Series representations for generalized harmonic functions in the case of three parameters

Abstract

We present a canonical series expansion for generalized harmonic functions in the open unit disc in the complex plane that generalizes that recently obtained for the class of $(p,q)$-harmonic functions.

Keywords:

• Harmonic function
• power series
• poisson kernel
• hypergeometric function

AMS Subject Classifications:

• Primary: 31A05
• Secondary: 33C05
• 35J15

1. Introduction

Denote by $\mathbb{D}$ the open unit disc in the complex plane $\mathbb{C}$ and by ${\mathrm{\partial }}_z=\mathrm{\partial }/\mathrm{\partial }z$ and ${\overline{\mathrm{\partial }}}_z=\mathrm{\partial }/\mathrm{\partial }\overline{z}$ the usual complex partial derivatives. We shall consider second order partial differential operators of the form (1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) where $p,q,r\in \mathbb{C}$ are complex parameters. Of particular concern are solutions to the homogeneous equation (2) $L_{p,q,r}u=0\mathrm{in}\mathbb{D}.$(2) A function u is said to be $(p,q,r)$-harmonic if u is twice continuously differentiable in $\mathbb{D}$ and satisfies Equation (2(2) $L_{p,q,r}u=0\mathrm{in}\mathbb{D}.$(2) ). Recall the Pochhammer symbol $(\cdot {)}_n$ defined by $(a{)}_n=a(a+1)\cdots (a+n-1)$ for $n=1,2,\dots$ and $(a{)}_0=1$. Commonly referred to below is the familiar Gaussian or hypergeometric function (3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) where $a,b\in \mathbb{C}$ and $c\in \mathbb{C}\setminus \{0,-1,-2,\dots \}$. A recent publication together with Olofsson [1] concerned a series expansion for $(p,q,\mathrm{pq})$-harmonic functions in $\mathbb{D}$, or $(p,q)$-harmonic for short, as they have been called in the case r = pq. ¹,²,³

As was shown in Theorem 5.1 of that paper, a function u in $\mathbb{D}$ is $(p,q)$-harmonic if and only if it has the form (4) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F(-p,m-q,m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F(-q,m-p,m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(4) for some sequence $\{c_m{\}}_{m=-\mathrm{\infty }}^{\mathrm{\infty }}$ of complex numbers such that $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1.$ The sums appearing in (4(4) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F(-p,m-q,m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F(-q,m-p,m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(4) ) are absolutely convergent in the space $C^{\mathrm{\infty }}(\mathbb{D})$ of smooth functions, by which we mean $C^{\mathrm{\infty }}(\mathbb{D})=\bigcap _{n=0}^{\mathrm{\infty }}{C}^n(\mathbb{D})$, where $C^n(\mathbb{D})$ is the space of n-times continuously differentiable functions in $\mathbb{D}$ for $n\in \mathbb{N}$. The space $C^{\mathrm{\infty }}(\mathbb{D})$ is complete in the topology induced by the semi-norms $\Vert u{\Vert }_{j,k;K}=\underset{z\in K}{max}|{\mathrm{\partial }}^j{\overline{\mathrm{\partial }}}^{k}u(z)|,$ where $j,k\in \mathbb{N}$ are non-negative integers and $K\subset \mathbb{D}$ is a compact subset of $\mathbb{D}$.

In this paper we extend these ideas and show that a similar such result holds for $(p,q,r)$-harmonic functions. For this purpose, we follow earlier practice and introduce the m-th homogeneous part for a suitably smooth function u in $\mathbb{D}$ and define it by the formula (5) $u_m(z)=\frac{1}{2\pi }{\int }_{\mathbb{T}}{\mathrm{e}}^{-\mathrm{im\theta }}u({\mathrm{e}}^{\mathrm{i\theta }}z)\mathrm{d}\theta ,z\in \mathbb{D},$(5) for $m\in \mathbb{Z}$.

We show that (see Theorem 4.3) the m-th homogeneous part of a $(p,q,r)$-harmonic function u in $\mathbb{D}$ has the form $u_m(z)=c_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$ for some $c_m\in \mathbb{C}$ when $m\in \mathbb{N}$ is a non-negative integer. The parameters ${\chi }_{p,q,r,m}^{+}$ and ${\chi }_{p,q,r,m}^{-}$ of the hypergeometric function F in this expression are the zeros of the quadratic polynomial (6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) The discriminant of this polynomial can be written as (7) ${\mathrm{\Delta }}_{p,q,r,m}=(m+p-q{)}^2-4(r-\mathrm{pq}).$(7) Note that the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ) are $-p$ and m−q in the case when r = pq.

A similar formula is given for the m-th homogeneous part $u_m$ of a $(p,q,r)$-harmonic function when $m\in {\mathbb{Z}}^{-}=\mathbb{Z}\setminus \mathbb{N}$ is a negative integer. These two results rely on the conclusion (Theorem 2.4) that for each of the homogeneous parts $u_m$ of a $(p,q,r)$-harmonic function, there exist complex numbers $p^{\mathrm{\prime }},q^{\mathrm{\prime }}\in \mathbb{C}$, perhaps depending on $m\in \mathbb{Z}$, such that $L_{p,q,r}{u}_m=L_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}}{u}_m$ with equality in $\mathbb{D}$. As a consequence, we will have that the m-th homogeneous part of a $(p,q,r)$-harmonic function is also harmonic in the former meaning of the word, or $(p^{\mathrm{\prime }},q^{\mathrm{\prime }})$-harmonic, for some $p^{\mathrm{\prime }},q^{\mathrm{\prime }}\in \mathbb{C}$. This will allow us to apply many of the results achieved in [1], and leads to the conclusion (Theorem 5.1) that a function u in $\mathbb{D}$ is $(p,q,r)$-harmonic if and only if it has the form (8) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(8) for some sequence $\{c_m{\}}_{m=-\mathrm{\infty }}^{\mathrm{\infty }}$ of complex numbers such that $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1.$ As for the particular case r = pq, the sums in (8(8) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(8) ) are shown to converge absolutely in the space $C^{\mathrm{\infty }}(\mathbb{D})$. This series characterization depends on a result concerning the asymptotic behaviour of the m-th homogeneous parts $u_m$ of a $(p,q,r)$-harmonic function. We show (see Theorem 4.6) that $\underset{|m|\to \mathrm{\infty }}{lim sup}\Vert u_m{\Vert }_{j,k;K}^{1/|m|}<1,$ for all $j,k\in \mathbb{N}$ and $K\subset \mathbb{D}$ compact. This result allows us to apply the root test for absolute convergence of the function series $\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m$ in $C^{\mathrm{\infty }}(\mathbb{D})$.

Further modelling on previous cases also shows (see Theorem 3.4) that $\underset{m\to \mathrm{\infty }}{lim}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;z)=(1-z{)}^p,z\in \mathbb{D}.$ This result retrieves that obtained for the hypergeometric functions associated with the m-th homogeneous parts of $(p,q)$-harmonic functions (see [1], Theorem 2.6).

