Möbius, Gauge, Discrete, and Confluence Transformations

A useful Heun transformation is more than a parameter identity. It consists of a coordinate map, a scalar gauge, a parameter map, and a rule for transporting normalized bases, branches, and continuation paths. Möbius maps and index gauges are invertible on suitable punctured domains. Confluence is different: it is a singular limit that changes the singularity pattern and generally discards information.

This page develops one reproducible calculation engine rather than listing 192 formulas. It then uses that engine to derive representative general-Heun transformations, explain the finite discrete orbit, and determine when a transformation survives the general-to-confluent limit.

Every transformation needs a passport

Start from

$$y_{zz}+p(z)y_z+r(z)y=0$$

and make the substitution

$$z=\phi(x), \qquad y\bigl(\phi(x)\bigr)=\chi(x)u(x).$$

Assume locally that $\phi'\neq0$ and $\chi\neq0$, and define

$$g=\frac{\chi'}{\chi}.$$

The coordinate pullback first produces

$$P_\phi = \phi' p(\phi)-\frac{\phi''}{\phi'},$$

and

$$R_\phi = (\phi')^2r(\phi).$$

Define also

$$Q_{\phi,\chi} = R_\phi+P_\phi g+g'+g^2.$$

All primes in the transformed equation now mean $x$-derivatives. Direct differentiation gives the compact result

$$u'' + \left( P_\phi+2g \right)u' + Q_{\phi,\chi}u =0.$$

This formula catches the two errors that cause most incorrect Heun identities: omitting $-\phi''/\phi'$ and transforming the singularity position without transforming the accessory coefficient.

Wronskians provide an independent audit. If $y_i(\phi)=\chi u_i$, then

$$\Wr_z[y_1,y_2]\bigl(\phi(x)\bigr) = \frac{\chi(x)^2}{\phi'(x)} \Wr_x[u_1,u_2].$$

The four operations that recur below should not be conflated.

Operation	What it changes	Is it locally invertible?
Coordinate pullback	Singular-point positions and local coordinates	Yes, where $\phi'\neq0$
Scalar gauge	Exponent representatives and Wronskian normalization	Yes, where $\chi\neq0$ on a chosen branch
Constant basis change	Only the coordinates on one solution space	Yes, if its matrix is nonsingular
Confluence	Singularity pattern, formal type, and often the basis problem	No; it is a singular limit

An equation transformation becomes a normalized-function identity only after the source and target germs have the same base point, exponent, leading coefficient, and branch.

The general-Heun ledger

The house convention is

$$\begin{aligned} y'' &+ \left( \frac{\gamma}{z} + \frac{\delta}{z-1} + \frac{\epsilon}{z-a} \right)y'\\ &+ \frac{\alpha\beta z-q} {z(z-1)(z-a)}y =0, \end{aligned}$$

with

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

The exponent pairs at $0$, $1$, $a$, and $\infty$ are

$$\begin{gathered} (0,1-\gamma), \qquad (0,1-\delta),\\ (0,1-\epsilon), \qquad (\alpha,\beta). \end{gathered}$$

At infinity the notation means that the two behaviors are $z^{-\alpha}$ and $z^{-\beta}$. The standard and normal-form page derives this ledger; here it serves as a rapid check on every parameter map.

Möbius maps relabel four singular points

A homography

$$x=\frac{Az+B}{Cz+D}, \qquad AD-BC\neq0,$$

can send any ordered triple of distinct points to $0$, $1$, and $\infty$. Consequently the four points $\{0,1,a,\infty\}$ admit $4!=24$ homographic relabelings. Only the quotient by the Klein four subgroup acts effectively on the cross-ratio, so a generic $a$ has the six-value orbit

$$\begin{gathered} a, \qquad 1-a, \qquad \frac1a,\\ \frac1{1-a}, \qquad \frac{a}{a-1}, \qquad \frac{a-1}{a}. \end{gathered}$$

The two involutions

$$s(u)=1-u, \qquad t(u)=\frac1u$$

satisfy $s^2=t^2=(st)^3=1$ and generate the effective $\mathfrak S_3$ action.

The generic cross-ratio orbit is the $\mathfrak S_4/V_4\simeq\mathfrak S_3$ orbit generated by $s$ and $t$. It collapses to three values on the harmonic orbit $\{-1,\tfrac12,2\}$ and to two values when $a^2-a+1=0$.

