Semi-commuting and commuting operators for the Heun family

TL;DR

This paper classifies and constructs differential operators that semi-commute and commute with the Heun class, enabling solutions to complex differential equations beyond standard computational tools.

Contribution

It introduces the most general semi-commuting and commuting operator families for the Heun class, including a new commuting generalized Heun equation.

Findings

Identified all semi-commuting families with Heun operators.

Discovered a generalized Heun equation that commutes with the Heun operator.

Constructed solutions to a complex fourth-order differential equation.

Abstract

We derive the most general families of differential operators of first and second degree semi-commuting with the differential operators of the Heun class. Among these families we classify all those families commuting with the Heun class. In particular, we discover that a certain generalized Heun equation commutes with the Heun differential operator allowing us to construct the general solution to a complicated fourth order linear differential equation with variable coefficients which Maple 16 cannot solve.

Tables

Table 1. Table 1: Heun family of differential operators P 𝑃 P as defined in ( 1 )

Equation	$p_1\left(x\right)$	$p_0\left(x\right)$	parameters
Heun	$\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{ϵ}{x-a}$	$\frac{\alpha \beta x-q}{x\left(x-1\right)\left(x-a\right)}$	$ϵ=\alpha +\beta +1-\delta -\gamma ,\alpha ,\beta ,\gamma ,\delta ,q\in ℂ,a\in ℝ\backslash \{0,1\}$
confluent Heun	$p+\frac{\gamma }{x}+\frac{\delta }{x-1}$	$\frac{p\alpha x-q}{x\left(x-1\right)}$	$p,q,\alpha ,\gamma ,\delta \in ℂ$
reduced confluent Heun	$\frac{\gamma }{x}+\frac{\delta }{x-1}$	$\frac{\kappa x+q}{x\left(x-1\right)}$	$\kappa ,\gamma ,\delta ,q\in ℂ$
biconfluent Heun	$\frac{\tau }{x}+\frac{\nu }{x^2}-1$	$-\frac{\alpha x+q}{x}$	$\tau ,\nu ,\alpha ,q\in ℂ$
double confluent Heun	$\frac{\tau }{x}+\frac{\nu }{x^2}-1$	$-\frac{\alpha x+q}{x^2}$	$\tau ,\nu ,\alpha ,q\in ℂ$
triconfluent Heun	$\sigma -x^2$	$\alpha x-q$	$\sigma ,\alpha ,q\in ℂ$
representative triconfluent Heun	$0$	$A_0+A_{1}x+A_2{x}^2-\frac{9}{4}{x}^4$	$A_0,A_1,A_2\in ℂ$

Equations

$$P=d_x^2+p_1(x)d_x+p_0(x),d_x:=\frac{d}{dx}.$$

$$Q=q_n(x)d_x^n+q_{n-1}(x)d_x^{n-1}+\cdots +q_1(x)d_x+q_0(x)$$

$$d_x{q}_n$$

$$d_x{q}_k$$

$$d_x^2{q}_0(x)+p_1(x)d_x{q}_0(x)=\sum _{k=1}^n{q}_k(x)d_x^k{p}_0(x)$$

$$\phi (x;{\beta }_0,{\beta }_1,\cdots ,{\beta }_n)=0.$$

$$[Q,P]=[2d_x{q}_0(x)-{\beta }_1{d}_x{p}_1(x)]d_x+d_x^2{q}_0(x)+p_1(x)d_x{q}_0(x)-{\beta }_1{d}_x{p}_0(x).$$

$$d_x{q}_0(x)=\frac{{\beta }_1}{2}{d}_x{p}_1(x),$$

$$d_x^2{q}_0(x)+p_1(x)d_x{q}_0(x)-{\beta }_1{d}_x{p}_0(x)=0.$$

$$d_x{q}_0(x)=\frac{{\beta }_1}{2}{d}_x{p}_1(x)-\frac{{\beta }_1}{2}{p}_0(x)$$

$$d_x{q}_1(x)$$

$$d_x{q}_0(x)$$

$$d_x^2{q}_0(x)+p_1(x)d_x{q}_0(x)$$

$$d_x{q}_1(x)={\beta }_2{d}_x{p}_1(x)-\frac{{\beta }_2}{2}{p}_0(x),d_x{q}_0(x)=\frac{1}{2}{q}_1(x)d_x{p}_1(x)+\frac{1}{2}{\beta }_2{d}_x^2{p}_1(x)-\frac{1}{2}P[q_1(x)].$$

$$Q={\beta }_1{d}_x+\frac{{\beta }_1}{2}\left(\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{ϵ}{x-a}\right)+{\beta }_0,ϵ=\alpha +\beta +1-\delta -\gamma$$

$$L={\beta }_1{d}_x^3+\left(\rho +\sum _{i=1}^3\frac{A_i}{x-x_i}\right)d_x^2+\left(\sum _{i=1}^3\frac{B_i}{x-x_i}\right)d_x+\sum _{i=1}^3\frac{C_i}{x-x_i}$$

$$f(x)=c_1+\frac{1}{x-a}\left(c_2+c_3{e}^{-\frac{{\beta }_0}{{\beta }_1}x}\right).$$

$$f(x)=c_1+c_{2}x+c_3{e}^{-\frac{{\beta }_0}{{\beta }_1}x}.$$

$$\rho ={\beta }_0,A_2=A_3=3{\beta }_1,B_2=2{\beta }_0-\frac{6{\beta }_1}{a-1},B_3=4{\beta }_0-B_2,C_2=-\frac{2{\beta }_0}{a-1},C_3=-C_2$$

$$f(x)=\frac{1}{(x-1)(x-a)}\left(c_1+c_{2}x+c_3{e}^{-\frac{{\beta }_0}{{\beta }_1}x}\right).$$

