A series expansion for generalized harmonic functions

Abstract

We consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.

1 Introduction

Let ${\mathbb {D}}$ be the open unit disc in the complex plane ${\mathbb {C}}$ and denote by $\partial _z=\partial /\partial z$ and $\bar{\partial }_z=\partial /\partial {\bar{z}}$ the usual complex partial derivatives. This work is concerned with second order partial differential operators of the form

$$\begin{aligned} L_{p,q}=(1-|z|^2)\partial _z\bar{\partial }_z +pz\partial _z+q{\bar{z}}\bar{\partial }_z-pq,\quad z\in {\mathbb {D}}, \end{aligned}$$

(0.1)

where $p,q\in {\mathbb {C}}$ are complex parameters. Of particular interest are solutions of the associated homogeneous equation

$$\begin{aligned} L_{p,q}u=0\quad \text {in}\ {\mathbb {D}}. \end{aligned}$$

We say that a function u is (p, q)-harmonic if u is twice continuously differentiable in ${\mathbb {D}}$ (in symbols $u\in C^2({\mathbb {D}})$) and $L_{p,q}u=0$ in ${\mathbb {D}}$, where $L_{p,q}$ is as in (0.1). Notice that a function u is (p, q)-harmonic if and only if its complex conjugate ${\bar{u}}$ is $({\bar{q}},{\bar{p}})$-harmonic. Observe also that a (0, 0)-harmonic function is a harmonic function in ${\mathbb {D}}$ in the usual sense.

An interesting example of a (p, q)-harmonic function is the function

$$\begin{aligned} u_{p,q}(z)= \frac{(1-|z|^2)^{p+q+1}}{(1-z)^{p+1} (1-{\bar{z}})^{q+1}}, \quad z \in {\mathbb {D}}\end{aligned}$$

(0.2)

(see Theorem 1.4). Here powers are defined in the usual way using the principal branch of the logarithm, that is, we require that $\log (1)=0$. Notice that the above functions $u_{p,q}$ have the hermitian symmetry property that ${\bar{u}}_{p,q}=u_{{\bar{q}},{\bar{p}}}$ for $p,q\in {\mathbb {C}}$.

Recall that elements of the unit circle ${\mathbb {T}}=\partial {\mathbb {D}}$ act on the unit disc ${\mathbb {D}}$ as rotations. On a function level we consider rotation operators

$$\begin{aligned} R_{e^{i\theta }}u(z)=u(e^{i\theta }z),\quad z\in {\mathbb {D}}, \end{aligned}$$

(0.3)

for $e^{i\theta }\in {\mathbb {T}}$ acting on functions u in ${\mathbb {D}}$. A basic observation concerning the differential operator $L_{p,q}$ is the commutativity relation

$$\begin{aligned} L_{p,q}R_{e^{i\theta }}u=R_{e^{i\theta }}L_{p,q}u,\quad u\in C^2({\mathbb {D}}), \end{aligned}$$

for $e^{i\theta }\in {\mathbb {T}}$. This latter commutativity relation suggests an analysis of (p, q)-harmonic functions using concepts natural to classical Fourier analysis on the unit circle.

Let ${\mathbb {Z}}$ be the set of integers. For a suitably smooth function u in ${\mathbb {D}}$ we define its m-th homogeneous part by the formula

$$\begin{aligned} u_m(z)=\frac{1}{2\pi }\int _{\mathbb {T}}e^{-im\theta }u(e^{i\theta }z)\, d\theta , \quad z\in {\mathbb {D}}, \end{aligned}$$

(0.4)

for $m\in {\mathbb {Z}}$. Notice that the m-th homogeneous part $u_m$ of u is the m-th Fourier coefficient of the vector-valued function

$$\begin{aligned} {\mathbb {T}}\ni e^{i\theta }\mapsto R_{e^{i\theta }}u, \end{aligned}$$

where $R_{e^{i\theta }}$ is as in (0.3).