The orbit of $a$ is only the geometric part. Exponent labels and $q$ must move with the punctures.

Moving the fourth point by a rescaling

Set

$$z=ax, \qquad y(z)=u(x), \qquad A=\frac1a.$$

After multiplying the pulled-back equation by $a^2$, one obtains

$$\begin{aligned} u'' &+ \left( \frac{\gamma}{x} + \frac{\epsilon}{x-1} + \frac{\delta}{x-A} \right)u'\\ &+ \frac{ \alpha\beta x-q/a }{ x(x-1)(x-A) }u =0. \end{aligned}$$

Therefore

$$\begin{aligned} a^\ast&=\frac1a, & q^\ast&=\frac qa,\\ \gamma^\ast&=\gamma, & \delta^\ast&=\epsilon, & \epsilon^\ast&=\delta, \end{aligned}$$

while $\alpha$ and $\beta$ are unchanged. The accessory transformation is forced by the numerator; it is not inferred from the cross-ratio.

Let $H\!\ell(a,q;\alpha,\beta,\gamma,\delta;z)$ denote the solution analytic at zero and normalized to one there. For the literal normalized germ identity below, assume $\gamma\notin\mathbb Z_{\leq0}$. Resonant values require a separately declared limiting or logarithmic prescription. In a common neighborhood of zero,

$$\begin{aligned} &H\!\ell \left( a,q;\alpha,\beta,\gamma,\delta;z \right)\\ &\quad= H\!\ell \left( \frac1a,\frac qa; \alpha,\beta,\gamma,\epsilon; \frac za \right). \end{aligned}$$

The first Taylor coefficient checks the normalization. The target Heun series has coefficient

$$\frac{q^\ast}{a^\ast\gamma} = \frac q\gamma$$

in its variable $x=z/a$, hence coefficient $q/(a\gamma)$ in $z$, exactly as on the left.

Moving infinity requires a prefactor

Six of the 24 homographies keep infinity at infinity. The other 18 move it to a finite point, so one of the former infinity exponents must be removed by a gauge. A representative is

$$x=\frac{z}{z-1}, \qquad y(z)=(1-z)^{-\alpha}u(x).$$

The transformed parameters are

$$\begin{aligned} a^\ast &= \frac{a}{a-1}, & q^\ast &= \frac{a\alpha\gamma-q}{a-1},\\ \alpha^\ast &= \alpha, & \beta^\ast &= \alpha+1-\delta,\\ \gamma^\ast &= \gamma, & \delta^\ast &= \alpha+1-\beta. \end{aligned}$$

The Fuchs relation gives $\epsilon^\ast=\epsilon$. With the branch of $(1-z)^{-\alpha}$ equal to one at zero, the normalized local identity is

$$\begin{aligned} &H\!\ell \left( a,q;\alpha,\beta,\gamma,\delta;z \right)\\ &\quad= (1-z)^{-\alpha} H\!\ell \left( a^\ast,q^\ast; \alpha^\ast,\beta^\ast,\gamma^\ast,\delta^\ast; x \right). \end{aligned}$$

It holds first as an identity of normalized germs and then along continuation paths on which the coordinate and logarithm branches are transported consistently.

Index gauges exchange Frobenius exponents

At zero, try

$$y=z^s u.$$

The coefficient of the unwanted $z^{-2}$ term in the transformed zeroth-order coefficient is

$$s(s+\gamma-1).$$

Thus $s=0$ leaves the chosen exponent unchanged, while $s=1-\gamma$ exchanges the two exponent representatives. For the nontrivial choice,

$$\begin{aligned} \gamma^\ast &= 2-\gamma,\\ \alpha^\ast &= \alpha+1-\gamma,\\ \beta^\ast &= \beta+1-\gamma,\\ q^\ast &= q+(1-\gamma)(a\delta+\epsilon), \end{aligned}$$

with $a$, $\delta$, and $\epsilon$ unchanged. In the older Heun literature this is called an F-homotopic transformation. “Index gauge” makes its action more transparent.