In this paper we have pursued the course of Olofsson with particular reference to [1,2]. We also refer to [3] by Olofsson and Wittsten. Other related sources include those of Wittsten [4] or Wittsten and Carlsson [5]. We also refer to Li with collaborators [6,7]. The case r = pq has also been treated by Ahern, Bruna and Cascante [8], who traced the corresponding class of operators for this particular case to Geller [9]. Related studies are also found in connection with conductivity problems, including Calderón [10] or Astala and Päivärinta [11]. Borichev and Hedenmalm also established a connection to polyharmonic theory [12]. Early traces also leads to Garabedian [13]. Connections to Bose-Einstein condensates and harmonic traps are found in Kling [14], as briefly mentioned in the concluding remarks.

2. Preliminaries and basic constructions

A function u in $\mathbb{D}\setminus \{0\}$ is said to be homogeneous of order $m\in \mathbb{Z}$ with respect to rotations if it has the property $u({\mathrm{e}}^{\mathrm{i\theta }}z)={\mathrm{e}}^{\mathrm{im\theta }}u(z),z\in \mathbb{D}\setminus \{0\},$ for ${\mathrm{e}}^{\mathrm{i\theta }}\in \mathbb{T}$. In particular, we will consider such functions of the form (9) $u(z)=f(|z{|}^2)z^m,z\in \mathbb{D}\setminus \{0\},$(9) for some $f\in C^2(0,1)$ and $m\in \mathbb{Z}$. Denote by (10) $H_{a,b,c}=(1-x)x\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+(c-[a+b+1]x)\frac{\mathrm{d}}{\mathrm{d}x}-\mathrm{ab},$(10) the ordinary hypergeometric differential operator, where $a,b,c\in \mathbb{C}$ are complex parameters. This operator is associated with the familiar ordinary hypergeometric differential equation $(1-x)x{y}^{\mathrm{\prime \prime }}(x)+(c-[a+b+1]x)y^{\mathrm{\prime }}(x)-\mathrm{aby}(x)=0,$ and takes the form $H_{a,b,c}y=0$ in terms of the operator $H_{a,b,c}$.

We will need the following result, and refer to [1], Theorem 2.1, for details.

Theorem 2.1

Let $L_{p,q,r}$ be as in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ) for some $p,q,r\in \mathbb{C}$. Let u be a function of the form (9(9) $u(z)=f(|z{|}^2)z^m,z\in \mathbb{D}\setminus \{0\},$(9) ) for some $f\in C^2(0,1)$ and $m\in \mathbb{Z}$. Then $L_{p,q,r}u(z)=z^m{H}_{a,b,c}f(|z{|}^2),z\in \mathbb{D}\setminus \{0\},$ where c = m + 1 and (11) $\{\begin{array}{l}a+b=m-p-q,\\ \mathrm{ab}=r-\mathrm{pm}.\end{array}$(11)

Equation (11(11) $\{\begin{array}{l}a+b=m-p-q,\\ \mathrm{ab}=r-\mathrm{pm}.\end{array}$(11) ) tells us that a and b are the zeros of the quadratic polynomial in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ). We denote these two zeros by ${\chi }_{p,q,r,m}^{+}$ and ${\chi }_{p,q,r,m}^{-}$, and get the following construction of $(p,q,r)$-harmonic functions.

Proposition 2.2

Let $p,q,r\in \mathbb{C}$. Consider the function (12) $u_m(z)=F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$(12) where $m\in \mathbb{N}$ and F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ). Then $u_m$ is a $(p,q,r)$-harmonic function.

Proof.

It is clear that $u_m\in C^{\mathrm{\infty }}(\mathbb{D})$. It is also known that the hypergeometric function $y=F(a,b,c;\cdot )$ satisfies the hypergeometric equation $H_{a,b,c}y=0$. The result now follows by Theorem 2.1.

We compare the last result with Proposition 2.2 in [1]. We also refer to [15], Section 2.3, for more on the hypergeometric equation and its solutions.

Corollary 2.3

Let $p,q,r\in \mathbb{C}$. Consider the function $u_m(z)=F({\chi }_{q,p,r,|m|}^{+},{\chi }_{q,p,r,|m|}^{-},|m|+1;|z{|}^2){\overline{z}}^{|m|},z\in \mathbb{D},$ where $m\in {\mathbb{Z}}^{-}$ and F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ). Then $u_m$ is a $(p,q,r)$-harmonic function.

Proof.

Consider the conjugated expression $\overline{u_m(z)}=F({\chi }_{\overline{q},\overline{p},\overline{r},|m|}^{+},{\chi }_{\overline{q},\overline{p},\overline{r},|m|}^{-},|m|+1;|z{|}^2)z^{|m|},z\in \mathbb{D}.$ By Proposition 2.2 we have that ${\overline{u}}_m$ is a $(\overline{q},\overline{p},\overline{r})$-harmonic function. It is then straightforward to check using the definition in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ) that $u_m$ is a $(p,q,r)$-harmonic function.

For coming purposes, we introduce the quadratic polynomial (13) $Q_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m+p-q)+r-\mathrm{pq},\lambda \in \mathbb{C}.$(13) Note that the polynomials $Q_{p,q,r,m}$ and $P_{p,q,r,m}$ have the same discriminant (7(7) ${\mathrm{\Delta }}_{p,q,r,m}=(m+p-q{)}^2-4(r-\mathrm{pq}).$(7) ).

We also state that a function $u\in C^{\mathrm{\infty }}(\mathbb{D})$ is homogeneous of order $m\ge 0$ with respect to rotations if and only if it has the form (9(9) $u(z)=f(|z{|}^2)z^m,z\in \mathbb{D}\setminus \{0\},$(9) ) for some $f\in C^{\mathrm{\infty }}[0,1)$, with care taken at the origin in the case that u is less smooth. We refer the reader to the fourth section of [16] and Theorem 4.2 for a more elaborative discussion.

Theorem 2.4

Let $p,q,r\in \mathbb{C}$ and let $m\in \mathbb{N}$. Let $u\in C^2(\mathbb{D})$ be homogeneous of order m with respect to rotations. Then there exist complex numbers $p^{\mathrm{\prime }},q^{\mathrm{\prime }}\in \mathbb{C}$ such that (14) $L_{p,q,r}u=L_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}}u,\text{in}\mathbb{D}.$(14) Particularly, the complex numbers (15) $\{\begin{array}{l}{p}^{\mathrm{\prime }}=p-\mu ,\\ q^{\mathrm{\prime }}=q+\mu ,\end{array}$(15) where μ is a zero of the polynomial $Q_{p,q,r,m}$ in (13(13) $Q_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m+p-q)+r-\mathrm{pq},\lambda \in \mathbb{C}.$(13) ), are admissible for (14(14) $L_{p,q,r}u=L_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}}u,\text{in}\mathbb{D}.$(14) ).

Proof.

