§31.3 Basic Solutions

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

$\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ 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 $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ exists, is analytic in the disk $|z|<1$, and has the Maclaurin expansion

 31.3.1 $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)=\sum_{j=0}^{% \infty}c_{j}z^{j},$ $|z|<1$, ⓘ Defines: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$: 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 See also: Annotations for §31.3(i), §31.3 and Ch.31

where $c_{0}=1$,

 31.3.2 $a\gamma c_{1}-qc_{0}=0,$ ⓘ 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 See also: Annotations for §31.3(i), §31.3 and Ch.31
 31.3.3 $R_{j}c_{j+1}-(Q_{j}+q)c_{j}+P_{j}c_{j-1}=0,$ $j\geq 1$,

with

 31.3.4 $\displaystyle P_{j}$ $\displaystyle=(j-1+\alpha)(j-1+\beta),$ $\displaystyle Q_{j}$ $\displaystyle=j\left((j-1+\gamma)(1+a)+a\delta+\epsilon\right),$ $\displaystyle R_{j}$ $\displaystyle=a(j+1)(j+\gamma).$ ⓘ 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, $j$: nonnegative integer, $a$: 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 $z=0$ is

 31.3.5 $z^{1-\gamma}\mathit{H\!\ell}\left(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-% \gamma,\beta+1-\gamma,2-\gamma,\delta;z\right).$

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 $z=0$.

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

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

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

 31.3.8 $\mathit{H\!\ell}\left(\frac{a}{a-1},\frac{\alpha\beta a-q}{a-1};\alpha,\beta,% \epsilon,\delta;\frac{a-z}{a-1}\right),$
 31.3.9 $\left(\frac{a-z}{a-1}\right)^{1-\epsilon}\mathit{H\!\ell}\left(\frac{a}{a-1},% \frac{(a(\delta+\gamma)-\gamma)(1-\epsilon)}{a-1}+\frac{\alpha\beta a-q}{a-1};% \alpha+1-\epsilon,\beta+1-\epsilon,2-\epsilon,\delta;\frac{a-z}{a-1}\right).$

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

 31.3.10 $z^{-\alpha}\mathit{H\!\ell}\left(\frac{1}{a},\alpha\left(\beta-\epsilon\right)% +\frac{\alpha}{a}\left(\beta-\delta\right)-\frac{q}{a};\alpha,\alpha-\gamma+1,% \alpha-\beta+1,\delta;\frac{1}{z}\right),$
 31.3.11 $z^{-\beta}\mathit{H\!\ell}\left(\frac{1}{a},\beta\left(\alpha-\epsilon\right)+% \frac{\beta}{a}\left(\alpha-\delta\right)-\frac{q}{a};\beta,\beta-\gamma+1,% \beta-\alpha+1,\delta;\frac{1}{z}\right).$

§31.3(iii) Equivalent Expressions

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

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

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

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

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).