The three finite points can be treated simultaneously. Set

$$\chi(z) = z^{s_0}(z-1)^{s_1}(z-a)^{s_a},$$

where

$$\begin{aligned} s_0&\in\{0,1-\gamma\},\\ s_1&\in\{0,1-\delta\},\\ s_a&\in\{0,1-\epsilon\}, \end{aligned}$$

and let

$$S=s_0+s_1+s_a.$$

If $y=\chi u$, the transformed equation is again in general-Heun form with

$$\begin{aligned} \gamma^\ast&=\gamma+2s_0,\\ \delta^\ast&=\delta+2s_1,\\ \epsilon^\ast&=\epsilon+2s_a,\\ \alpha^\ast&=\alpha+S,\\ \beta^\ast&=\beta+S. \end{aligned}$$

The complete accessory map is

$$\begin{aligned} q^\ast ={}& q+(a\delta+\epsilon)s_0 +a\gamma s_1+\gamma s_a\\ &+2a s_0s_1+2s_0s_a. \end{aligned}$$

There is no $s_1s_a$ term. One way to see the asymmetry is to recover $q^\ast$ from the residue at the distinguished point $z=0$. Equivalently, compose the three elementary transformations in a fixed order, always using the already transformed parameters. The individual accessory shifts are

$$\begin{aligned} q_0-q &= (a\delta+\epsilon)(1-\gamma),\\ q_1-q &= a\gamma(1-\delta),\\ q_a-q &= \gamma(1-\epsilon). \end{aligned}$$

Choosing either exponent at each of three finite points gives $2^3=8$ index gauges, including the identity.

The shifted normal-form accessory coordinate from the standard-to-normal form map,

$$\kappa_{\mathrm H} = q-\frac{\gamma}{2} \left( a\delta+\epsilon \right),$$

is invariant under all eight gauges:

$$\kappa_{\mathrm H}^\ast = \kappa_{\mathrm H}.$$

At the same time, the selected exponent differences $1-\gamma$, $1-\delta$, and $1-\epsilon$ change sign. Thus the eight standard-form representatives reduce to the same centered normal equation; the index gauges record which exponent representatives were used before centering.

These gauges are generally multivalued. On a fixed simply connected punctured domain they are invertible, but a positive loop around a selected point multiplies the gauge by $\ee^{2\pi\ii s_j}$. If $M_u$ is the corresponding monodromy matrix for the transformed equation, then

$$M_y = \ee^{2\pi\ii s_j}M_u.$$

Projective monodromy is unchanged, while $GL(2)$ monodromy is twisted by a scalar character. Returning to an $SL(2)$ normalization requires a separate determinant-line retwist; it is not automatic. The branch factor is therefore harmless neither for linear monodromy nor for a normalized local function.

The discrete orbit has 192 automorphisms

The count is now immediate:

$$\begin{aligned} 24 &\quad\text{homographic relabelings},\\ 8 &\quad\text{index gauges},\\ 24\cdot8 &=192 \quad\text{composite automorphisms}. \end{aligned}$$

For generic parameters the group is

$$D_4 \simeq (\mathbb Z_2)^3\rtimes\mathfrak S_4,$$

the Coxeter group of type $D_4$. This $D_4$ is not the order-eight dihedral group that is sometimes denoted by the same symbol.

192 expressions are not 192 independent solutions

The solution space is still two-dimensional. There are eight distinguished local Frobenius branches—two at each of four singular points—and each has 24 equivalent transformed expressions. Resonance, special parameters, or geometric stabilizers can make nominally distinct expressions coincide.

The order-24 stabilizer of one normalized local branch is abstractly isomorphic to $\mathfrak S_4$, but it should not be identified naively with the literal $\mathfrak S_4$ that permutes the four punctures. The equation is also unchanged by $\alpha\leftrightarrow\beta$. The conventional count fixes an ordering of the infinity labels and does not count this parameter relabeling as a new automorphism. Adjoining it can double a formal list, but it creates neither a new differential operator nor a new solution-space dimension.

A family automorphism need not fix one equation

A generic element of the 192-element group changes the parameter tuple. It is an automorphism of the canonical family, not a symmetry of one fixed operator. Fixed-operator symmetry requires the transformed tuple to equal the original one.

For example, the rescaling $x=z/a$ fixes the tuple on the locus

$$a=-1, \qquad q=0, \qquad \delta=\epsilon.$$

There $x=-z$, and the equation is invariant under reflection. Whenever the analytic solution normalized by $y(0)=1$ is unique, that germ is even. A harmonic or equianharmonic value of $a$ alone enlarges only the geometric stabilizer; exponent and accessory data must satisfy their own fixed-point conditions before the differential equation gains a symmetry.