By the comments preceding this statement we can write $u\in C^2(\mathbb{D})$ on the form $u(z)=z^{m}f(|z{|}^2),z\in \mathbb{D},$ for some $f\in C^2(0,1)$. By Theorem 2.1 we have that (16) $L_{p,q,r}u(z)=z^m{H}_{a,b,c}f(|z{|}^2),z\in \mathbb{D},$(16) where $a,b\in \mathbb{C}$ satisfy the system of equations in (11(11) $\{\begin{array}{l}a+b=m-p-q,\\ \mathrm{ab}=r-\mathrm{pm}.\end{array}$(11) ) and c = m + 1. This means that (17) $L_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},r^{\mathrm{\prime }}}u=L_{p,q,r}u,\mathrm{in}\mathbb{D},$(17) for $p^{\mathrm{\prime }},q^{\mathrm{\prime }},r^{\mathrm{\prime }}\in \mathbb{C}$ satisfying (18) $\{\begin{array}{l}m-p^{\mathrm{\prime }}-q^{\mathrm{\prime }}=m-p-q,\\ r^{\mathrm{\prime }}-p^{\mathrm{\prime }}m=r-\mathrm{pm}.\end{array}$(18) In vector terms, we can express the general solution to this system of equations as (19) $(p^{\mathrm{\prime }}-p,q^{\mathrm{\prime }}-q,r^{\mathrm{\prime }}-r)=\mu (-1,1,-m),\mu \in \mathbb{C}.$(19) This can be seen by noticing that the complex vector on the left of this expression is in the orthogonal complement of the set $\{w_1,w_2\}$ formed by the two vectors $w_1=(1,1,0)$ and $w_2=(-m,0,1)$. We can then let $r^{\mathrm{\prime }}=p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}$ in (19(19) $(p^{\mathrm{\prime }}-p,q^{\mathrm{\prime }}-q,r^{\mathrm{\prime }}-r)=\mu (-1,1,-m),\mu \in \mathbb{C}.$(19) ) for a quadratic equation in μ that is satisfied by the zeros of the polynomial $Q_{p,q,r,m}$ in (13(13) $Q_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m+p-q)+r-\mathrm{pq},\lambda \in \mathbb{C}.$(13) ). This gives the conclusion of the statement.

The relations in (18(18) $\{\begin{array}{l}m-p^{\mathrm{\prime }}-q^{\mathrm{\prime }}=m-p-q,\\ r^{\mathrm{\prime }}-p^{\mathrm{\prime }}m=r-\mathrm{pm}.\end{array}$(18) ) can be retrieved from (15(15) $\{\begin{array}{l}{p}^{\mathrm{\prime }}=p-\mu ,\\ q^{\mathrm{\prime }}=q+\mu ,\end{array}$(15) ) and the polynomial $Q_{p,q,r,m}$ in the form of $r^{\mathrm{\prime }}=p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}$ by noticing that (20) $Q_{p,q,r,m}(\lambda )=r-\mathrm{pm}-(p-\lambda )(q+\lambda -m),\lambda \in \mathbb{C}.$(20) We also note that every triple of complex numbers $p^{\mathrm{\prime }},q^{\mathrm{\prime }},r^{\mathrm{\prime }}\in \mathbb{C}$ of the form (19(19) $(p^{\mathrm{\prime }}-p,q^{\mathrm{\prime }}-q,r^{\mathrm{\prime }}-r)=\mu (-1,1,-m),\mu \in \mathbb{C}.$(19) ) is admissible for (17(17) $L_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},r^{\mathrm{\prime }}}u=L_{p,q,r}u,\mathrm{in}\mathbb{D},$(17) ). However, the statement of Theorem (2.4) as it stands will do for our purposes, and we will proceed accordingly.

We will need the following result and refer to [1], Theorem 2.4, for details. We also remind the reader that the naming convention $(p,q,r)$-harmonic is an adaptation to our situation and stems from the terminology used in the particular case r = pq, in which case the $(p,q,\mathrm{pq})$-harmonic functions are simply called $(p,q)$-harmonic.

Theorem 2.5

Let $p,q\in \mathbb{C}$ and $m\in \mathbb{N}$. Let $u\in C^2(\mathbb{D})$ be homogeneous of order m with respect to rotations. Then u is $(p,q)$-harmonic if and only if it has the form (21) $u(z)=\mathrm{cF}(-p,m-q,m+1;|z{|}^2)z^m,z\in \mathbb{D},$(21) for some $c\in \mathbb{C}$, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ).

The last Theorem adapted to the general situation reads as follows.

Theorem 2.6

Let $p,q,r\in \mathbb{C}$ and $m\in \mathbb{N}$. Let $u\in C^2(\mathbb{D})$ be homogeneous of order m with respect to rotations. Then u is $(p,q,r)$-harmonic if and only if it has the form $u(z)=\mathrm{cF}({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$ for some $c\in \mathbb{C}$, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ).

Proof.

The ”if” part of this statement is a consequence of Proposition 2.2. For the converse, let $u\in C^2(\mathbb{D})$ be a $(p,q,r)$-harmonic function homogeneous of order m with respect to rotations. Let $p^{\mathrm{\prime }},q^{\mathrm{\prime }}\in \mathbb{C}$ be of the form (15(15) $\{\begin{array}{l}{p}^{\mathrm{\prime }}=p-\mu ,\\ q^{\mathrm{\prime }}=q+\mu ,\end{array}$(15) ) for some $\mu \in \mathbb{C}$ such that $Q_{p,q,r,m}(\mu )=0$, with the latter defined as in (13(13) $Q_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m+p-q)+r-\mathrm{pq},\lambda \in \mathbb{C}.$(13) ). Then Theorem 2.4 tells us that $L_{p,q,r}u=L_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}}u,\mathrm{in}\mathbb{D}.$ From Theorem 2.5, we have that $u(z)=\mathrm{cF}(-p^{\mathrm{\prime }},m-q^{\mathrm{\prime }},m+1;|z{|}^2)z^m,z\in \mathbb{D},$ for some $c\in \mathbb{C}$. Note that $-p^{\mathrm{\prime }}$ and $m-q^{\mathrm{\prime }}$ are the zeros of the quadratic polynomial $P_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},p^{\mathrm{\prime }}{q}^{\mathrm{\prime }},m}(\lambda )={\lambda }^2-\lambda (m-p^{\mathrm{\prime }}-q^{\mathrm{\prime }})+p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}-p^{\mathrm{\prime }}m,\lambda \in \mathbb{C}.$ Recall the expression for $Q_{p,q,r,m}$ in the form of (20(20) $Q_{p,q,r,m}(\lambda )=r-\mathrm{pm}-(p-\lambda )(q+\lambda -m),\lambda \in \mathbb{C}.$(20) ). From this expression, we get that $p^{\mathrm{\prime }}{q}^{\mathrm{\prime }}-p^{\mathrm{\prime }}m=r-\mathrm{pm},$ for $p^{\mathrm{\prime }},q^{\mathrm{\prime }}\in \mathbb{C}$ as above. This shows that $P_{p^{\mathrm{\prime }},q^{\mathrm{\prime }},p^{\mathrm{\prime }}{q}^{\mathrm{\prime }},m}$ and $P_{p,q,r,m}$ define the same polynomial, where $P_{p,q,r,m}$ is the polynomial in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ). We conclude with the statement of this theorem.

The next result gives an indication of how we will proceed in view of the comments made in the introduction and was given in [1], Theorem 3.2.