$$B_1$$

$$C_1$$

$$f(x)=\frac{1}{x(x-1)(x-a)}\left(c_1+c_{2}x+c_3{e}^{-\frac{{\beta }_0}{{\beta }_1}x}\right).$$

$$A_1=A_3=3{\beta }_1,B_1=2{\beta }_0-\frac{6{\beta }_1}{a},B_3=4{\beta }_0-B_1,C_1=-\frac{2{\beta }_0}{a},C_3=-C_1$$

$$f(x)=\frac{1}{x(x-a)}\left(c_1+c_{2}x+c_3{e}^{-\frac{{\beta }_0}{{\beta }_1}x}\right).$$

$$Q={\beta }_2{d}_x^2+\left[{\beta }_2\left(\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{ϵ}{x-a}\right)+{\beta }_1\right]d_x+\frac{a_3{x}^3+a_2{x}^2+a_{1}x+a_0}{x(x-1)(x-a)}$$

$$a_3={\beta }_0,a_2=-{\beta }_0(a+1)+\frac{{\beta }_1}{2}(\delta +ϵ+\gamma ),a_1={\beta }_{0}a+\alpha \beta {\beta }_2-\frac{{\beta }_1}{2}\left[a(\delta +\gamma )+ϵ+\gamma \right],a_0=\frac{1}{2}{\beta }_1\gamma a-{\beta }_{2}q$$

$$L=d_x^4+\left(\nu +\sum _{i=1}^3\frac{{\mathfrak{A}}_i}{x-x_i}\right)d_x^3+\left[\mu +\sum _{k=1}^2\sum _{i=1}^3\frac{{\mathfrak{B}}_{i,k}}{(x-x_i{)}^k}\right]d_x^2+\left(\sum _{k,i=1}^3\frac{{\mathfrak{C}}_{i,k}}{(x-x_i{)}^k}\right)d_x+\sum _{k,i=1}^3\frac{{\mathfrak{D}}_{i,k}}{(x-x_i{)}^k}$$

$$\psi (x)=c_{1}H(a,q,\alpha ,\beta ,\gamma ,\delta ;x)+c_2{x}^{1-\gamma }H(a,q-(\gamma -1)(a\delta +ϵ),\beta -\gamma +1,\alpha -\gamma +1,2-\gamma ,\delta ;x),$$

$$d_x^{2}f+\left(\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{ϵ}{x-a}+\kappa \right)d_{x}f+\frac{b_2{x}^2+b_{1}x+b_0}{x(x-1)(x-a)}f=0$$

$${\mathfrak{A}}_1$$

$${\mathfrak{B}}_{3,1}$$

$${\mathfrak{C}}_{1,1}$$

Full text

Keywords: semi-commuting operators, commuting operators, Heun equation, confluent Heun equation, biconfluent Heun equation, double confluent Heun equation, triconfluent Heun equation, generalized Heun equation, factorizations

I Introduction

The interest in commuting differential expressions started almost $140$ years ago with the work of Floquet followed by contributions of Wallenberg and Schur FWS but the decisive input came from the work of Burchnall and Chaundy which allowed to connect this mathematical area to algebraic geometry BC123 . Since many operators in mathematical physics do not commute, the most celebrated example being represented by the position and momentum operators in Quantum Mechanics, the theory of non-commuting operators is also interesting in its own. Furthermore, there can be cases when certain symmetries or degeneracies are present leading differential operators to almost commute, or semi-commute but what does it mean that two operators semi-commute? Let us consider two operators $P$ and $T$ of degree $m$ and $n$, respectively. The commutator $[P,T]$ will be in general an operator of degree $m+n-1$. Let us suppose that $P$ is given and $Q$ be an arbitrary operator of degree $n$. We say that $P$ and $Q$ semi-commute if the highest order term in $[P,Q]$ vanish, i.e. the commutator of these two operators is an operator of degree $m+n-2$. The problem of finding operators with analytic coefficients that semi-commute with a given monic differential operator having analytic coefficients has been thoroughly treated in Gorder . The interest in the construction of a class of operators semi-commuting with a given operator $P$ associated to some equation of mathematical physics relies in the fact that such a class may contain a subclass of commuting operators. Hence, starting with a certain differential operator and after having computed the corresponding class of semi-commuting operators, we can impose additional constraints that if fulfilled they can lead to a class of commuting operators. If two operators commute, then the Burchnall and Chaundy theory ensures that they must share a solution which in turn we may use to reduce the order of the ODE associated to the operator $P$. In what follows we consider a given differential operator $P$ with analytic coefficients, namely

[TABLE]

The operator

[TABLE]

semi-commute with $P$ if we can find coefficients $q_{0},\cdots,q_{n}$ such that Gorder

[TABLE]

If the additional condition

[TABLE]

is satisfied for all $x$, then $P$ and $Q$ commute. Note that in principle the unknown coefficients $q_{k}$’s can be obtained from (3) and will depend on some integration constants $\beta_{i}$. Substituting these solutions in (4) yield the relation

[TABLE]