We set $C^\infty ({\mathbb {D}})=\bigcap _{n=0}^\infty C^n({\mathbb {D}})$, where $C^n({\mathbb {D}})$ is the space of n-times continuously differentiable functions in ${\mathbb {D}}$ for $n\in {\mathbb {N}}=\{0,1,2,\dots \}$. We topologize these spaces in the usual way using the semi-norms

$$\begin{aligned} \Vert u\Vert _{j,k;K}= \max _{z \in K}|\partial ^{j} \bar{\partial }^{k}u(z)|, \end{aligned}$$

where $j,k\in {\mathbb {N}}$ are non-negative integers and $K\subset {\mathbb {D}}$ is a compact subset of ${\mathbb {D}}$.

Let us recall the classical hypergeometric function defined by

$$\begin{aligned} F(a,b;c;z)=\sum _{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!}, \quad z\in {\mathbb {D}}, \end{aligned}$$

(0.5)

for $a,b,c\in {\mathbb {C}}$ such that $c\ne -1,-2,\dots$. Here $(a)_0=1$ and

$$\begin{aligned} (a)_n=a(a+1)\dots (a+n-1) \end{aligned}$$

for $n=1,2,\dots$ are Pochhammer symbols.

Let us return to a (p, q)-harmonic function u. We show that the m-th homogeneous part of u has the form

$$\begin{aligned} u_m(z)= c_mF(-p,m-q;m+1;|z|^2)z^m,\quad z\in {\mathbb {D}}, \end{aligned}$$

for some $c_m\in {\mathbb {C}}$ when $m\in {\mathbb {N}}$ is a non-negative integer (see Theorem 4.3). Here F is the hypergeometric function (0.5). Hermitian symmetry leads to a similar formula for $u_m$ when $m\in {\mathbb {Z}}^-={\mathbb {Z}}{\setminus }{\mathbb {N}}$ is a negative integer (see Corollary 4.4). Notice that $u_m\in C^\infty ({\mathbb {D}})$ for $m\in {\mathbb {Z}}$.

A principal result of the present paper concerns the asymptotic behavior of the m-th homogeneous part $u_m$ of a (p, q)-harmonic function u. We show that

$$\begin{aligned} \limsup _{|m|\rightarrow \infty }\Vert u_m\Vert _{j,k;K}^{1/|m|}<1 \end{aligned}$$

for all $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact (see Theorem 4.6). This result enables us to use the classical root test to establish absolute convergence of the function series $\sum _{m=-\infty }^\infty u_m$ in $C^\infty ({\mathbb {D}})$ (see Corollary 4.7).

A further analysis leads to a function series characterization of (p, q)-harmonic functions. A function u in ${\mathbb {D}}$ is (p, q)-harmonic if and only if it has the form

$$\begin{aligned} u(z)&=\sum _{m=0}^\infty c_mF(-p,m-q;m+1;|z|^2)z^m\nonumber \\&\quad +\,\sum _{m=1}^\infty c_{-m}F(-q,m-p;m+1;|z|^2){\bar{z}}^m, \quad z\in {\mathbb {D}}, \end{aligned}$$

(0.6)

for some sequence $\{c_m\}_{m=-\infty }^\infty$ of complex numbers such that

$$\begin{aligned} \limsup _{|m|\rightarrow \infty } |c_m|^{1/|m|}\le 1 \end{aligned}$$

(see Theorem 5.1). As mentioned above, the sums in (0.6) are absolutely convergent in the space $C^\infty ({\mathbb {D}})$. As a consequence we have that $u\in C^\infty ({\mathbb {D}})$ if u is (p, q)-harmonic (see Corollary 4.9). This characterization of (p, q)-harmonic functions improves on a result by Ahern et al. [1, Theorem 2.1] when specialized to our setting.