Theorem 2.7

Let $\{f_m{\}}_{m=0}^{\mathrm{\infty }}$ be a sequence in $C^{\mathrm{\infty }}[0,1)$ such that $\underset{m\to \mathrm{\infty }}{lim sup}(\underset{0\le x\le r}{max}|f_m^{(n)}(x)|{)}^{1/m}\le 1,$ for $n\in \mathbb{N}$ and 0<r<1. Set $u_m(z)=f_m(|z{|}^2)z^m,z\in \mathbb{D},$ for $m=0,1,2,\dots$. Then $\underset{m\to \mathrm{\infty }}{lim sup}\Vert u_m{\Vert }_{j,k;K}^{1/m}<1$ for all $j,k\in \mathbb{N}$ and $K\subset \mathbb{D}$ compact.

3. Asymptotics

The space $H(\mathbb{D})$ of analytic functions in $\mathbb{D}$ is topologized in the usual manner using the semi-norms (22) $\Vert f{\Vert }_K=\underset{z\in K}{max}|f(z)|,$(22) for $K\subset \mathbb{D}$ compact. We refer to convergence in $H(\mathbb{D})$ as normal convergence in $\mathbb{D}$, and say that a subset $\mathcal{F}$ of $H(\mathbb{D})$ is normal if every sequence of functions of $\mathcal{F}$ has a subsequence which converges in $H(\mathbb{D})$. The limit function is not required to belong to $\mathcal{F}$.

Within our context, the Binomial series is suitably expressed in terms of the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) as (23) $F(a,b,b;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n}{n!}{z}^n=\frac{1}{(1-z{)}^a},$(23) for $z\in \mathbb{D}$.

On a last note before we proceed, we also refer to Conway [17], Theorem VII.2.9, for Montel's Theorem, characterizing normal families of analytic functions.

Lemma 3.1

Let $p,q,r\in \mathbb{C}$. Consider the functions $f_m(z)=F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;z),z\in \mathbb{D},$ for $m\in \mathbb{N}$, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ). Then $\mathcal{F}=\{f_m:m\in \mathbb{N}\}$ is a normal family of analytic functions in $\mathbb{D}$.

Proof.

We will show that the functions in $\mathcal{F}$ are uniformly bounded on compact subsets of $\mathbb{D}$. An application of Montel's Theorem then gives the conclusion of the lemma. Let $K\subset \mathbb{D}$ be compact and let $0<\rho <1$ be such that ${\max }_{z\in K}|z|<\rho$. It is straightforward to check that (24) $({\chi }_{p,q,r,m}^{+}+k)({\chi }_{p,q,r,m}^{-}+k)=-\mathrm{mp}+r+k(m+1+k-(1+p+q)),$(24) where $k,m\in \mathbb{N}$. This gives us $\frac{({\chi }_{p,q,r,m}^{+}+k)({\chi }_{p,q,r,m}^{-}+k)}{m+1+k}=-\mathrm{mp}\frac{1}{m+1+k}+r\frac{1}{m+1+k}+k(1-\frac{1+p+q}{m+1+k}),$ for $k,m\in \mathbb{N}$. Applying the triangle inequality to this expression grants $\left|\frac{({\chi }_{p,q,r,m}^{+}+k)({\chi }_{p,q,r,m}^{-}+k)}{m+1+k}\right|\le |p|\frac{m}{m+1+k}+|r|\frac{1}{m+1+k}+k|1-\frac{1+p+q}{m+1+k}|,$ for $k,m\in \mathbb{N}$. Choose N>0 such that $|1-\frac{1+p+q}{m+1}|\le \frac{1}{\rho },$ for m>N. Then $\left|\frac{({\chi }_{p,q,r,m}^{+}+k)({\chi }_{p,q,r,m}^{-}+k)}{m+1+k}\right|\le \frac{|p|+|r|+k}{\rho },$ for m>N and $k\in \mathbb{N}$. Note that $\frac{({\chi }_{p,q,r,m}^{+}{)}_n({\chi }_{p,q,r,m}^{-}{)}_n}{(m+1{)}_n}=\prod _{k=0}^{n-1}-p\frac{m}{m+1+k}+\frac{r}{m+1+k}+k(1-\frac{1+p+q}{m+1+k}),$ where $m,n\in \mathbb{N}$. By the last inequality we get that (25) $\left|\frac{({\chi }_{p,q,r,m}^{+}{)}_n({\chi }_{p,q,r,m}^{-}{)}_n}{(m+1{)}_n}\right|\le \frac{(|p|+|r|{)}_n}{{\rho }^n},$(25) for $n\in \mathbb{N}$ when m>N. This shows that the family $\mathcal{F}$ is uniformly bounded on compact subsets $K\subset \mathbb{D}$. Indeed, by another application of the triangle inequality, we get from Equation (25(25) $\left|\frac{({\chi }_{p,q,r,m}^{+}{)}_n({\chi }_{p,q,r,m}^{-}{)}_n}{(m+1{)}_n}\right|\le \frac{(|p|+|r|{)}_n}{{\rho }^n},$(25) ) that $|f_m(z)|\le \sum _{n=0}^{\mathrm{\infty }}\frac{(|p|+|r|{)}_n}{n!}\frac{|z{|}^n}{{\rho }^n}=\frac{1}{(1-|z|/\rho {)}^{|p|+|r|}},$ for $|z|<\rho$ and m>N, where the last equality follows by the binomial series in (23(23) $F(a,b,b;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n}{n!}{z}^n=\frac{1}{(1-z{)}^a},$(23) ).

In the particular case when $p,q\in \mathbb{C}$ are such that $\mathrm{Re}(p)+\mathrm{Re}(q)>-1$ and r = 0, we obtain the following asymptotic bound for the sequence $\mathcal{F}$ in Lemma 3.1.

Proposition 3.2

Let $p,q\in \mathbb{C}$ be such that $\text{R}e(p)+\text{R}e(q)>-1$ and let r = 0. Let $\mathcal{F}$ be the normal family in Lemma 3.1. Then there is N>0 such that $|f_m(z)|\le \frac{1}{(1-|z|{)}^{|p|}},z\in \mathbb{D},$ for m>N.

Proof.

From (24(24) $({\chi }_{p,q,r,m}^{+}+k)({\chi }_{p,q,r,m}^{-}+k)=-\mathrm{mp}+r+k(m+1+k-(1+p+q)),$(24) ) we see that $\frac{({\chi }_{p,q,r,m}^{+}+k)({\chi }_{p,q,r,m}^{-}+k)}{m+1+k}=-p\frac{m}{m+1+k}+k\frac{m-(1+p+q)+1+k}{m+1+k},$ for $k,m\in \mathbb{N}$. We can choose N>0 such that $|1-\frac{1+p+q}{m}|\le 1,$ for m>N. Following a similar line of argument to that in Lemma 3.1, we obtain $\left|\frac{({\chi }_{p,q,r,m}^{+}{)}_n({\chi }_{p,q,r,m}^{-}{)}_n}{(m+1{)}_n}\right|\le (|p|{)}_n,$ for $n\in \mathbb{N}$ when m>N. Use of the triangle inequality then shows that $|f_m(z)|\le \sum _{n=0}^{\mathrm{\infty }}\frac{(|p|{)}_n}{n!}|z{|}^n=\frac{1}{(1-|z|{)}^{|p|}},$ for $z\in \mathbb{D}$ and m>N.

The last two results are modelled on Lemma 2.5 in [1].