In some cases the $\beta_{i}$’s can be chosen so as to permit commutativity. If this is not the case, we can look instead for a solution $x_{0}=x_{0}(\beta_{0},\beta_{1},\cdots,\beta_{n})$ to (5). If such a solution exists, we say that $P$ and $Q$ locally commute at $x=x_{0}$. In this work we extend the treatment of Airy, Bessel, and hypergeometric operators presented in Gorder to differential operators associated to the Heun equation and its confluent forms Ronveaux ; Slavyanov displayed in Table 1. We recall that the Heun equation(HE) has been originally constructed by the German mathematician Karl Heun (1889) Heun as a generalization of the hypergeometric equation. To underline the importance of the HE in mathematical physics we recall that it contains the generalized spheroidal equation, the Coulomb spheroidal equation, Lam e, Mathieau, and Ince equations as special cases. The fields of applications of the HE in physics are so large that it is not possible to describe them here in detail. However, a review of many general situations relevant to physics, chemistry, and engineering where the HE and its confluent forms occur can be found in Ronveaux ; Slavyanov . Since we will show in the next section that a certain generalized Heun equation (GHE) commutes with the Heun differential operator, thus allowing us to construct the general solution of a rather complicated fourth order linear ODE with variable coefficients, some comments on the GHE are in order. The GHE is a second order differential equation with three regular singular points and one irregular singular point at infinity. It generalizes the ellipsoidal wave equation as well as the Heun equation. SS obtained under certain conditions all connection coefficients between the Floquet solutions at the finite singularities, thus determining the full monodromy group of the GHE. This equation plays an important role in applications in the context of Quantum Field Theory in curved space-times. More precisely, it describes the radial spinors of an electron or neutrino immersed in the gravitational potential of a Kerr-Newman black hole and also the static perturbations for the non-extremal Reissner-Nordström solution to the Einstein field equations BSW .

Before we derive the classes of semi-commuting and commuting operators of first and second degree, we show that equation (3) derived in Gorder is not correct. This can be already seen in the case $n=1$. Let $P$ be given as in (1). Suppose that we want to construct the class of operators $Q$ of degree one commuting with $P$. Let $Q=\beta_{1}d_{x}+q_{0}(x)$ with $q_{0}$ a yet-to-be determined function. It is not difficult to verify that

[TABLE]

Hence, the function $q_{0}$ will be represented by the solution of the first order ODE

[TABLE]

whereas the commutativity condition reads

[TABLE]

However, even though in the case $n=1$ equation (4) coincides with our (7), equation (3) becomes instead

[TABLE]

and as a result $Q$ and $P$ cannot commute. Hence, $(27)$ in Gorder should be taken with some caution. In the case $n=2$ we consider the same operator $P$ as before but now $Q=\beta_{2}d_{xx}+q_{1}(x)d_{x}+q_{0}(x)$. It can be easily verified that the operators $Q$ and $P$ will commute whenever

[TABLE]

Notice again that equation (4) for $n=2$ reproduces correctly our equation (10) but (3) or equivalently $(27)$ in Gorder leads to the wrong results

[TABLE]

II Semi-commuting operators for the Heun family

We construct classes of semi-commuting operators for the differential operators associated to different Heun-like differential equations. We also investigate the problem of the existence of subclasses of commuting operators among the aforementioned classes. For each differential operator associated to the equations presented in Table 1 we derive when possible all operators of first and second degree commuting with it.

II.1 The Heun operator

II.1.1 The case $n=1$

With the help of (6) we find the following family of first degree operators

[TABLE]

semi-commuting with the operator $P_{H}$ associated to the Heun equation. The commutativity condition (7) will be satisfied for certain choices of the parameters that we list here below. Note that if $L:=QP_{H}=P_{H}Q$, the solutions to the equations $Q\varphi=0$ and $P_{H}\psi=0$ allow to construct the general solution to the third order ODE $Lf=0$ with

[TABLE]

where $x_{1}=0$, $x_{2}=1$, and $x_{3}=a$.

Case $\alpha=\gamma=\delta=q=0$, and $\beta=1$ or $\beta=\gamma=\delta=q=0$, and $\alpha=1$. The general solutions to the ODEs $P_{H}\psi=0$ and $Q\varphi=0$ are $\psi(x)=c_{1}+c_{2}(x-a)^{-1}$ and $\varphi(x)=c_{3}(x-a)^{-1}\mbox{exp}(-\beta_{0}x/\beta_{1})$, respectively. Moreover, the general solution to the ODE $Lf=0$ with $L$ as in (12), $\rho=\beta_{0}$, $A_{3}=3\beta_{1}$ and $B_{3}=2\rho$, while all other coefficients are zero, is given by

[TABLE] 2. 2.

Case $\alpha=\gamma=q=0$, $\beta=1$ and $\delta=2$ or $\beta=\gamma=q=0$, $\alpha=1$, and $\delta=2$. See case $1.$ with $a=1$. If $\alpha=\delta=q=0$, $\beta=1$, and $\gamma=2$ or $\beta=\delta=q=0$, $\alpha=1$, and $\gamma=2$, see case $1.$ with $a=0$. 3. 3.

Case $\alpha=\gamma=\delta=q=0$, and $\beta=-1$ or $\beta=\gamma=\delta=q=0$, and $\alpha=-1$. The general solutions to the ODEs $P_{H}\psi=0$ and $Q\varphi=0$ are $\psi(x)=c_{1}+c_{2}x$ and $\varphi(x)=c_{3}\mbox{exp}(-\beta_{0}x/\beta_{1})$, respectively. Moreover, the general solution to the ODE $Lf=0$ with $L$ as in (12), $\rho=\beta_{0}$, and $A_{i}=B_{i}=C_{i}=0$ for all $i=1,2,3$ is

[TABLE] 4. 4.

Case $\gamma=q=0$, $\alpha=\delta=2$, and $\beta=1$ or $\gamma=q=0$, $\beta=\delta=2$, and $\alpha=1$. The general solutions to the ODEs $P_{H}\psi=0$ and $Q\varphi=0$ are $\psi(x)=(x-1)^{-1}(x-a)^{-1}(c_{1}+c_{2}x)$ and $\varphi(x)=(x-1)^{-1}(x-a)^{-1}\mbox{exp}(-\beta_{0}x/\beta_{1})$, respectively. Moreover, the general solution to the ODE $Lf=0$ with $L$ as in (12), $A_{1}=B_{1}=C_{1}=0$

[TABLE]

is

[TABLE] 5. 5.