We then turn to coefficient formulas for (p, q)-harmonic functions. We show that

$$\begin{aligned} c_m=\partial ^m u(0)/m! \quad \text {and}\quad c_{-m}=\bar{\partial }^m u(0)/m! \end{aligned}$$

for $m\in {\mathbb {N}}$, where the $c_m$’s are as in (0.6) (see Theorem 5.3). As a consequence, we have that

$$\begin{aligned} u(z)&=\sum _{m=0}^\infty \frac{\partial ^m u(0)}{m!} F(-p,m-q;m+1;|z|^2)z^m\\&\quad \ +\sum _{m=1}^\infty \frac{\bar{\partial }^m u(0)}{m!} F(-q,m-p;m+1;|z|^2){\bar{z}}^m, \quad z\in {\mathbb {D}}, \end{aligned}$$

whenever u is a (p, q)-harmonic function (see Theorem 5.4) as well as a corresponding uniqueness result for such functions (see Corollary 5.5).

Let us return to the function $u_{p,q}$ in (0.2). A calculation of partial derivatives at the origin of the function $u_{p,q}$ leads to the series expansion

$$\begin{aligned} u_{p,q}(z)&=\sum _{m=0}^\infty \frac{(p+1)_m}{m!}F(-p,m-q;m+1;|z|^2)z^m\\&\quad +\sum _{m=1}^\infty \frac{(q+1)_m}{m!}F(-q,m-p;m+1;|z|^2){\bar{z}}^m, \quad z\in {\mathbb {D}}\end{aligned}$$

(see Theorem 6.3). This latter series expansion generalizes a well-known partial fraction decomposition formula for the classical Poisson kernel for ${\mathbb {D}}$ which is obtained for $p=q=0$.

A main contribution of this paper concerns series expansion of (p, q)-harmonic functions. Of particular mention is a limit theorem for associated hypergeometric functions:

$$\begin{aligned} \lim _{m\rightarrow \infty } F(-p,m-q;m+1;z)=(1-z)^p,\quad z\in {\mathbb {D}}\end{aligned}$$

(see Theorem 2.6). Apart from its intrinsic interest, this limit theorem provides an efficient tool for the study of limit properties of homogeneous parts of (p, q)-harmonic functions.

The results of this paper have applications to Poisson integral representations of (p, q)-harmonic functions which is possible when $p,q\in {\mathbb {C}}{\setminus }{\mathbb {Z}}^-$ are such that ${\text {Re}}(p)+{\text {Re}}(q)>-1$. In the final section of the paper we comment briefly on the connection to such theory.

We have traced the study of (p, q)-harmonic functions back to Daryl Geller [13]. Other significant contributions are those of Ahern and collaborators [1, 2]. Our interest in those topics [18,19,20,21] arose in connection to standard weights and earlier work of Garabedian [12]. Borichev and Hedenmalm [6] have established a connection to polyharmonic theory. Other recent related papers are [5, 8,9,10, 14, 16, 17, 22].

The authors thank the referee for a careful reading of the manuscript.

2 The function $u_{p,q}$ is (p, q)-harmonic

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Observe that the functions $u_{p,q}$ have the hermitian symmetry property ${\bar{u}}_{p,q}=u_{{\bar{q}}, {\bar{p}}}$. Indeed,

$$\begin{aligned} {\bar{u}}_{p,q}(z)=\overline{u_{p,q}(z)}= \frac{(1-|z|^2)^{{\bar{p}}+{\bar{q}}+1}}{(1-{\bar{z}})^{{\bar{p}}+1} (1-z)^{{\bar{q}}+1}} = u_{{\bar{q}}, {\bar{p}}}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. We shall discuss in this section some differentiation formulas for the functions $u_{p,q}$. In particular, we shall show that the function $u_{p,q}$ is (p, q)-harmonic (see Theorem 1.4).

Lemma 1.1

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} z\partial u_{p,q}(z)=\Big (-(p+q+1)\frac{|z|^2}{1-|z|^2} +(p+1)\frac{z}{1-z}\Big ) u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$.