We mention here briefly that the restriction $\mathrm{Re}(p)+\mathrm{Re}(q)>-1$ on the parameters $p,q\in \mathbb{C}$ appears naturally in the $(p,q)$-harmonic Dirichlet boundary value problem: (26) $\{\begin{array}{l}{L}_{p,q}u=0\mathrm{in}\mathbb{D},\\ u=\phi \mathrm{on}\mathbb{T},\end{array}$(26) where φ is a given function continuous on the unit circle $\mathbb{T}$ and with $L_{p,q}:=L_{p,q,\mathrm{pq}}$ as in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ). It can be shown that this problem admits a solution in terms of a Poisson type integral that includes the classical case corresponding to p = q = 0, in which case the operator $L_{p,q}$ reduces to that of Laplace. For the statement of this result, we define the $(p,q)$-harmonic Poisson kernel by (27) $K_{p,q}(z)=\frac{(1-|z{|}^2{)}^{p+q+1}}{(1-z{)}^{p+1}(1-\overline{z}{)}^{q+1}},z\in \mathbb{D}.$(27) Let $p,q\in \mathbb{C}$ be such that $\mathrm{Re}(p)+\mathrm{Re}(q)>-1$. Associated with this kernel is the $(p,q)$-harmonic Poisson integral (28) $K_{p,q}[f](z)=\frac{c_{p,q}}{2\pi }{\int }_{\mathbb{T}}{K}_{p,q}(z{\mathrm{e}}^{-\mathrm{i\theta }})f({\mathrm{e}}^{\mathrm{i\theta }})\mathrm{d}\theta ,z\in \mathbb{D},$(28) defined for integrable functions $f\in L^1(\mathbb{T})$ on $\mathbb{T}$, where $c_{p,q}=\frac{\mathrm{\Gamma }(p+1)\mathrm{\Gamma }(q+1)}{\mathrm{\Gamma }(p+q+1)}.$

Theorem 3.3

Let $p,q\in \mathbb{C}\setminus {\mathbb{Z}}^{-}$ be such that $\mathrm{Re}(p)+\mathrm{Re}(q)>-1$. Let $\phi \in C(\mathbb{T})$. Then a function u in $\mathbb{D}$ satisfies (26(26) $\{\begin{array}{l}{L}_{p,q}u=0\mathrm{in}\mathbb{D},\\ u=\phi \mathrm{on}\mathbb{T},\end{array}$(26) ) if and only if it has the form $u(z)=K_{p,q}[\phi ](z),z\in \mathbb{D}.$

We refer to [1–3] for proofs and more on the Dirichlet problem. We should also mention that our definition of the kernel in (27(27) $K_{p,q}(z)=\frac{(1-|z{|}^2{)}^{p+q+1}}{(1-z{)}^{p+1}(1-\overline{z}{)}^{q+1}},z\in \mathbb{D}.$(27) ) differs slightly from that given in [1], where this function is instead denoted $u_{p,q}$ and $K_{p,q}=c_{p,q}{u}_{p,q}$. We have chosen to do so for the underscoring of certain relations between the more familiar elements associated with $(p,q)$-harmonics that we will give later on (Theorem 5.3).

To proceed, we give the following limit theorem for the family in Lemma 3.1, the proof details of which agrees in form to the proof of Theorem 2.6 in [1].

Theorem 3.4

Let $p,q,r\in \mathbb{C}$. Then (29) $\underset{m\to \mathrm{\infty }}{lim}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;z)=(1-z{)}^p,z\in \mathbb{D},$(29) with normal convergence, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ).

Proof.

Let the sequence of functions $f_m$ be as in Lemma 3.1. From Lemma 3.1 we have that the set $\mathcal{F}=\{f_m:m\in \mathbb{N}\}$ is a normal family. From (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) we see that (30) $f_m^{(n)}(0)=\frac{({\chi }_{p,q,r,m}^{+}{)}_n({\chi }_{p,q,r,m}^{-}{)}_n}{(m+1{)}_n},$(30) for $m,n\in \mathbb{N}$. Recall that the Pochhammer symbol $(\cdot {)}_n$ is a monic polynomial of degree n. In passing to the limit we obtain (31) $\underset{m\to \mathrm{\infty }}{lim}{f}_m^{(n)}(0)=(-p{)}_n,$(31) for $n\in \mathbb{N}$. From (23(23) $F(a,b,b;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n}{n!}{z}^n=\frac{1}{(1-z{)}^a},$(23) ) we have that $f^{(n)}(0)=(-p{)}_n$ for $n\in \mathbb{N}$, where $f(z)=(1-z{)}^p,z\in \mathbb{D}.$ A standard argument invoking Lemma 3.1 now gives that $f_m\to f$ in $H(\mathbb{D})$ as $m\to \mathrm{\infty }$. We refer the reader to [1], Theorem 2.6, for details.

We end this section with the following result, related to the asymptotic behaviour of the m-th homogeneous parts $u_m$ of a $(p,q,r)$-harmonic function, as further described in Theorem 4.6.

Proposition 3.5

Let $p,q,r\in \mathbb{C}$. Then $\underset{m\to \mathrm{\infty }}{lim sup}(\underset{0\le x\le r}{max}|F^{(n)}({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;x)|{)}^{1/m}\le 1,$ for $n\in \mathbb{N}$ and 0<r<1, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ).

Proof.

We note that the complex derivative $f\mapsto f^{\mathrm{\prime }}$ is continuous in the topology of normal convergence of analytic functions. In view of this and Montel's Theorem, the proposition now follows from Theorem 3.4.

This last result should be compared with Proposition 3.3 in [1].

4. Homogeneous parts of the generalized harmonic function

We will need the following result and refer to [1], Proposition 4.1, for details.

Proposition 4.1

Let $u_m$ be the m-th homogeneous part of $u\in C^n(\mathbb{D})$ for some $m\in \mathbb{Z}$, where $n\in \mathbb{N}\cup \{\mathrm{\infty }\}$. Then $u_m\in C^n(\mathbb{D})$ and $u_m$ is homogeneous of order m with respect to rotations.

The next result shows that each of the homogeneous parts of a $(p,q,r)$-harmonic function is also $(p,q,r)$-harmonic.

Proposition 4.2

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function and denote by $u_m$ its m-th homogeneous part for some $m\in \mathbb{Z}$. Then $u_m$ is $(p,q,r)$-harmonic.

Proof.

The proof is identical in form to Proposition 4.2, [1]. We leave out the details.

The reader is asked to compare the following discussion to that of the fourth section in [1].

Theorem 4.3

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function and denote by $u_m$ its m-th homogeneous part for some $m\in \mathbb{N}$. Then (32) $u_m(z)=c_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$(32) for some $c_m\in \mathbb{C}$, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ).

Proof.

By Proposition 4.1, the function $u_m$ is homogeneous of order m with respect to rotations. By Proposition 4.2 the function $u_m$ is $(p,q,r)$-harmonic. An application of Theorem 2.6 now gives the conclusion of this statement.

Corollary 4.4

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function and denote by $u_m$ its m-th homogeneous part for some $m\in \mathbb{Z}\setminus {\mathbb{Z}}^{+}$. Then (33) $u_m(z)=c_{m}F({\chi }_{q,p,r,|m|}^{+},{\chi }_{q,p,r,|m|}^{-},|m|+1;|z{|}^2){\overline{z}}^{|m|},z\in \mathbb{D},$(33) for some $c_m\in \mathbb{C}$, where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ).