Case $\beta=\gamma=\delta=2$, $\alpha=3$, and $q=2a+2$ or $\alpha=\gamma=\delta=2$, $\beta=3$, and $q=2a+2$. The general solutions to the ODEs $P_{H}\psi=0$ and $Q\varphi=0$ are $\psi(x)=[x(x-1)(x-a)]^{-1}(c_{1}+c_{2}x)$ and $\varphi(x)=c_{3}[x(x-1)(x-a)]^{-1}\mbox{exp}(-\beta_{0}x/\beta_{1})$, respectively. Furthermore, the general solution to the ODE $Lf=0$ with $L$ as in (12), $\rho=\beta_{0}$, $A_{i}=3\beta_{1}$ for all $i=1,2,3$ and

[TABLE]

is given by

[TABLE] 6. 6.

Case $\alpha=\gamma=q=2$, $\beta=1$, and $\delta=0$ or $\beta=\gamma=q=2$, $\alpha=1$, and $\delta=0$. The general solutions to the ODEs $P_{H}\psi=0$ and $Q\varphi=0$ are $\psi(x)=[x(x-a)]^{-1}(c_{1}+c_{2}x)$ and $\varphi(x)=c_{3}[x(x-a)]^{-1}\mbox{exp}(-\beta_{0}x/\beta_{1})$. Moreover, the general solution to the ODE $Lf=0$ with $L$ as in (12), $\rho=\beta_{0}$, $A_{2}=B_{2}=C_{2}=0$ and

[TABLE]

is

[TABLE]

II.1.2 The case $n=2$

With the help of (8) and (9) we find the following family of second degree operators

[TABLE]

with $\epsilon=\alpha+\beta+1-\delta-\gamma$ and

[TABLE]

semi-commuting with the operator $P_{H}$. The commutativity condition (10) will be satisfied for certain choices of the parameters that we list here below. Observe that if $L:=QP_{H}=P_{H}Q$, the general solution to the fourth order ODE $Lf=0$ with

[TABLE]

can be immediately constructed from the solutions of the equations $Q\varphi=0$ and $P_{H}\psi=0$.

Case $\beta_{1}=0$, and $\beta_{2}\neq 0$. The operator $P_{H}$ is represented by the general Heun operator given in Table 1 and $Q$ is given by (13) with $a_{3}=\mu$, $a_{2}=-\mu(a+1)$, $a_{1}=\mu a+\alpha\beta$, $a_{0}=-q$ and $\mu=\beta_{0}/\beta_{2}$. Moreover, the general solution to the ODE $P_{H}\psi=0$ is

[TABLE]

where $H(\cdot)$ denotes the Heun function. Regarding the equation $Q\varphi=0$ observe that it has an irregular singular point at infinity and three finite regular singular points. The transformation $\varphi(x)=e^{Ax}f(x)$ with $A^{2}=-a_{3}$ brings the aforementioned equation into the generalized Heun equation (GHE)BSW

[TABLE]

with $\kappa=2A$, $b_{0}=Aa\gamma+a_{0}$, $b_{1}=Aa(A-\delta-\gamma)-A(\epsilon+\gamma)+a_{1}$, $b_{2}=-A^{2}(a+1)+A(\alpha+\beta+1)+a_{2}$. Then, the general solution to the equation $Q\varphi=0$ in a neighbourhood of the singularities $x_{i}$ of the GHE can be written as $\phi_{i}(x)=e^{Ax}\left[c_{3,i}h_{1,i}(x)+c_{4,i}h_{2,i}(x)\right]$, where $h_{1,i}$ and $h_{2,i}$ are two particular solutions to the GHE defined up to the next singularity. Then, the general solution to the ODE $Lf=0$ with $L$ as in (14), $\nu=0$ and

[TABLE]

[TABLE]

is

[TABLE]

with $i=1,2,3$. It is interesting to observe that Maple $16$ fails to solve the fourth order ODE $Lf=0$. Furthermore, note that $Q=\beta_{2}P_{H}$ whenever $\beta_{0}=0$. In this case the general solution to the ODE $Lf=0$ is expressed in terms of Heun functions only. 2. 2.

Case $\alpha=\gamma=\delta=q=0$ and $\beta=1$, or $\beta=\gamma=\delta=q=0$, and $\alpha=1$. The operator $P_{H}$ and the solution to $P_{H}\psi=0$ are given as in case $1.$ ($n=1$) of the Heun operator. Moreover, the solution to $Q\varphi=0$ is

[TABLE]

Then, the general solution to the ODE $Lf=0$ with $L$ as in (14) and non-vanishing coefficients $\mu=\beta_{0}/\beta_{2}$, $\nu=\beta_{1}/\beta_{2}$, $\mathfrak{A}_{3}=4$, $\mathfrak{B}_{3,1}=3\nu$ and $\mathfrak{C}_{3,1}=2\mu$ is

[TABLE] 3. 3.

Case $\alpha=\gamma=q=0$, $\beta=1$, and $\delta=2$, or $\beta=\gamma=q=0$, $\alpha=1$, and $\delta=2$. We find that $P_{H}$ and the solution to $P_{H}\psi=0$ are given as in case $2.$ ($n=1$) of the Heun operator. Moreover, $Q$, the solution to $Q\varphi=0$, the operator $L$, and the solution of $Lf=0$ can be obtained from the previous case with $a=1$. If instead $\alpha=\delta=q=0$, $\beta=1$, and $\gamma=2$, or $\beta=\delta=q=0$, $\alpha=1$, and $\gamma=2$, we find that $P_{H}$ and the solution to $P_{H}\psi=0$ are given as in case $1.$ ($n=1$) of the Heun operator with $a=0$. Moreover, $Q$, the solution to $Q\varphi=0$, the operator $L$, and the solution to $Lf=0$ can be obtained from case $2.$ ($n=2$) of the Heun operator with $a=0$. 4. 4.

