink:: http://dlmf.nist.gov/31.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.4 and Ch.31

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31.4.3

(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),

m=0,1,2,\dots

,

ⓘ

Symbols:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.4, §31.6
Permalink:: http://dlmf.nist.gov/31.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.4 and Ch.31

…

9: 31.6 Path-Multiplicative Solutions

… ►

31.6.1

(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),

m=0,1,2,\dots

,

ⓘ

Symbols:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\nu$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $s_{j}$ : parameters
Permalink:: http://dlmf.nist.gov/31.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.6 and Ch.31

…

10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.1

\begin{bmatrix}0&a\gamma&0&\dots&0\\ P_{1}&-Q_{1}&R_{1}&\dots&0\\ 0&P_{2}&-Q_{2}&&\vdots\\ \vdots&\vdots&&\ddots&R_{n-1}\\ 0&0&\dots&P_{n}&-Q_{n}\end{bmatrix},

ⓘ

Symbols:: $\gamma$ : real or complex parameter, $n$ : nonnegative integer, $a$ : complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/31.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

… ►

31.5.2

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

ⓘ

Defines:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials
Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

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