with complex parameters

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11: 15.1 Special Notation

$x$	real variable.
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$a, b, c$	real or complex parameters.
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15.1.1

{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),

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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, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.1.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

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15.1.2

\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z% \right)=\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,% b;c;z\right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.1 and Ch.15

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12: 31.9 Orthogonality

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31.9.1

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

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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
Permalink:: http://dlmf.nist.gov/31.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

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31.9.2

\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\zeta$ : point and $\theta_{m}$ : normalization constant
Permalink:: http://dlmf.nist.gov/31.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

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31.9.3

\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},

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Defines:: $\theta_{m}$ : normalization constant (locally)
Symbols:: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\zeta$ : point and $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients
Permalink:: http://dlmf.nist.gov/31.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

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31.9.6

\rho(s,t)=(s-t)(st)^{\gamma-1}\left((s-1)(t-1)\right)^{\delta-1}\*\left((s-a)(% t-a)\right)^{\epsilon-1},

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Defines:: $\rho(s,t)$ : weight function (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter
Permalink:: http://dlmf.nist.gov/31.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(ii), §31.9 and Ch.31

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13: 28.14 Fourier Series

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28.14.1

\operatorname{me}_{\nu}\left(z,q\right)=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(% q)e^{\mathrm{i}(\nu+2m)z},

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Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $q=h^{2}$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
A&S Ref:: 20.3.8 (in slightly different form)
Referenced by:: §28.22(ii)
Permalink:: http://dlmf.nist.gov/28.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

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28.14.2

\operatorname{ce}_{\nu}\left(z,q\right)=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(% q)\cos(\nu+2m)z,

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $m$ : integer, $q=h^{2}$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

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28.14.5

\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;

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Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d)
Permalink:: http://dlmf.nist.gov/28.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

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28.14.7

c_{-2m}^{-\nu}(q)=c_{2m}^{\nu}(q),

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Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

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28.14.8

c_{2m}^{\nu}(-q)=(-1)^{m}c_{2m}^{\nu}(q).

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Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

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14: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.3

\lambda+\mu=\gamma+\delta-1=\alpha+\beta-\epsilon.

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Defines:: $\lambda$ (locally) and $\mu$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.6

K_{j}=\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j-\mu)}{(2j+% \lambda-\mu-1)(2j+\lambda-\mu-2)},

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Symbols:: $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\lambda$ , $\mu$ and $K_{j}$ : coefficients
Referenced by:: Erratum (V1.1.7) for Equation (31.11.6)
Permalink:: http://dlmf.nist.gov/31.11.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: The sign has been corrected and the final term in the numerator $(j+\lambda-1)$ has been corrected to be $(j-\mu)$ .
Suggested 2022-09-14 by Hans Volkmer
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.7

L_{j}=a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma-\mu)(j+% \lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)(j-\beta% +\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)},

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Symbols:: $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\lambda$ , $\mu$ and $L_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/31.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.8

M_{j}=\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+1)(j+% \lambda)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)}.

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Symbols:: $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\lambda$ , $\mu$ and $M_{j}$ : coefficients
Referenced by:: Erratum (V1.1.7) for Equation (31.11.8)
Permalink:: http://dlmf.nist.gov/31.11.E8
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: The sign has been corrected and the final term in the numerator $(j-\mu+1)$ has been corrected to be $(j+\lambda)$ .
Suggested 2022-09-14 by Hans Volkmer
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12

P_{j}^{5}=\frac{{\left(\alpha\right)_{j}}{\left(1-\gamma+\alpha\right)_{j}}}{{% \left(1+\alpha-\beta+\epsilon\right)_{2j}}}z^{-\alpha-j}\*{{}_{2}F_{1}}\left({% \alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-\beta+\epsilon+2j};\frac{1}{z}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $P_{j}^{5}$ : first Kummer solution at $\infty$ of generalized hypergeometric differential equation
Referenced by:: §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)
Permalink:: http://dlmf.nist.gov/31.11.E12
Encodings:: pMML, png, TeX
Notational Change (effective with 1.1.7):: We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.
See also:: Annotations for §31.11(iii), §31.11 and Ch.31

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15: 31.16 Mathematical Applications

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31.16.1

\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A% _{j}{\sin}^{2j}\theta\*P^{(\gamma+\delta+2j-1,\epsilon-1)}_{n-j}\left(\cos% \left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}_{j}\left(\cos\left(2\phi% \right)\right),

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31.16.3

A_{0}=\frac{n!}{{\left(\gamma+\delta\right)_{n}}}\mathit{Hp}_{n,m}\left(1% \right),\quad Q_{0}A_{0}+R_{0}A_{1}=0,

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Symbols:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\beta$ : real or complex parameter, $A_{j}$ : coefficients, $Q_{j}$ and $R_{j}$
Referenced by:: §31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)
Permalink:: http://dlmf.nist.gov/31.16.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.21):: The initial data $A_{0}$ , previously missing, has now been included.
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.5

P_{j}=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+\delta+2j-3% )(\gamma+\delta+2j-2)},

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Defines:: $P_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.6

Q_{j}=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma+\delta-% 1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1)j(j+% \delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta-2)},

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Defines:: $Q_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.7

R_{j}=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+\delta+2j)(% \gamma+\delta+2j+1)}.