Case $\alpha=\gamma=\delta=q=0$, and $\beta=-1$, or $\beta=\gamma=\delta=q=0$, and $\alpha=-1$. The operator $P_{H}$ and the solution to $P_{H}\psi=0$ are given as in case $3.$ ($n=1$) of the Heun operator. Furthermore, $Q=\beta_{2}d^{2}_{x}+\beta_{1}d_{x}+\beta_{0}$ and the solution to $Q\varphi=0$ is $\varphi(x)=c_{3}e^{m_{-}x}+c_{4}e^{m_{+}x}$ with $m_{\pm}$ given in (15). Then, the general solution to the equation $Lf=0$ with $L=\beta_{2}d^{4}_{x}+\beta_{1}d^{3}_{x}+\beta_{0}d_{x}$ is

[TABLE] 5. 5.

Case $\gamma=q=0$, $\alpha=\delta=2$, and $\beta=1$, or $\gamma=q=0$, $\beta=\delta=2$, and $\alpha=1$. We find that $P_{H}$ and the solution of $P_{H}\psi=0$ are given as in case $4.$ ($n=1$) of the Heun operator. Moreover, the solution to $Q\varphi=0$ is given by $\varphi(x)=[(x-1)(x-a)]^{-1}\left[c_{3}e^{m_{-}x}+c_{4}e^{m_{+}x}\right]$ with $m_{\pm}$ as in (15). Then, the general solution to the ODE $Lf=0$ with $L$ as in (14) and non-vanishing coefficients $\mu=\beta_{0}/\beta_{2}$, $\nu=\beta_{1}/\beta_{2}$ and

[TABLE]

is

[TABLE] 6. 6.

Case $\beta=\gamma=\delta=2$, $\alpha=3$, and $q=2a+2$, or $\alpha=\gamma=\delta=2$, $\beta=3$, and $q=2a+2$. The solutions to the equations $P_{H}\psi=0$ and $Q\varphi=0$ are $\psi(x)=[x(x-1)(x-a)]^{-1}(c_{1}+c_{2}x)$ and $\varphi(x)=[x(x-1)(x-a)]^{-1}\left(c_{3}e^{m_{-}x}+c_{4}e^{m_{+}x}\right)$, respectively. Then, the general solution to the ODE $Lf=0$ with $L$ as in (14) and non-vanishing coefficients $\mu=\beta_{0}/\beta_{2}$, $\nu=\beta_{1}/\beta_{2}$, $\mathfrak{A}_{1}=\mathfrak{A}_{2}=\mathfrak{A}_{3}=4$ and

[TABLE]

is

[TABLE] 7. 7.

Case $\alpha=\gamma=q=2$, $\beta=1$, and $\delta=0$, or $\beta=\gamma=q=2$, $\alpha=1$, and $\delta=0$. The operator $P_{H}$ and the solution to $P_{H}\psi=0$ are given as in $7.$ ($n=1$) of the Heun operator. Moreover, the solution to $Q\varphi=0$ is $\phi(x)=[x(x-a)]^{-1}\left(c_{3}e^{m_{-}x}+c_{4}e^{m_{+}x}\right)$. Finally, the solution to the ODE $Lf=0$ with $L$ as in (14) and non-vanishing coefficients $\mu=\beta_{0}/\beta_{2}$, $\nu=\beta_{1}/\beta_{2}$, $\mathfrak{A}_{1}=\mathfrak{A}_{3}=4$ and

[TABLE]

is represented by

[TABLE]

II.2 The confluent Heun operator

II.2.1 The case $n=1$

Using (6) yields the following family of first degree operators

[TABLE]

semi-commuting with the operator $P_{CH}$ associated to the confluent Heun equation. The commutativity condition (7) will be satisfied for certain choices of the parameters that we list here below. Note that if $L:=QP_{CH}=P_{CH}Q$, the solutions to the equations $Q\varphi=0$ and $P_{CH}\psi=0$ allow to construct the general solution to the third order ODE $Lf=0$ with

[TABLE]

where $x_{1}=0$ and $x_{2}=1$.

Cases $\gamma=p=q=\delta=0$; $\gamma=p=q=0$ and $\delta=2$; $p=q=\delta=0$ and $\gamma=2$; $\gamma=\delta=2$, $p=0$ and $q=-2$ have been already analyzed in Section II.1.1. 2. 2.

Case $\alpha=\gamma=q=\delta=0$. The solution to $P_{H}\psi=0$ is $\psi(x)=c_{1}+c_{2}e^{-px}$. Furthermore, $Q$ is given as in case $3.$ ($n=1$) of the Heun operator. Finally, the solution of the third order ODE $Lf=0$ with $L=\beta_{1}d^{3}_{x}+(\beta_{1}p+\beta_{0})d^{2}_{x}+\beta_{0}pd_{x}$ is

[TABLE] 3. 3.

Case $\alpha=1$, $\gamma=q=0$ and $\delta=2$. The solution to the equation $P_{CH}\psi=0$ is $\psi(x)=(x-1)^{-1}\left(c_{1}+c_{2}e^{-px}\right)$. Furthermore, $Q$ and the solution to $Q\varphi=0$ can be obtained from Section II.1.1. The general solution to the equation $Lf=0$ with $L$ as in (17) and non-vanishing coefficients $\xi=\beta_{1}p+\beta_{0}$, $\eta=\beta_{0}p$, $\mathfrak{a}_{2}=3\beta_{1}$, $\mathfrak{b}_{2}=2\xi$, $\mathfrak{c}_{2}=\eta$ is

[TABLE] 4. 4.