Proof.

The complex conjugate ${\overline{u}}_m$ is the $-m=|m|$-th homogeneous part of the function $\overline{u}$. It is straightforward to check that $\overline{u}$ is a $(\overline{q},\overline{p},\overline{r})$-harmonic function from the definition in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ). From Theorem 4.3 we have that $\overline{u_m(z)}=k_{m}F({\chi }_{\overline{q},\overline{p},\overline{r},|m|}^{+},{\chi }_{\overline{q},\overline{p},\overline{r},|m|}^{-},|m|+1;|z{|}^2)z^{|m|},z\in \mathbb{D},$ for some $k_m\in \mathbb{C}$. A complex conjugation now yields (33(33) $u_m(z)=c_{m}F({\chi }_{q,p,r,|m|}^{+},{\chi }_{q,p,r,|m|}^{-},|m|+1;|z{|}^2){\overline{z}}^{|m|},z\in \mathbb{D},$(33) ) with $c_m={\overline{k}}_m$.

It is also clear from Theorem 4.3 and Corollary 4.4 that $u_m\in C^{\mathrm{\infty }}(\mathbb{D})$ for $m\in \mathbb{Z}$ if u is $(p,q,r)$-harmonic. We will now proceed with summation of the homogeneous parts $u_m$ of a $(p,q,r)$-harmonic function. From the triangle inequality we have that (34) $\underset{|z|=r}{sup}|u_m(z)|\le \underset{|z|=r}{sup}|u(z)|,$(34) for 0<r<1 and $m\in \mathbb{Z}$.

Lemma 4.5

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function and consider its m-th homogeneous part $u_m$ for $m\in \mathbb{Z}$. Let $c_m$ be as in (32(32) $u_m(z)=c_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$(32) ) or (33(33) $u_m(z)=c_{m}F({\chi }_{q,p,r,|m|}^{+},{\chi }_{q,p,r,|m|}^{-},|m|+1;|z{|}^2){\overline{z}}^{|m|},z\in \mathbb{D},$(33) ) depending on whether $m\in \mathbb{N}$ or $m\in {\mathbb{Z}}^{-}$. Then $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1$.

Proof.

Let $m\in \mathbb{N}$ and let 0<r<1. From Theorem 4.3 and (34(34) $\underset{|z|=r}{sup}|u_m(z)|\le \underset{|z|=r}{sup}|u(z)|,$(34) ) we have that (35) $|c_m||F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;r^2)|r^m\le \underset{|z|=r}{max}|u(z)|,$(35) for $m\in \mathbb{N}$. From Theorem 3.4 we have that $\underset{m\to \mathrm{\infty }}{lim}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;r^2)=(1-r^2{)}^p\ne 0.$ Therefore, a passage to the limit in (35(35) $|c_m||F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;r^2)|r^m\le \underset{|z|=r}{max}|u(z)|,$(35) ) gives that $\underset{m\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/m}\le 1/r.$ Since 0<r<1 was arbitrarily chosen we conclude that $\underset{m\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/m}\le 1$. The case $m\in {\mathbb{Z}}^{-}$ follows by a similar argument to that above. We conclude with the statement of this lemma.

Theorem 4.6

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function and denote by $u_m$ its m-th homogeneous part for $m\in \mathbb{Z}$. Then $u_m\in C^{\mathrm{\infty }}(\mathbb{D})$ for $m\in \mathbb{Z}$ and $\underset{|m|\to \mathrm{\infty }}{lim sup}\Vert u_m{\Vert }_{j,k;K}^{1/|m|}<1,$ for all $j,k\in \mathbb{N}$ and $K\subset \mathbb{D}$ compact.

Proof.

Recall Theorem 4.3 and Corollary 4.4. In the case of $m\in \mathbb{N}$, Proposition 3.5 and Lemma 4.5 applies for the fulfilment of the assumptions in Theorem 2.7. An application of Theorem 2.7 shows that $\underset{m\to \mathrm{\infty }}{lim sup}\Vert u_m{\Vert }_{j,k;K}^{1/m}<1$ for $j,k\in \mathbb{N}$ and $K\subset \mathbb{D}$ compact.

This last argument can also be applied to the case of negative integer values $m\in {\mathbb{Z}}^{-}$ by noting that $u_m$ is the $(-m)$-th homogeneous part of the $(\overline{q},\overline{p},\overline{r})$-harmonic function $\overline{u}$.

Corollary 4.7

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function with m-th homogeneous part $u_m$ for $m\in \mathbb{Z}$. Then the function series $\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m$ is absolutely convergent in $C^{\mathrm{\infty }}(\mathbb{D})$.

Proof.

Theorem 4.6 allows us to apply the root test to conclude that $\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\Vert u_m{\Vert }_{j,k;K}<+\mathrm{\infty },$ for $j,k\in \mathbb{N}$ and $K\subset \mathbb{D}$ compact.

We will need the following lemma due to Fejér and refer the reader to [1], Lemma 4.8, and Katznelson [18], Section I.2, for proofs and a more elaborative description.

Lemma 4.8

Let $u\in C^n(\mathbb{D})$ for some $n\in \mathbb{N}$ and denote by $u_m$ its m-th homogeneous part for $m\in \mathbb{Z}$. Then $u=\underset{N\to +\mathrm{\infty }}{lim}\sum _{m=-N}^N(1-\frac{|m|}{N+1})u_m,$ in $C^n(\mathbb{D})$.

We now show that $(p,q,r)$-harmonic functions are in fact smooth.

Corollary 4.9

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function. Then $u\in C^{\mathrm{\infty }}(\mathbb{D})$ and $u=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m$ in $C^{\mathrm{\infty }}(\mathbb{D})$, where $u_m$ is the m-th homogeneous part of u for $m\in \mathbb{Z}$.

Proof.

From Lemma 4.8 we have that (36) $u=\underset{N\to +\mathrm{\infty }}{lim}\sum _{m=-N}^N(1-\frac{|m|}{N+1})u_m,$(36) in $C^2(\mathbb{D})$. From Corollary 4.7 we know that the function series $\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m$ is absolutely convergent in $C^{\mathrm{\infty }}(\mathbb{D})$. Following a standard argument we can do away with the convergence factors in (36(36) $u=\underset{N\to +\mathrm{\infty }}{lim}\sum _{m=-N}^N(1-\frac{|m|}{N+1})u_m,$(36) ) and deduce that $u=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m$ in $C^{\mathrm{\infty }}(\mathbb{D})$.

5. Series representations for generalized harmonic functions

We are now in a position to give the following function series representation for $(p,q,r)$-harmonic functions.

Theorem 5.1

Let $p,q,r\in \mathbb{C}$. Then u is a $(p,q,r)$-harmonic function if and only if it has the form (37) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(37) for some sequence $\{c_m{\}}_{m=-\mathrm{\infty }}^{\mathrm{\infty }}$ of complex numbers such that (38) $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1,$(38) where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ). Moreover, the sums in (37(37) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(37) ) are absolutely convergent in $C^{\mathrm{\infty }}(\mathbb{D})$ when (38(38) $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1,$(38) ) holds.

