§31.3 Basic Solutions

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

§31.3(i) Fuchs–Frobenius Solutions at $z=0$
§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
§31.3(iii) Equivalent Expressions

§31.3(i) Fuchs–Frobenius Solutions at $z=0$

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 $z=0$ 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 $|z|<1$ , and has the Maclaurin expansion

31.3.1 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)=\sum _{{j=0}}^{{\infty}}c_{j}z^{j},$ $|z|<1$ ,

Defines:: $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions
Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : 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 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, $a$ : complex parameter, $q$ : 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:: $j$ : nonnegative integer, $q$ : 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, $j$ : nonnegative integer, $a$ : 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 $z=0$ 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, $z$ : 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:: 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 $z=0$ .

§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 $z=1$ 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, $z$ : 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:: 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, $z$ : 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:: TeX, pMML, png

Solutions of (31.2.1) corresponding to the exponents 0 and $1-\epsilon$ at $z=a$ 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, $z$ : 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:: 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, $z$ : 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.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, $z$ : 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.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, $z$ : 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)
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 $192/8=24$ 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, $z$ : 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.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, $z$ : 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.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).

§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

§31.3(i) Fuchs–Frobenius Solutions at $z=0$