Case $\alpha=1$, $\gamma=2$, $q=p$ and $\delta=0$. The solution to the equation $P_{CH}\psi=0$ is $\psi(x)=x^{-1}\left(c_{1}+c_{2}e^{-px}\right)$. Moreover, $Q$ and the solution to $Q\varphi=0$ can be obtained from Section II.1.1. The general solution to the equation $Lf=0$ with $L$ as in (17) and non-vanishing coefficients $\xi=\beta_{1}p+\beta_{0}$, $\eta=\beta_{0}p$, $\mathfrak{a}_{1}=3\beta_{1}$, $\mathfrak{b}_{1}=2\xi$, $\mathfrak{c}_{1}=\eta$ is

[TABLE] 5. 5.

Case $\alpha=\gamma=\delta=2$ and $q=p-2$. The solution to $P_{CH}\psi=0$ is $\psi(x)=[x(x-1)]^{-1}\left(c_{1}+c_{2}e^{-px}\right)$. Moreover, $Q$ and the solution to $Q\varphi=0$ can be obtained from Section II.1.1. The general solution to the equation $Lf=0$ with $L$ as in (17) and non-vanishing coefficients $\xi=\beta_{1}p+\beta_{0}$, $\eta=\beta_{0}p$, $\mathfrak{a}_{1}=\mathfrak{a}_{2}=3\beta_{1}$, $\mathfrak{b}_{1}=\xi-\mathfrak{a}_{1}$, $\mathfrak{b}_{2}=2\xi-\mathfrak{b}_{1}$, $\mathfrak{c}_{1}=\eta-2\xi$, $\mathfrak{c}_{2}=2\eta-\mathfrak{c}_{1}$ is

[TABLE]

In the case of the reduced confluent Heun differential operator $P_{RCH}$ we find that the most general first degree differential operator $Q$ semi-commuting with $P_{RCH}$ is given by (16). By means of (7) we can verify that the operators $P_{RCH}$ and $Q$ will commute whenever $\gamma=k=q=\delta=0$, or $\gamma=k=q=0$, and $\delta=2$, or $\gamma=2$, and $k=q=\delta=0$, or $\gamma=q=\delta=2$, and $k=0$ but these cases reduce to one of the cases treated above.

II.3 The case $n=2$

By means of (8) and (9) we find the following family of second degree operators

[TABLE]

semi-commuting with the operator $P_{CH}$ associated to the confluent Heun equation. The commutativity condition (10) will be satisfied for certain choices of the parameters that we list here below. Notice that if $L:=QP_{CH}=P_{CH}Q$, the solutions to the equations $Q\varphi=0$ and $P_{CH}\psi=0$ allow to construct the general solution the the fourth order ODE $Lf=0$ with

[TABLE]

with $\eta=\beta_{0}p$.

Case $\beta_{1}=\beta_{2}p$. The solution to $P_{CH}\psi=0$ is Ronveaux

[TABLE]

with

[TABLE]

Moreover, the solution to $Q\varphi=0$ is given by

[TABLE]

with

[TABLE]

Then, the general solution to the equation $Lf=0$ with $L$ as in (18) and non-vanishing coefficients $\Gamma=p$, $\mathfrak{A}_{1}=\gamma$, $\mathfrak{A}_{2}=\delta$ and

[TABLE]

[TABLE]

[TABLE]

is given by

[TABLE]

[TABLE]

It is interesting to observe that also in this case Maple $16$ cannot solve the fourth order ODE $Lf=0$ discussed above. 2. 2.

The cases $\alpha=\gamma=\delta=2$ and $q=p=-2$, or $\gamma=\delta=2$, $p=0$, and $q=-2$, or $\delta=p=q=0$, and $\gamma=2$, or $\gamma=p=q=0$, and $\delta=2$, or $\gamma=p=q=\delta=0$ have been already discussed in Section II.1.2. 3. 3.

Case $\alpha=1$, $\gamma=2$, $q=p$, and $\delta=0$. The solutions to $P_{CH}\psi=0$ and $Q\varphi=0$ have been already computed in Section II.2.1 and Section II.1.2, respectively. Finally, the solution to the equation $Lf=0$ with $L$ as in (18) and non-vanishing coefficients $\Gamma=\beta_{2}p+\beta_{1}$, $\mathfrak{A}_{2}=4\beta_{2}$, $\lambda=\beta_{1}p+\beta_{0}$, $\mathfrak{B}_{1,1}=3\Gamma$, $\mathfrak{C}_{1,1}=2\Gamma$ and $\mathfrak{D}_{1,1}=\eta$ is

[TABLE]

with $m_{\pm}$ defined in (15). 4. 4.

Case $\alpha=1$, $\gamma=q=0$, and $\delta=2$. The solutions to $P_{CH}\psi=0$ and $Q\varphi=0$ have been already computed in Section II.2.1 and Section II.1.2, respectively. Moreover, the solution to the equation $Lf=0$ with $L$ as in (18) and coefficients $\Gamma$, $\mathfrak{A}_{2}$ and $\lambda$ as in the case above and $\mathfrak{B}_{2,1}=3\Gamma$, $\mathfrak{C}_{2,1}=2\Gamma$ and $\mathfrak{D}_{2,1}=\eta$ is

[TABLE] 5. 5.

Case $\alpha=\gamma=\delta=q=0$. The solutions to $P_{CH}\psi=0$ and $Q\varphi=0$ have been already computed in Section II.2.1 and Section II.1.2, respectively. Moreover, the solution to the equation $Lf=0$ with $L$ as in (18) and non-vanishing coefficients $\Gamma$, $\lambda$ and $\eta$ given as in the previous case is

[TABLE]

In the case of the reduced confluent Heun differential operator $P_{RCH}$ we find that the most general second degree differential operator $Q$ semi-commuting with $P_{RCH}$ is given by

[TABLE]

By means of (7) we can verify that the operators $P_{RCH}$ and $Q$ will commute whenever $\gamma=q=\delta=2$ and $k=0$, or $\gamma=2$ and $k=q=\delta=0$, or $\gamma=k=q=0$ and $\delta=2$, or $\gamma=k=q=\delta=0$ but these cases reduce to one of the cases treated above.