Proof

Differentiating using the product rule for differentiation we have that

$$\begin{aligned} \partial u_{p,q}(z)&=\frac{1}{ (1-{\bar{z}})^{q+1}}\Big ( (p+q+1)(1-|z|^2)^{p+q}(-{\bar{z}}) \frac{1}{(1-z)^{p+1}}\\&\quad +(1-|z|^2)^{p+q+1}(-(p+1)) \frac{1}{(1-z)^{p+2}}(-1)\Big )\\&= \Big (-(p+q+1)\frac{{\bar{z}}}{1-|z|^2} +(p+1)\frac{1}{1-z}\Big ) u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$, where the last equality is straightforward to check. This yields the conclusion of the lemma. $\square$

We next turn to the $\bar{\partial }$-derivative of $u_{p,q}$.

Lemma 1.2

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} {\bar{z}}\bar{\partial }u_{p,q}(z)= \Big (-(p+q+1)\frac{|z|^2}{1-|z|^2} +(q+1)\frac{{\bar{z}}}{1-{\bar{z}}}\Big ) u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$.

Proof

We shall use the hermitian symmetry property ${\bar{u}}_{p,q}=u_{{\bar{q}}, {\bar{p}}}$. From Lemma 1.1 we have that

$$\begin{aligned} z\partial u_{{\bar{q}},{\bar{p}}}(z) =\Big (-({\bar{p}}+{\bar{q}}+1)\frac{|z|^2}{1-|z|^2} +({\bar{q}}+1)\frac{z}{1-z}\Big ) u_{{\bar{q}},{\bar{p}}}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. A complex conjugation now yields the conclusion of the lemma. $\square$

Following earlier practice from [20] we denote by $A=z\partial -{\bar{z}}\bar{\partial }$ the angular derivative. We next calculate the angular derivative of $u_{p,q}$.

Corollary 1.3

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} (z\partial -{\bar{z}}\bar{\partial }) u_{p,q}(z)= \Big ((p+1)\frac{z}{1-z}-(q+1)\frac{{\bar{z}}}{1-{\bar{z}}}\Big ) u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$.

Proof

The result is evident from Lemmas 1.1 and 1.2. $\square$

We mention that Corollary 1.3 generalizes [20, Theorem 1.11].

Recall the partial fraction formula

$$\begin{aligned} \frac{1-|z|^2}{|1-z|^2}=\frac{z}{1-z} +\frac{{\bar{z}}}{1-{\bar{z}}}+1 \end{aligned}$$

(1.1)

for the classical Poisson kernel for the unit disc. Formula (1.1) is straightforward to check.

Theorem 1.4

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Then $L_{p, q}u_{p, q}=0$ in ${\mathbb {D}}$, where $L_{p,q}$ is as in (0.1).

Proof

Recall Lemma 1.2. Notice that the differential operator $z\partial$ satisfies the product rule for differentiation. It is straightforward to check that

$$\begin{aligned} z\frac{\partial }{\partial z} \Big (\frac{|z|^2}{1-|z|^2} \Big ) =\frac{|z|^2}{(1-|z|^2)^2} \end{aligned}$$

whenever the formula makes sense. Differentiating using the product rule we have that

$$\begin{aligned} |z|^2\partial \bar{\partial }u_{p,q}(z)&= -(p+q+1)\frac{|z|^2}{(1-|z|^2)^2}u_{p,q}(z)\\&\quad +\Big (-(p+q+1)\frac{|z|^2}{1-|z|^2} +(q+1)\frac{{\bar{z}}}{1-{\bar{z}}}\Big ) z\partial u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. From Lemma 1.1 we now have that

$$\begin{aligned} |z|^2\partial \bar{\partial }u_{p,q}(z)&=-(p+q+1)\frac{|z|^2}{(1-|z|^2)^2}u_{p,q}(z)\\&\quad +\Big (-(p+q+1)\frac{|z|^2}{1-|z|^2} +(q+1)\frac{{\bar{z}}}{1-{\bar{z}}}\Big )\\&\quad \times \Big (-(p+q+1)\frac{|z|^2}{1-|z|^2} +(p+1)\frac{z}{1-z}\Big )u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. Expanding the above product we see that