A practical transformation algorithm is shorter and safer than a catalogue:

1. choose the source point and one of its two exponents;
2. choose a homography that carries it to the desired normalized point;
3. add an index gauge if the selected target exponent is not zero;
4. substitute into the differential equation and match all coefficients;
5. enforce the Fuchs relation and compute $q^\ast$ explicitly;
6. normalize the target germ and transport branches and paths;
7. check a local coefficient and the Wronskian.

Normalized bases carry matrices

The basis atlas on the preceding page fixed ordered local frames rather than unnamed solutions. A transformation must carry that additional data.

Let $\mathcal B_j$ be the source frame at point $j$, let $\widetilde{\mathcal B}_{\sigma(j)}$ be a target frame at the relabeled point, and write

$$\mathcal B_j(z) = \chi(x) \widetilde{\mathcal B}_{\sigma(j)}(x) N_j.$$

The constant matrix $N_j$ restores the house normalization. It may contain leading-coefficient factors, a column permutation, branch phases, or—at resonance—a triangular Levelt renormalization.

Using the house connection convention

$$\mathcal B_k = \mathcal B_j C_{jk},$$

and the analogous target relation

$$\widetilde{\mathcal B}_{\sigma(k)} = \widetilde{\mathcal B}_{\sigma(j)} \widetilde C_{\sigma(j)\sigma(k)},$$

one finds

$$C_{jk} = N_j^{-1} \widetilde C_{\sigma(j)\sigma(k)} N_k.$$

This formula explains why a correct equation transformation can still give a wrong connection coefficient: the endpoint normalization matrices were dropped. At an irregular endpoint, $N_j$ is only part of the passport. The coordinate map also transports angular lifts, Stokes sectors, and lateral summation directions. Labels such as “ingoing”, “outgoing”, and “recessive” must be checked after the transformation.

Confluence is a singular limit, not another automorphism

The confluence hierarchy page derived an exact general-to-confluent family. Fix confluent-Heun parameters $(q,\alpha,\gamma,\delta,\epsilon)$ with $\epsilon\neq0$, set $a=A$, and choose

$$\begin{aligned} \epsilon_{\mathrm G} &= -A\epsilon,\\ \alpha_{\mathrm G} &= \frac{\alpha}{\epsilon},\\ \beta_{\mathrm G} &= -A\epsilon +\gamma+\delta-1-\frac{\alpha}{\epsilon},\\ q_{\mathrm G} &= -Aq. \end{aligned}$$

As $A\to\infty$ on a fixed outer $z$-domain, the GHE tends to

$$\begin{aligned} y'' &+ \left( \frac{\gamma}{z} + \frac{\delta}{z-1} + \epsilon \right)y'\\ &+ \frac{\alpha z-q}{z(z-1)}y =0. \end{aligned}$$

The limit changes a regular singularity into rank-one irregular data at infinity, so it has no inverse within the Heun family.

An index gauge commutes with this limit

Let

$$s=1-\gamma$$

and apply the zero-index gauge to every member of the general-Heun family. The exact transformed GHE parameters include

$$\begin{aligned} \gamma^\ast &= 2-\gamma,\\ \alpha_{\mathrm G}^\ast &= \frac{\alpha+s\epsilon}{\epsilon},\\ q_{\mathrm G}^\ast &= -A \left[ q+s(\epsilon-\delta) \right]. \end{aligned}$$

The transformed $\beta_{\mathrm G}^\ast$ has precisely the scaling required by the Fuchs relation:

$$\beta_{\mathrm G}^\ast = -A\epsilon +\gamma^\ast+\delta-1 -\frac{\alpha+s\epsilon}{\epsilon}.$$

Consequently the lower horizontal transformation is the CHE index gauge

$$y=z^{1-\gamma}u$$

with

$$\begin{aligned} \gamma^\ast &= 2-\gamma,\\ \alpha^\ast &= \alpha+(1-\gamma)\epsilon,\\ q^\ast &= q+(1-\gamma)(\epsilon-\delta), \end{aligned}$$

and $\delta^\ast=\delta$, $\epsilon^\ast=\epsilon$. For this generator, “transform then confluence” and “confluence then transform” commute exactly.

Horizontal arrows are invertible index gauges at fixed $A$. Vertical arrows are singular limits on a declared outer domain; the right-hand limit also requires a controlled transformed basis normalization.