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Defines:: $R_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/31.16.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

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16: 31.10 Integral Equations and Representations

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31.10.2

\rho(t)=t^{\gamma-1}(t-1)^{\delta-1}(t-a)^{\epsilon-1},

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Defines:: $\rho(t)$ : weight function (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter
Permalink:: http://dlmf.nist.gov/31.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10 and Ch.31

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31.10.4

\mathcal{D}_{z}=z(z-1)(z-a)(\ifrac{{\partial}^{2}}{{\partial z}^{2}})+\left(% \gamma(z-1)(z-a)+\delta z(z-a)+\epsilon z(z-1)\right)(\ifrac{\partial}{% \partial z})+\alpha\beta z.

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Defines:: $\mathcal{D}_{z}$ : Heun’s operator (locally)
Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.10(ii)
Permalink:: http://dlmf.nist.gov/31.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10 and Ch.31

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31.10.6

p(t)=t^{\gamma}(t-1)^{\delta}(t-a)^{\epsilon}.

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Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter
Referenced by:: §31.10(ii)
Permalink:: http://dlmf.nist.gov/31.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10 and Ch.31

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31.10.10

\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\*{{}_{2}F_{1}}\left({\frac% {1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-\sigma+\beta\atop\gamma};\frac{% zt}{a}\right)\*{{}_{2}F_{1}}\left({-\frac{1}{2}+\delta+\sigma,-\frac{1}{2}+% \epsilon-\sigma\atop\delta};\frac{a(z-1)(t-1)}{(a-1)(zt-a)}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel and $\sigma$ : separation constant
Permalink:: http://dlmf.nist.gov/31.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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31.10.13

\rho(s,t)=(s-t)(st)^{\gamma-1}\left((1-s)(1-t)\right)^{\delta-1}\*\left((1-(s/% a))(1-(t/a))\right)^{\epsilon-1},

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Defines:: $\rho(s,t)$ : weight function (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter
Permalink:: http://dlmf.nist.gov/31.10.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(ii), §31.10 and Ch.31

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17: 31.2 Differential Equations

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31.2.1

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

\alpha+\beta+1=\gamma+\delta+\epsilon

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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, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8
Permalink:: http://dlmf.nist.gov/31.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.2(i), §31.2 and Ch.31

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31.2.2

w(z)=z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z),

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Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.2(ii), §31.2 and Ch.31

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31.2.6

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\theta}^{2}}+\left({(2\gamma-1)\cot\theta-% (2\delta-1)\tan\theta}-\frac{\epsilon\sin\left(2\theta\right)}{a-{\sin}^{2}% \theta}\right)\frac{\mathrm{d}w}{\mathrm{d}\theta}+4\frac{\alpha\beta{\sin}^{2% }\theta-q}{a-{\sin}^{2}\theta}w=0.

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Symbols:: $\cot\NVar{z}$ : cotangent function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\gamma$ : 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/31.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.2(iii), §31.2 and Ch.31

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31.2.8

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\left((2\gamma-1)\frac{% \operatorname{cn}\zeta\operatorname{dn}\zeta}{\operatorname{sn}\zeta}-(2\delta% -1)\frac{\operatorname{sn}\zeta\operatorname{dn}\zeta}{\operatorname{cn}\zeta}% -(2\epsilon-1)k^{2}\frac{\operatorname{sn}\zeta\operatorname{cn}\zeta}{% \operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}+4k^{2}(% \alpha\beta{\operatorname{sn}}^{2}\zeta-q)w=0.

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31.2.10

w(\xi)=\left(\wp\left(\xi\right)-e_{3}\right)^{(1-2\gamma)/4}\left(\wp\left(% \xi\right)-e_{2}\right)^{(1-2\delta)/4}\*\left(\wp\left(\xi\right)-e_{1}\right% )^{(1-2\epsilon)/4}W(\xi),

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathbb{L}$ : lattice and $W(\xi)$ : solution
Permalink:: http://dlmf.nist.gov/31.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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18: 15.11 Riemann’s Differential Equation

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15.11.2

a_{1}+a_{2}+b_{1}+b_{2}+c_{1}+c_{2}=1.

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Symbols:: $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

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15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

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15.11.4

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

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15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

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15.11.8

z^{\lambda}(1-z)^{\mu}P\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}0&1&\infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\lambda$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/15.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

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19: 31.7 Relations to Other Functions

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31.7.1

{{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=\mathit{H\!\ell}\left(1,\alpha% \beta;\alpha,\beta,\gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha% \beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\right).

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

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31.7.2

\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

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31.7.3

\mathit{H\!\ell}\left(4,\alpha\beta;\alpha,\beta,\tfrac{1}{2},\tfrac{2}{3}(% \alpha+\beta);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta% ;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

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31.7.4

\mathit{H\!\ell}\left(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2},\alpha\beta(\tfrac{1}{% 2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}(\alpha+\beta+1),\tfrac{1}{3% }(\alpha+\beta+1);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}% \beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-\left(\tfrac{3}{2}-i\tfrac{\sqrt{% 3}}{2}\right)z\right)^{3}\right).

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

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20: 12.1 Special Notation

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$x, y$	real variables.
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$a,\nu$	real or complex parameters.
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