$$\begin{aligned} |z|^2 \partial \bar{\partial }u_{p,q}(z)&= \Big (-(p+q+1)\frac{|z|^2}{(1-|z|^2)^2} +(p+q+1)^2\frac{|z|^4}{(1-|z|^2)^2} \Big )u_{p,q}(z)\\&\quad -(p+q+1)\frac{|z|^2}{1-|z|^2} \Big ((p+1)\frac{z}{1-z}+(q+1)\frac{{\bar{z}}}{1-{\bar{z}}}\Big ) u_{p,q}(z)\\&\quad +(p+1)(q+1)\frac{|z|^2}{|1-z|^2}u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. Multiplying by a factor $(1-|z|^2)/|z|^2$ we see that

$$\begin{aligned} (1-|z|^2) \partial \bar{\partial }u_{p,q}(z)&= \Big ((p+q+1)^2\frac{|z|^2}{1-|z|^2} -(p+q+1)\frac{1}{1-|z|^2}\Big )u_{p,q}(z)\\&\quad -(p+q+1)\Big ((p+1)\frac{z}{1-z}+(q+1)\frac{{\bar{z}}}{1-{\bar{z}}}\Big ) u_{p,q}(z)\\&\quad +\,(p+1)(q+1)\frac{1-|z|^2}{|1-z|^2}u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. Notice an appearance of the classical Poisson kernel in the rightmost term above. Using the partial fraction formula (1.1) we have that

$$\begin{aligned} (1-|z|^2) \partial \bar{\partial }u_{p,q}(z)&=(p+q+1) \Big ((p+q)\frac{|z|^2}{1-|z|^2}-1\Big )u_{p,q}(z)\\&\quad +\,((p+1)(q+1) -(p+q+1)(p+1))\frac{z}{1-z}u_{p,q}(z)\\&\quad +\,((p+1)(q+1)-(p+q+1)(q+1))\frac{{\bar{z}}}{1-{\bar{z}}} u_{p,q}(z)\\&\quad +\,(p+1)(q+1)u_{p,q}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. A simplification of terms now leads to the formula

$$\begin{aligned} (1-|z|^2)&\partial \bar{\partial }u_{p,q}(z) = (p+q+1)(p+q)\frac{|z|^2}{1-|z|^2}u_{p,q}(z)\nonumber \\&-p(p+1)\frac{z}{1-z}u_{p,q}(z) -q(q+1)\frac{{\bar{z}}}{1-{\bar{z}}} u_{p,q}(z) +pq u_{p,q}(z) \end{aligned}$$

(1.2)

for $z\in {\mathbb {D}}$. Recall Lemmas 1.1 and 1.2. In view of these two lemmas our latter formula (1.2) says that $L_{p,q}u_{p,q}=0$ in ${\mathbb {D}}$. $\square$

3 A sequence of hypergeometric functions

Let us first consider a second order partial differential operator of the form

$$\begin{aligned} L_{p,q;r}=(1-|z|^2)\partial _z \bar{\partial }_z+pz\partial _z+ q\bar{z}\bar{\partial }_z-r, \quad z \in {\mathbb {D}}, \end{aligned}$$

(2.1)

where $p,q,r \in {\mathbb {C}}$ are complex parameters. A principal case is when $r=pq$. Notice that $L_{p,q;pq}=L_{p,q}$, where $L_{p,q}$ is as in (0.1). The introduction of an additional parameter $r\in {\mathbb {C}}$ allows for more general operators appearing in the study of conductivity problems, see for instance Calderón [7] or Astala and Päivärinta [4].