Not every general-Heun automorphism has such a limit. For example, the homography $x=z/A$ sends every fixed outer $z$-domain toward $x=0$ as $A\to\infty$. It describes an inner scaling chart, not a nondegenerate outer CHE symmetry. A transformation survives confluence only when its coordinate, gauge, parameter map, domain, branches, and basis normalizers all have controlled limits.

The confluent equation has a different finite symmetry

Part of the four-point symmetry disappears when one puncture becomes irregular, and part reappears as an exponential gauge. Besides the two finite-point index flips, useful CHE generators include

$$x=1-z$$

with

$$\begin{aligned} q^\ast&=q-\alpha, & \alpha^\ast&=-\alpha,\\ \gamma^\ast&=\delta, & \delta^\ast&=\gamma, & \epsilon^\ast&=-\epsilon, \end{aligned}$$

and

$$y=\ee^{-\epsilon z}u$$

with

$$\begin{aligned} q^\ast &= q-\gamma\epsilon,\\ \alpha^\ast &= \alpha-\epsilon(\gamma+\delta),\\ \epsilon^\ast &= -\epsilon, \end{aligned}$$

while $\gamma$ and $\delta$ remain unchanged. Together with the two index flips, these generate the standard 16-element CHE transformation group. This is a new symmetry problem adapted to two regular points and one irregular point, not a literal copy of the 192-element general-Heun group.

Exceptional rational pullbacks

A Möbius map has degree one. A rational map of degree greater than one usually creates additional singular preimages and critical points, so it does not belong to the generic 192-element automorphism group. Exceptional nontrivial Heun-to-hypergeometric reductions by rational substitution require compatible cross-ratio, exponent, and accessory data. In Maier’s classification the admissible substitutions are polynomial. Harmonic and equianharmonic configurations give quadratic and cubic cases, but a symmetric value of $a$ alone does not guarantee a reduction.

This distinction matters computationally. A generic automorphism rewrites the same Heun-class problem. An exceptional pullback exists only on a lower-dimensional parameter locus and can genuinely reduce the equation to a simpler class.

A reusable transformation audit

Before using a transformed formula, record:

1. the source equation convention and ordered parameters;
2. the coordinate $\phi$, its inverse, and the working domain;
3. the gauge $\chi$ and every logarithm or root branch;
4. the full transformed coefficient functions;
5. the image of each singular point and its ordered exponents;
6. the transformed accessory parameter;
7. the local or asymptotic basis normalization matrices;
8. the continuation paths, angular lifts, and boundary labels;
9. for confluence, the fixed variable, parameter ray, domain, and limiting basis renormalization.

A Taylor coefficient tests a normalized regular germ. Abel’s identity and the Wronskian rule test the gauge. Formal exponential parts and equal-magnitude rays test an irregular endpoint.

Common pitfalls

Transforming only the cross-ratio. A Möbius map also permutes exponent labels and changes $q$. Substitute into the differential equation; do not guess the accessory map from geometry.

Promoting an equation equivalence to a function identity. A map can send the source normalization point to a different singularity. A connection coefficient, branch phase, or leading factor is then required.

Reading 192 as a dimension. The number counts transformed representations. The solution space remains two-dimensional, with eight distinguished local branches in the generic four-point problem.

Ignoring multivalued gauges. Factors such as $z^s$ and $(1-z)^s$ carry scalar monodromy and branch-dependent constants. The equation may be correct while the claimed normalized solution is wrong.

Assuming every symmetry survives confluence. A finite-$A$ automorphism may collapse the outer domain or make transformed parameters diverge. Check the complete transformation passport before taking the limit.

Exercises

1. Derive the master substitution law

Starting from $z=\phi(x)$ and $y(\phi(x))=\chi(x)u(x)$, derive the transformed equation and the Wronskian rule.

Solution

Let

$$Y(x)=y\bigl(\phi(x)\bigr).$$

The chain rule gives

$$y_z=\frac{Y'}{\phi'}$$

and

$$y_{zz} = \frac{Y''}{(\phi')^2} -\frac{\phi''Y'}{(\phi')^3}.$$

Multiplication by $(\phi')^2$ gives

$$Y''+P_\phi Y'+R_\phi Y=0,$$

where

$$P_\phi = \phi'p(\phi)-\frac{\phi''}{\phi'}, \qquad R_\phi = (\phi')^2r(\phi).$$