II.4 The biconfluent Heun operator

II.4.1 The case $n=1$

Using (6) yields the following family of first degree operators

[TABLE]

semi-commuting with the operator $P_{BCH}$ associated to the biconfluent Heun equation. The commutativity condition (7) will be satisfied for certain choices of the parameters that we list here below. Note that if $L:=QP_{BCH}=P_{BCH}Q$, the solutions to the equations $Q\varphi=0$ and $P_{BCH}\psi=0$ allow to construct the general solution to the third order ODE $Lf=0$ with

[TABLE]

Case $\nu=q=\tau=0$. The solution to $P_{BCH}\psi=0$ is $\psi(x)=c_{1}e^{\alpha_{+}x}+c_{2}e^{\alpha_{-}x}$ with $\alpha_{\pm}=\left(1\pm\sqrt{1+4\alpha}\right)/2$. Moreover, the solution to $Q\varphi=0$ has been already obtained in Section II.1.1. Finally, the general solution to the ODE $Lf=0$ with $L$ as in (19) and non-vanishing coefficients $K=\beta_{0}-\beta_{1}$, $R=-(\beta_{1}\alpha+\beta_{0})$ and $C=-\beta_{0}\alpha$ is

[TABLE] 2. 2.

Case $\nu=0$, $q=1$, and $\tau=2$. The solution to $P_{BCH}\psi=0$ is $\psi(x)=\frac{1}{x}\left(c_{1}e^{\alpha_{+}x}+c_{2}e^{\alpha_{-}x}\right)$ with $\alpha_{\pm}$ defined in the previous case. Furthermore, the solution to $Q\varphi=0$ has been obtained in Section II.1.1. The general solution to the ODE $Lf=0$ with $L$ as in (19), $K$ and $R$ as in the case above and $A=3\beta_{1}$, $B=2K$ and $D=R$ is

[TABLE]

II.4.2 The case $n=2$

By means of (8) and (9) we find the following family of second degree operators

[TABLE]

semi-commuting with the operator $P_{BCH}$ associated to the biconfluent Heun equation. The commutativity condition (10) will be satisfied for certain choices of the parameters that we list here below. Notice that if $L:=QP_{BCH}=P_{BCH}Q$, the solutions to the equations $Q\varphi=0$ and $P_{BCH}\psi=0$ allow to construct the general solution to the fourth order ODE $Lf=0$ with

[TABLE]

Case $\beta_{1}=-\beta_{2}$. The solution to the equation $P_{BCH}\psi=0$ is

[TABLE]

where $H_{D}$ denotes the double confluent Heun function Ronveaux ,

[TABLE]

and

[TABLE]

Here, $\epsilon(\cdot)$ denotes the sign function. If we let $\mu=\beta_{0}/\beta_{2}$, the solution to the equation $Q\varphi=0$ is given by

[TABLE]

with $\sigma_{-}$, $\sigma_{+}$ and $r$ formally given by $\rho_{-}$, $\rho_{+}$ and $\omega$ with $\alpha$ replaced by $-\mu$ and

[TABLE]

Then, the general solution to the equation $Lf=0$ with $L$ as in (20) and non-vanishing coefficients

[TABLE]

is

[TABLE]

[TABLE]

Also in this case Maple $16$ is not able to solve the above fourth order ODE. 2. 2.

Case $\nu=q=\tau=0$. The solution to $P_{BCH}\psi=0$ is given in case $1.$ ($n=1$) of the biconfluent Heun operator. Moreover, the solution to the equation $Q\varphi=0$ has been already studied in Section‘II.1.1. Finally, the solution of to the ODE $Lf=0$ with with $L$ as in (19) and non-vanishing coefficients $A_{0}=\beta_{1}-\beta_{2}$, $B_{0}=\beta_{0}-\beta_{1}-\beta_{2}\alpha$, $C_{0}=-\beta_{0}-\beta_{1}\alpha$ and $D_{0}=-\beta_{0}\alpha$ is

[TABLE]

where $m_{\pm}$ and $\alpha_{\pm}$ have been defined in (15) and in case 1 Section II.4.1, respectively. 3. 3.

Case $\nu=0$, $q=1$, and $\tau=2$. The solution to $P_{BCH}\psi=0$ is given as in case $2.$ ($n=1$) of the biconfluent Heun operator. Moreover, the solution to $Q\varphi=0$ has been studied in Section II.1.2. Finally, the general solution to the ODE $Lf=0$ with $L$ as in (19) and non-vanishing coefficients $A_{0}$, $B_{0}$, $C_{0}$, $D_{0}$ given in the case above and $A_{1}=4\beta_{2}$, $B_{1}=3A_{0}$, $C_{1}=-2B_{0}$ and $D_{1}=C_{0}$ is

[TABLE]

where $m_{\pm}$ and $\alpha_{\pm}$ have been defined in (15) and in case 1 Section II.4.1, respectively.