Proof.

Let $\{c_m{\}}_{m=-\mathrm{\infty }}^{\mathrm{\infty }}$ be a sequence of complex numbers satisfying (38(38) $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1,$(38) ). Set (39) $u_m(z)=c_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$(39) for $m\in \mathbb{N}$ and (40) $u_m(z)=c_{m}F({\chi }_{q,p,r,|m|}^{+},{\chi }_{q,p,r,|m|}^{-},|m|+1;|z{|}^2){\overline{z}}^{|m|},z\in \mathbb{D},$(40) for $m\in {\mathbb{Z}}^{-}$. Consider the formal $(p,q,r)$-harmonic series expression on $\mathbb{D}$ of the form (41) $u\sim \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m.$(41) From Theorem 2.7 and Proposition 3.5 we have that $\underset{|m|\to \mathrm{\infty }}{lim sup}\Vert u_m{\Vert }_{j,k;K}^{1/|m|}<1,$ for $j,k\in \mathbb{N}$ and $K\subset \mathbb{D}$ compact. The root test now applies to show that the series in (41(41) $u\sim \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m.$(41) ) is absolutely convergent in $C^{\mathrm{\infty }}(\mathbb{D})$, and we can write (41(41) $u\sim \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m.$(41) ) with equality. Since our series converges in $C^{\mathrm{\infty }}(\mathbb{D})$, we have that $u\in C^{\mathrm{\infty }}(\mathbb{D})$. From Proposition 2.2 and Corollary 2.3 we have that each term $u_m$ is $(p,q,r)$-harmonic. An evaluation of u in (41(41) $u\sim \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m.$(41) ) with respect to the operator $L_{p,q,r}$ shows that u is a $(p,q,r)$-harmonic function. This gives the first part of the statement.

For the converse, let u be a $(p,q,r)$-harmonic function and denote by $u_m$ its m-th homogeneous part. From Theorem 4.3 and Corollary 4.4 we know the form of each of the functions $u_m$ for $m\in \mathbb{N}$ and ${\mathbb{Z}}_{-}$ as given in (39(39) $u_m(z)=c_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m,z\in \mathbb{D},$(39) ) and (40(40) $u_m(z)=c_{m}F({\chi }_{q,p,r,|m|}^{+},{\chi }_{q,p,r,|m|}^{-},|m|+1;|z{|}^2){\overline{z}}^{|m|},z\in \mathbb{D},$(40) ), respectively. From Corollary 4.9 we have (37(37) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(37) ) in the form $u=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_m$ in $C^{\mathrm{\infty }}(\mathbb{D})$. By Corollary 4.7, this sum converges absolutely in $C^{\mathrm{\infty }}(\mathbb{D})$. From Lemma 4.5 we have that (38(38) $\underset{|m|\to \mathrm{\infty }}{lim sup}|c_m{|}^{1/|m|}\le 1,$(38) ) holds.

We note that Theorem 5.1 generalizes Theorem 5.1 in [1], which in turn is an improvement of Ahern, Bruna and Cascante's Theorem 2.1 in [8] when specialized to the current setting. Theorem 5.1 also includes Theorem 1.2 in Olofsson and Wittsten [3], and Theorem 2.2 in Olofsson [2].

We claim that it is possible to retrieve a formula for the expansion coefficients $c_m$ appearing in (37(37) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}{c}_{m}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}{c}_{-m}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(37) ). We will not delve into the details of this topic however, and instead refer to the fifth section of [1]. The form in which the series expansion appears below can be deduced from the results of this section following a slight change of wording. We will content ourselves with stating the result.

Theorem 5.2

Let $p,q,r\in \mathbb{C}$. Let u be a $(p,q,r)$-harmonic function. Then (42) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}\frac{{\mathrm{\partial }}^{m}u(0)}{m!}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}\frac{{\overline{\mathrm{\partial }}}^{m}u(0)}{m!}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(42) where F is the hypergeometric function (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ) with parameters ${\chi }_{p,q,r,m}^{\pm }$ being the zeros of the polynomial $P_{p,q,r,m}$ in (6(6) $P_{p,q,r,m}(\lambda )={\lambda }^2-\lambda (m-p-q)+r-\mathrm{pm},\lambda \in \mathbb{C}.$(6) ).

Note that the last result also settles the question on uniqueness of the series representation.

On a side note and before we conclude, we also wish to include an alternative form of the series representation in (42(42) $\begin{array}{rl}u(z)& =\sum _{m=0}^{\mathrm{\infty }}\frac{{\mathrm{\partial }}^{m}u(0)}{m!}F({\chi }_{p,q,r,m}^{+},{\chi }_{p,q,r,m}^{-},m+1;|z{|}^2)z^m\\ & +\sum _{m=1}^{\mathrm{\infty }}\frac{{\overline{\mathrm{\partial }}}^{m}u(0)}{m!}F({\chi }_{q,p,r,m}^{+},{\chi }_{q,p,r,m}^{-},m+1;|z{|}^2){\overline{z}}^m,z\in \mathbb{D},\end{array}$(42) ), valid in the restriction to $(p,q)$-harmonic functions. In particular, we show that the hypergeometric function in (21(21) $u(z)=\mathrm{cF}(-p,m-q,m+1;|z{|}^2)z^m,z\in \mathbb{D},$(21) ) can be retrieved from the $(p,q)$-harmonic Poisson kernel in (27(27) $K_{p,q}(z)=\frac{(1-|z{|}^2{)}^{p+q+1}}{(1-z{)}^{p+1}(1-\overline{z}{)}^{q+1}},z\in \mathbb{D}.$(27) ) and the basic elements of $(p,0)$-harmonic functions via integration of their product over $\mathbb{T}$. This peculiar integral representation is given by (43) $F(-p,m-q,m+1;r^2)=\frac{1}{2\pi }{\int }_{\mathbb{T}}{K}_{p,q}(r{\mathrm{e}}^{\mathrm{i\theta }})F(-p,m,m+1;r{\mathrm{e}}^{\mathrm{i\theta }})\mathrm{d}\theta ,$(43) for $0\le r<1$ and $m\in \mathbb{N}$. Although we have found no natural fit for this formula, it can be applied to retrieve some of the earlier results, such as Theorem 6.6 in [1]. We also note here that the hypergeometric function appearing under the integral sign in (43(43) $F(-p,m-q,m+1;r^2)=\frac{1}{2\pi }{\int }_{\mathbb{T}}{K}_{p,q}(r{\mathrm{e}}^{\mathrm{i\theta }})F(-p,m,m+1;r{\mathrm{e}}^{\mathrm{i\theta }})\mathrm{d}\theta ,$(43) ) regularly occurs in α-harmonic theory, for which we refer to [3], where instances of this formula also appears. We have chosen the following form for this representation, where we consider $f_r({\mathrm{e}}^{\mathrm{i\theta }})=f(r{\mathrm{e}}^{\mathrm{i\theta }})$ as functions on $\mathbb{T}$.

