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

§31.3 Basic Solutions

Referenced by:: §31.18
Permalink:: http://dlmf.nist.gov/31.3

§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

Keywords:: Heun’s equation
Referenced by:: §31.3(ii), §31.4, §31.5
Permalink:: http://dlmf.nist.gov/31.3.i

$\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ 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 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ exists, is analytic in the disk , and has the Maclaurin expansion

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

Defines:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

where $c_{0}=1$ ,

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

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

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

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

with

31.3.4

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

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

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

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, $\beta$ : real or complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/31.3.E4
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

When $\gamma\in\Integer$ , 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

Notes:: See Ronveaux (1995, Part A, pp. 36–41).
Keywords:: Heun’s equation
Referenced by:: §31.7(ii)
Permalink:: http://dlmf.nist.gov/31.3.ii

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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

31.3.8 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(\frac{a}{a-1},\frac{\alpha\beta a-% q}{a-1};\alpha,\beta,\epsilon,\delta;\frac{a-z}{a-1}\right),$

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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.E10
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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)
Permalink:: http://dlmf.nist.gov/31.3.E11
Encodings:: TeX, pMML, png

§31.3(iii) Equivalent Expressions

Notes:: See Snow (1952).
Keywords:: Heun’s equation
Permalink:: http://dlmf.nist.gov/31.3.iii

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, $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ is equal to

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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

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

Symbols:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : 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

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 31.2 Differential Equations 31.4 Solutions Analytic at Two Singularities: Heun Functions

§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