Now put $Y=\chi u$. Since

$$\frac{\chi''}{\chi} = g'+g^2,$$

division by $\chi$ yields

$$Q_{\phi,\chi} = R_\phi+P_\phi g+g'+g^2$$

and

$$u'' + (P_\phi+2g)u' + Q_{\phi,\chi}u =0.$$

For two solutions,

$$\Wr_x[\chi u_1,\chi u_2] = \chi^2\Wr_x[u_1,u_2].$$

On the other hand,

$$\Wr_x \left[ y_1(\phi),y_2(\phi) \right] = \phi' \Wr_z[y_1,y_2](\phi).$$

Equating these expressions gives

$$\Wr_z[y_1,y_2](\phi) = \frac{\chi^2}{\phi'} \Wr_x[u_1,u_2].$$

2. Move the fourth singularity

Use $z=ax$ to derive the map

$$(a,q,\gamma,\delta,\epsilon) \longmapsto \left( a^{-1},q/a,\gamma,\epsilon,\delta \right)$$

and verify the normalized Heun identity through its linear Taylor term.

Solution

With $y(z)=u(x)$,

$$y_z=\frac1a u_x, \qquad y_{zz}=\frac1{a^2}u_{xx}.$$

After multiplication by $a^2$, the first-derivative coefficient is

$$\frac{\gamma}{x} + \frac{\epsilon}{x-1} + \frac{\delta}{x-1/a}.$$

The zeroth-order coefficient becomes

$$\frac{ \alpha\beta x-q/a }{ x(x-1)(x-1/a) }.$$

This proves the parameter map. A normalized Heun germ has

$$H\!\ell(a,q;\ldots;z) = 1+\frac{q}{a\gamma}z+O(z^2)$$

when $a\gamma\neq0$. The transformed germ has coefficient

$$\frac{q/a}{(1/a)\gamma} = \frac q\gamma$$

in $x$. Substituting $x=z/a$ recovers $q/(a\gamma)$ in $z$.

3. Exchange the exponents at zero

Insert $y=z^s u$ into the general Heun equation. Derive the admissible values of $s$, the nontrivial parameter map, and the scalar monodromy twist.

Solution

Here

$$g=\frac{s}{z}, \qquad g'+g^2=\frac{s(s-1)}{z^2}.$$

The coefficient of $z^{-2}$ in $r+pg+g'+g^2$ is

$$s\gamma+s(s-1) = s(s+\gamma-1).$$

Hence

$$s=0 \qquad\text{or}\qquad s=1-\gamma.$$

For the second choice, matching residues and the behavior at infinity gives

$$\begin{aligned} \gamma^\ast&=2-\gamma,\\ \alpha^\ast&=\alpha+1-\gamma,\\ \beta^\ast&=\beta+1-\gamma,\\ q^\ast&=q+(1-\gamma)(a\delta+\epsilon), \end{aligned}$$

with the remaining parameters unchanged. A positive loop around zero sends $z^s$ to $\ee^{2\pi\ii s}z^s$, so

$$M_y = \ee^{2\pi\ii(1-\gamma)}M_u.$$

The two monodromies therefore have the same projective class but can differ as linear lifts.

4. Explain the number 192

Count the homographic and index choices. Explain why the result does not give 192 linearly independent solutions, and determine the sizes of the cross-ratio orbits at harmonic and equianharmonic values.

Solution

There are

$$4!=24$$

permutations of the four singular points and

$$2^3=8$$

choices of finite-point exponent index. Therefore

$$24\cdot8=192.$$

These transformations reorganize eight distinguished local branches, two at each singularity. Each branch has 24 equivalent representations, and all are solutions of second-order equations with two-dimensional solution spaces.

For $a=-1$, the six generic anharmonic expressions reduce to

$$\left\{ -1,\frac12,2 \right\},$$

so the harmonic orbit has size three. If $a^2-a+1=0$, then $1/a=1-a$ and all six expressions reduce to the two conjugate roots. The equianharmonic orbit has size two.

5. Transport a connection matrix

Suppose

$$\mathcal B_j = \chi\widetilde{\mathcal B}_{\sigma(j)}N_j$$

at each endpoint. Derive the transformation law for $C_{jk}$ in the house convention $\mathcal B_k=\mathcal B_jC_{jk}$.