Theorem 5.3

Let $p,q\in \mathbb{C}$. Let u be a $(p,q)$-harmonic function. Then $\begin{array}{rl}{u}_r({\mathrm{e}}^{\mathrm{i\theta }})& =\sum _{m=0}^{\mathrm{\infty }}\frac{{\mathrm{\partial }}^{m}u(0)r^m}{m!}(\frac{1}{2\pi }{\int }_{\mathbb{T}}{K}_{p,q,r}({\mathrm{e}}^{\mathrm{i\tau }})F_r(-p,m,m+1;{\mathrm{e}}^{\mathrm{i\tau }})\mathrm{d}\tau ){\mathrm{e}}^{\mathrm{im\theta }}\\ & +\sum _{m=1}^{\mathrm{\infty }}\frac{{\overline{\mathrm{\partial }}}^{m}u(0)r^m}{m!}(\frac{1}{2\pi }{\int }_{\mathbb{T}}{K}_{q,p,r}({\mathrm{e}}^{\mathrm{i\tau }})F_r(-q,m,m+1;{\mathrm{e}}^{\mathrm{i\tau }})\mathrm{d}\tau ){\mathrm{e}}^{-\mathrm{im\theta }},\end{array}$ for $0\le r<1$, where $K_{p,q}$ is the $(p,q)$-harmonic Poisson kernel in (27(27) $K_{p,q}(z)=\frac{(1-|z{|}^2{)}^{p+q+1}}{(1-z{)}^{p+1}(1-\overline{z}{)}^{q+1}},z\in \mathbb{D}.$(27) ) and F is the hypergeometric function in (3(3) $F(a,b,c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!},z\in \mathbb{D},$(3) ).

Proof.

We will give the formula in (43(43) $F(-p,m-q,m+1;r^2)=\frac{1}{2\pi }{\int }_{\mathbb{T}}{K}_{p,q}(r{\mathrm{e}}^{\mathrm{i\theta }})F(-p,m,m+1;r{\mathrm{e}}^{\mathrm{i\theta }})\mathrm{d}\theta ,$(43) ). The statement then follows from Theorem 5.4 in [1]. By Euler's transformation formula for hypergeometric functions (see [15], Theorem 2.2.5), we can write $F(-p,m-q,m+1;r^2)=(1-r^2{)}^{p+q+1}F(m+1+p,1+q,m+1;r^2),$ for $0\le r<1$ and $m\in \mathbb{N}$. The hypergeometric function on the right of this expression is given by $F(m+1+p,1+q,m+1;r^2)=\sum _{n=0}^{\mathrm{\infty }}\frac{(m+1+p{)}_n}{(m+1{)}_n}\frac{(1+q{)}_n}{n!}{r}^{2n},$ for $0\le r<1$ and $m\in \mathbb{N}$. Note that $F(m+1+p,1,m+1;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(m+1+p{)}_n}{(m+1{)}_n}{z}^n,z\in \mathbb{D},$ for $m\in \mathbb{N}$. We also recall that $F(1,1+\overline{q},1;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(1+\overline{q}{)}_n}{n!}{z}^n=\frac{1}{(1-z{)}^{\overline{q}+1}},z\in \mathbb{D}.$ Following an application of Parseval's formula, we obtain $F(-p,m-q,m+1;r^2)=(1-r^2{)}^{p+q+1}\frac{1}{2\pi }{\int }_{\mathbb{T}}\frac{F(m+1+p,1,m+1;r{\mathrm{e}}^{\mathrm{i\theta }})}{(1-r{\mathrm{e}}^{-\mathrm{i\theta }}{)}^{q+1}}\mathrm{d}\theta ,$ for $0\le r<1$ and $m\in \mathbb{N}$. Another use of Euler's transformation formula gives that $F(m+1+p,1,m+1;z)=(1-z{)}^{-(1+p)}F(-p,m,m+1;z),z\in \mathbb{D},$ for $m\in \mathbb{N}$. This gives the formula in (43(43) $F(-p,m-q,m+1;r^2)=\frac{1}{2\pi }{\int }_{\mathbb{T}}{K}_{p,q}(r{\mathrm{e}}^{\mathrm{i\theta }})F(-p,m,m+1;r{\mathrm{e}}^{\mathrm{i\theta }})\mathrm{d}\theta ,$(43) ).

6. Final remarks

Questions concerning the class of second order differential operators in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ) and the mapping $(p,q,r)\mapsto (a,b,c)$ of this class onto the subclass of hypergeometric operators as determined through $z^m{H}_{a,b,c}f(|z{|}^2)=L_{p,q,r}{z}^{m}f(|z{|}^2),z\in \mathbb{D},$ in accordance with Theorem 2.1, has led us to Theorem 2.4. It was this identification of elements $(p,q,r)$ in the underlying parameter space of the operator class in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ), with elements $(p,q,\mathrm{pq})$ in the underlying parameter realm of the subclass $L_{p,q,\mathrm{pq}}$, that allowed us to recycle the developments in [1] concerned with the latter. This made possible the short cut in the upgrade from two degrees of freedom to three, and the generalization of Theorem 5.1 in [1] as presented in Theorem 5.1. We will maintain our interest for the operator class in (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ) and its corresponding class of hypergeometric operators.

We also briefly discussed the associated Dirichlet boundary value problem for the case r = pq, and mentioned in section two that a solution to this problem has been obtained in terms of a Poisson type integral [1–3,8]. The case considered here in the addition of a third parameter is left open and we are unmindful of any such formulas.

As mentioned in the concluding remarks of [1], earlier work also concerns pointwise boundary limits, Green functions and Lipschitz continuity of generalized harmonic functions [16,19,20].

Before we finish, we also give a (formal) version of the operator class (1(1) $L_{p,q,r;z}=(1-|z{|}^2){\mathrm{\partial }}_z{\overline{\mathrm{\partial }}}_z+\mathrm{pz}{\mathrm{\partial }}_z+q\overline{z}{\overline{\mathrm{\partial }}}_z-r,z\in \mathbb{D},$(1) ) in polar terms, and consider the expression $\frac{1}{4}(1-{\rho }^2){\mathrm{\nabla }}^2+\frac{p}{2}(\rho {\mathrm{\partial }}_{\rho }-i{\mathrm{\partial }}_{\theta })+\frac{q}{2}(\rho {\mathrm{\partial }}_{\rho }+i{\mathrm{\partial }}_{\theta })-r.$ The homogeneous equation associated with this class of operators for the particular choice of $p=q=-1/2$ and $r=-{\gamma }^2/2$ corresponds to $\rho {\mathrm{\partial }}_{\rho }u-\frac{1}{2}(1-{\rho }^2){\mathrm{\nabla }}^{2}u={\gamma }^{2}u.$ We compare the form of this equation to that associated with harmonic traps in connection with Bose-Einstein condensates, as laid out in the third chapter of Kling [14].⁴

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is the author's opinion that the results of this paper should be viewed as an application of the methods presented in [1], and be attributed as such. For this very reason, the author has chosen to preserve the earlier format to any suitable extent, and the reader is advised to consult the former text for reference. The author has also tried to highlight the connection between the two whenever applicable, and where a comparison can be made between the special and the more general.

2 The author wishes to thank Anders Olofsson for his constructive criticism of the manuscript.

3 The author would like to thank the referee for a thorough reading of the text.

4 The author thanks Felix Tellander for drawing his attention to this source.