31 Heun Functions Properties 31.2 Differential Equations 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.3 Basic Solutions

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Referenced by:: §31.18
Permalink:: http://dlmf.nist.gov/31.3
See also:: Annotations for Ch.31

§31.3(i) Fuchs–Frobenius Solutions at
§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
§31.3(iii) Equivalent Expressions

§31.3(i) Fuchs–Frobenius Solutions at

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Keywords:: Fuchs–Frobenius, Heun’s equation, basic solutions
Referenced by:: §31.3(ii), §31.4, §31.5
Permalink:: http://dlmf.nist.gov/31.3.i
See also:: Annotations for §31.3 and Ch.31

$\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ denotes the solution of (31.2.1) that corresponds to the exponent 0 at and assumes the value 1 there. If the other exponent is not a positive integer, that is, if $\gamma\neq 0,-1,-2,\dots$ , then from §2.7(i) it follows that $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ exists, is analytic in the disk , and has the Maclaurin expansion

31.3.1 $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)=\sum_{j=0}^{% \infty}c_{j}z^{j},$

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Defines:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions
Symbols:: : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $c_{j}$ : coefficients
Referenced by:: §31.3(iii), §31.7(ii)
Permalink:: http://dlmf.nist.gov/31.3.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(i), §31.3 and Ch.31

where $c_{0}=1$ ,

31.3.2 $a\gamma c_{1}-qc_{0}=0,$

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

31.3.3 $R_{j}c_{j+1}-(Q_{j}+q)c_{j}+P_{j}c_{j-1}=0,$ $j\geq 1$ ,

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Symbols:: : nonnegative integer, : real or complex parameter, $c_{j}$ : coefficients, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/31.3.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(i), §31.3 and Ch.31

with

31.3.4

$P_{j}=(j-1+\alpha)(j-1+\beta),$

$Q_{j}=j\left((j-1+\gamma)(1+a)+a\delta+\epsilon\right),$

$R_{j}=a(j+1)(j+\gamma).$

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Defines:: $P_{j}$ : coefficient (locally), $Q_{j}$ : coefficient (locally) and $R_{j}$ : coefficient (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : nonnegative integer, : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E4
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §31.3(i), §31.3 and Ch.31

Similarly, if $\gamma\neq 1,2,3,\dots$ , then the solution of (31.2.1) that corresponds to the exponent $1-\gamma$ at is

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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, : 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:: TeX, pMML, png
See also:: Annotations for §31.3(i), §31.3 and Ch.31

When $\gamma\in\mathbb{Z}$ , linearly independent solutions can be constructed as in §2.7(i). In general, one of them has a logarithmic singularity at .

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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Keywords:: Fuchs–Frobenius, Heun’s equation, basic solutions
Notes:: See Ronveaux (1995, Part A, pp. 36–41).
Referenced by:: §31.7(ii)
Permalink:: http://dlmf.nist.gov/31.3.ii
See also:: Annotations for §31.3 and Ch.31

With similar restrictions to those given in §31.3(i), the following results apply. Solutions of (31.2.1) corresponding to the exponents 0 and $1-\delta$ at are respectively,

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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

31.3.7 $(1-z)^{1-\delta}\*\mathit{H\!\ell}\left(1-a,((1-a)\gamma+\epsilon)(1-\delta)+% \alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-z\right).$

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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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

Solutions of (31.2.1) corresponding to the exponents 0 and $1-\epsilon$ at are respectively,

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, : complex variable, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

31.3.9 $\left(\frac{a-z}{a-1}\right)^{1-\epsilon}\mathit{H\!\ell}\left(\frac{a}{a-1},% \frac{(a(\delta+\gamma)-\gamma)(1-\epsilon)}{a-1}+\frac{\alpha\beta a-q}{a-1};% \alpha+1-\epsilon,\beta+1-\epsilon,2-\epsilon,\delta;\frac{a-z}{a-1}\right).$

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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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

Solutions of (31.2.1) corresponding to the exponents $\alpha$ and $\beta$ at $z=\infty$ are respectively,

31.3.10 $z^{-\alpha}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\alpha\left(\beta-% \epsilon\right)-\frac{\alpha}{a}\left(\beta-\delta\right);\alpha,\alpha-\gamma% +1,\alpha-\beta+1,\delta;\frac{1}{z}\right),$

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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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11), Erratum (V1.1.7) for Subsection 31.11(iii)
Permalink:: http://dlmf.nist.gov/31.3.E10
Encodings:: TeX, pMML, png
Correction (effective with 1.1.7):: The second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

31.3.11 $z^{-\beta}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\beta\left(\alpha-% \epsilon\right)-\frac{\beta}{a}\left(\alpha-\delta\right);\beta,\beta-\gamma+1% ,\beta-\alpha+1,\delta;\frac{1}{z}\right).$

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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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.3(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11)
Permalink:: http://dlmf.nist.gov/31.3.E11
Encodings:: TeX, pMML, png
Correction (effective with 1.1.7):: The second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

§31.3(iii) Equivalent Expressions

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Keywords:: Heun’s equation, automorphisms, basic solutions, equivalent expressions
Notes:: See Snow (1952).
Permalink:: http://dlmf.nist.gov/31.3.iii
See also:: Annotations for §31.3 and Ch.31

Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). There are 192 automorphisms in all, so there are equivalent expressions for each of the 8. For example, $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is equal to

31.3.12 $\mathit{H\!\ell}\left(1/a,q/a;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma-\delta% ;z/a\right),$

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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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(iii), §31.3 and Ch.31

which arises from the homography $\tilde{z}=z/a$ , and to

31.3.13 $(1-z)^{-\alpha}\*\mathit{H\!\ell}\left(\frac{a}{a-1},-\frac{q-a\alpha\gamma}{a% -1};\alpha,\alpha+1-\delta,\gamma,\alpha+1-\beta;\frac{z}{z-1}\right),$

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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, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §31.3(iii), §31.3 and Ch.31

which arises from $\tilde{z}=z/(z-1)$ , and also to 21 further expressions. The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).

31.2 Differential Equations 31.4 Solutions Analytic at Two Singularities: Heun Functions

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§31.3 Basic Solutions

Contents

§31.3(i) Fuchs–Frobenius Solutions at

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

§31.3(iii) Equivalent Expressions