Solution

Substituting

$$\widetilde{\mathcal B}_{\sigma(k)} = \widetilde{\mathcal B}_{\sigma(j)} \widetilde C_{\sigma(j)\sigma(k)}$$

gives

$$\begin{aligned} \mathcal B_k &= \chi \widetilde{\mathcal B}_{\sigma(k)}N_k\\ &= \chi \widetilde{\mathcal B}_{\sigma(j)} \widetilde C_{\sigma(j)\sigma(k)}N_k. \end{aligned}$$

But

$$\mathcal B_jC_{jk} = \chi \widetilde{\mathcal B}_{\sigma(j)} N_jC_{jk}.$$

Therefore

$$C_{jk} = N_j^{-1} \widetilde C_{\sigma(j)\sigma(k)} N_k.$$

If $N_j$ permutes columns, it changes the ordering of the basis. If it is triangular, as can occur in a resonant Levelt basis, the transformed connection matrix is not obtained by diagonal rescaling alone.

6. Commute an index gauge with confluence

Apply the zero-index gauge with $s=1-\gamma$ to the exact GHE-to-CHE scaling. Show that the transformed family again has the required scaling form and recover the CHE parameter map.

Solution

The general-Heun zero-index map gives

$$\alpha_{\mathrm G}^\ast = \alpha_{\mathrm G}+s = \frac{\alpha+s\epsilon}{\epsilon}.$$

For the accessory parameter,

$$\begin{aligned} q_{\mathrm G}^\ast &= q_{\mathrm G} +s(A\delta+\epsilon_{\mathrm G})\\ &= -Aq+s(A\delta-A\epsilon)\\ &= -A \left[ q+s(\epsilon-\delta) \right]. \end{aligned}$$

Also $\gamma^\ast=2-\gamma$. The transformed $\beta_{\mathrm G}^\ast=\beta_{\mathrm G}+s$ can be rewritten as

$$\beta_{\mathrm G}^\ast = -A\epsilon +\gamma^\ast+\delta-1 -\frac{\alpha+s\epsilon}{\epsilon},$$

so the Fuchs relation and the exact confluence scaling are preserved. Taking $A\to\infty$ yields

$$\begin{aligned} \gamma^\ast &= 2-\gamma,\\ \alpha^\ast &= \alpha+(1-\gamma)\epsilon,\\ q^\ast &= q+(1-\gamma)(\epsilon-\delta). \end{aligned}$$

7. Diagnose a false local-function identity

The reflection $x=1-z$ gives the valid equation-level map

$$\begin{aligned} a^\ast&=1-a,\\ q^\ast&=\alpha\beta-q,\\ \gamma^\ast&=\delta,\\ \delta^\ast&=\gamma. \end{aligned}$$

Why is the naive identity

$$H\!\ell(a,q;\ldots;z) = H\!\ell(a^\ast,q^\ast;\ldots;1-z)$$

generically false?

Solution

The symbol $H\!\ell$ on the left denotes the solution normalized at $z=0$. The target Heun function on the right is normalized at $x=0$, which corresponds to the source point

$$z=1.$$

Thus the right-hand side is a particular local branch based at the wrong source singularity. To express the zero-normalized source solution in that basis, one needs the connection coefficients between the bases at $z=0$ and $z=1$, together with a continuation path and compatible branches. The equation parameter map is correct; the normalized-function identity is not.

References

• NIST Digital Library of Mathematical Functions, §31.2(v) gives the three index transformations, the 24 homographies, and the 192 composite automorphisms; §31.3(iii), especially equations 31.3.12 and 31.3.13, gives normalized equivalent expressions and the eight sets of 24; and §31.12 states the confluent canonical equations and singularity patterns.
• R. S. Maier, “The 192 solutions of the Heun equation”, Mathematics of Computation 76 (2007), 811–843, especially Theorem 4.3, Corollary 4.6, Theorem 4.7, and §5, identifies the Coxeter groups and the eight-by-24 organization.
• A. Ronveaux (ed.), Heun’s Differential Equations, Oxford University Press, 1995, Part A, Chapter 2, pp. 14–29, develops homographic and F-homotopic transformations in the classical convention.
• L. Jaccoud El-Jaick and B. D. B. Figueiredo, “Transformations of Heun’s equation and its integral relations”, Journal of Physics A: Mathematical and Theoretical 44 (2011), 075204, records systematic general and confluent Heun transformation generators and emphasizes ordered parameter maps.
• R. S. Maier, “On reducing the Heun equation to the hypergeometric equation”, Journal of Differential Equations 213 (2005), 171–203, classifies exceptional Heun-to-hypergeometric reductions by polynomial substitutions, including the harmonic and equianharmonic cases.