II.5 The double confluent Heun operator

II.5.1 The case $n=1$

Using (6) yields the same family of first degree operators $Q$ obtained for the biconfluent Heun operator in the case $n=1$. The commutativity condition (7) will be satisfied for $\alpha=\tau/2$, $\nu=0$, and $q=(\tau/2)-(\tau^{2}/4)$. For this choice of the parameters we have

[TABLE]

Then, the operator $L:=QP_{DCH}=P_{DCH}Q$ is of the form

[TABLE]

with $a_{0}=\beta_{0}-\beta_{1}$, $a_{1}=3\beta_{1}\tau/2$, $b_{0}=-\beta_{0}$, $b_{1}=\tau a_{0}$, $b_{2}=3\beta_{1}\tau(\tau-2)$, $c_{1}=b_{0}\tau$ and $c_{2}=a_{0}\tau(\tau-2)$ and the solution to the ODE $Lf=0$ reads

[TABLE]

II.5.2 The case $n=2$

By means of (8) and (9) we find the following family of second degree operators

[TABLE]

semi-commuting with $P_{DCH}$. The commutativity condition (10) will be satisfied for certain choices of the parameters that we list here below. Notice that if $L:=QP_{DCH}=P_{DCH}Q$, the solutions to the equations $Q\varphi=0$ and $P_{DCH}\psi=0$ allow to construct the general solution to the fourth order ODE $Lf=0$ with

[TABLE]

Case $\beta_{1}=-\beta_{2}$. The solution to the equation $P_{DCH}\psi=0$ is

[TABLE]

where $H_{D}$ denotes the double confluent Heun function Ronveaux ; Slavyanov and $m_{1}=-4\sqrt{\nu}$, $m_{2}=4(\alpha-1)\sqrt{\nu}-\tau^{2}-4q+4\nu+2\tau-1$, $m_{3}=8\sqrt{\nu}(\alpha-\tau+1)$, $m_{4}=8(\alpha-1)\sqrt{\nu}-m_{2}$. Furthermore, the solution to the equation $Q\varphi=0$ is given by

[TABLE]

where the functions $\sigma_{\pm}(\cdot)$ and $r(\cdot)$ have been defined in Section II.4.2 and

[TABLE]

Finally, the general solution to the ODE $Lf=0$ with $L$ as in (21) and non-vanishing coefficients

[TABLE]

is

[TABLE]

[TABLE]

Also in this case Maple $16$ fails to find the general solution to the ODE $Lf=0$. 2. 2.

Case $\alpha=\tau/2$, $\nu=0$, $q=(\tau/2)-(\tau/2)^{2}$. By means of the results obtained in Section II.5.1 the general solution to the ODE $Lf=0$ with $L$ as in (21) and non-vanishing coefficients

[TABLE]

is

[TABLE]

where $m_{\pm}$ has been defined in (15).

II.6 The triconfluent Heun operator

II.6.1 The case $n=1$

Let $P_{TCH}$ and $P_{RTCH}$ be the operators associated to the triconfluent Heun equation and the reduced triconfluent Heun equation, respectively. Using (6) yields the following family of first degree operators

[TABLE]

semi-commuting with the operators $P_{TCH}$ and $P_{RTCH}$, respectively. In both cases, the commutativity condition (7) will be satisfied whenever $\beta_{1}=0$ but then, $Q_{1}$ and $Q_{2}$ become trivial and therefore, there are no nontrivial first degree operators commuting with $P_{TCH}$ or $P_{RTCH}$.

II.6.2 The case $n=2$

By means of (8) and (9) we find the following family of second degree operators

[TABLE]

semi-commuting with the operator $P_{TCH}$. The commutativity condition (10) will be satisfied whenever $\beta_{1}=\sigma\beta_{2}$. Furthermore, the solution to the equation $P_{TCH}\psi=0$ is

[TABLE]

where $H_{T}$ denotes the triconfluent Heun function Ronveaux and the solution to $Q\varphi=0$ is given by

[TABLE]

Then, the general solution to the fourth order equation $Lf=0$ with

[TABLE]

where

[TABLE]

is $f(x)=\psi(x)+\varphi(x)$. Also in this case Maple $16$ is not able to solve the ODE $Lf=0$. We conclude by observing that the following family of second degree operators

[TABLE]

semi-commutes with the operator $P_{RTCH}$. The commutativity condition (10) will be satisfied for $\beta_{1}=0$. Then, the general solution to the equation $P_{RTCH}\psi=0$ is

[TABLE]

and the general solution to $Q\varphi=0$ is given by

[TABLE]

with $\mu$ defined as above. Hence, the general solution to the fourth order equation $Lf=0$ with

[TABLE]

where

[TABLE]

is simply $f(x)=\psi(x)+\varphi(x)$. Also in this case Maple $16$ fails to find the general solution to the ODE $Lf=0$.

III Conclusions

The study of the Heun equation, its generalizations and solutions is one of the next challenges in the area differential equations connected to mathematical physics. Being a generalization of the hypergeometric equation it is not surprising that its appearance in (mathematical) physics is manifold. A quite exhaustive review of its applications in physics and quantum chemistry is offered by Ronveaux . More recent and unexpected applications of the Heun family of equations in physics can be found in Slavyanov ; Hortascu ; Fiziev . In general relativity the Heun equation has a well defined connection to the black hole physics BSW ; CKTSK ; BS and is used in quantum mechanics as well Bay ; Tolstikhan ; Hall . In particular, one can connect it in an elegant way to the Schrödinger equation U1 ; U2 ; U3 , the Stark effect Epstein and other specific quantum mechanical problems BIWTRL .

From a mathematical point of view the study of the Heun equation and its confluent forms is far from being complete. Even though we can easily construct pairs of Frobenius series solutions around each regular singular point and derive the asymptotics of the solutions for the point at infinity, the so-called connection problem for these local solutions is still open mainly due to the fact that the construction of integral representations for the solutions results to be a very difficult task. Some steps towards the resolution of this problem have being taken in RD where a modification of the methods used in SS allowed to solve the two-point connection problem for a subclass of the Heun equation. Other attempts can be found in Ronveaux ; Slavyanov ; PF .

Here, we have taken a slightly different point of view. We considered the Heun family of differential operators given in Table 1 and for each member of this family we constructed the corresponding most general class of commuting differential operators of degree one and degree two. This in turn allowed to show that using this method solutions of complicated higher order linear homogeneous differential equations with variable coefficients can be found analytically even though the software package Maple $16$ was not able to compute them.