We shall evaluate the operator $L_{p,q;r}$ on a complex-valued function u in the punctured disc ${\mathbb {D}}{\setminus } \{ 0 \}$ of the form

$$\begin{aligned} u(z)=f(|z|^2) z^m,\quad z\in {\mathbb {D}}{\setminus } \{ 0 \}, \end{aligned}$$

(2.2)

for some $f\in C^2(0,1)$ and $m\in {\mathbb {Z}}$.

We introduce also the ordinary differential operator

$$\begin{aligned} H_{a,b;c}=(1-x)x\frac{d^2}{dx^2}+(c-[a+b+1]x)\frac{d}{dx}-ab, \end{aligned}$$

(2.3)

where $a,b,c\in {\mathbb {C}}$ are complex parameters. Notice that the famous hypergeometric ordinary differential equation

$$\begin{aligned} (1-x)xy''(x)+(c-[a+b+1]x)y'(x)-aby(x)=0 \end{aligned}$$

takes the form $H_{a,b;c}y=0$ using the operator $H_{a,b;c}$.

Theorem 2.1

Let $L_{p,q;r}$ be as in (2.1) for some $p,q,r \in {\mathbb {C}}$. Let u be a function of the form (2.2) for some $f\in C^2(0,1)$ and $m \in {\mathbb {Z}}$. Then

$$\begin{aligned} L_{p,q;r}u(z)=z^m H_{a,b;c} f(|z|^2), \quad z \in {\mathbb {D}}{\setminus }\{ 0 \}, \end{aligned}$$

where $c=m+1$ and

$$\begin{aligned} \left\{ \begin{array}{ccc} a+b&{} = &{}m-p-q, \\ ab &{}= &{} r-pm. \end{array}\right. \end{aligned}$$

(2.4)

Proof

A differentiation shows that

$$\begin{aligned} \partial u(z)=mz^{m-1}f(|z|^2)+z^{m}{\bar{z}} f'(|z|^2) \end{aligned}$$

for $z \in {\mathbb {D}} {\setminus } \{ 0 \}$, and similarly that

$$\begin{aligned} \bar{\partial }u(z)=z^{m+1}f'(|z|^2) \end{aligned}$$

for $z \in {\mathbb {D}} {\setminus } \{ 0 \}$. Another differentiation gives that

$$\begin{aligned} \partial \bar{\partial }u(z)=(m+1)z^{m}f'(|z|^2)+z^{m+1} {\bar{z}} f''(|z|^2) \end{aligned}$$

for $z \in {\mathbb {D}} {\setminus } \{ 0 \}$. A calculation using these formulas gives that

$$\begin{aligned} L_{p,q;r}u(z)&=(1-|z|^2)((m+1)z^{m}f'(|z|^2)+z^{m}|z|^2f''(|z|^2)) +q\bar{z} z^{m+1}f'(|z|^2) \\&\quad +pz (mz^{m-1}f(|z|^2)+z^{m}\bar{z}f'(|z|^2)) - rz^mf(|z|^2) \\&= z^m \Big (|z|^2(1-|z|^2)f''(|z|^2)+(m+1-[m+1-p-q]|z|^2)f'(|z|^2) \\&\qquad -\,(r-pm)f(|z|^2)\Big )=z^m H_{a,b;c}f(|z|^2) \end{aligned}$$

for $z \in {\mathbb {D}} {\setminus } \{ 0 \}$, where $c=m+1$, the numbers a and b are as in (2.4) and $H_{a,b;c}$ is as in (2.3). This yields the conclusion of the theorem. $\square$

Equations (2.4) say that a and b are the zeros of the quadratic polynomial

$$\begin{aligned} P_{p,q;r;m}(\lambda )=\lambda ^2-(m-p-q)\lambda +r-pm, \quad \lambda \in {\mathbb {C}}. \end{aligned}$$

Notice that

$$\begin{aligned} P_{p,q;r;m}(\lambda )=(\lambda +p)(\lambda +q-m)+r-pq, \quad \lambda \in {\mathbb {C}}. \end{aligned}$$

In particular, the zeros of $P_{p,q;r;m}$ are $-p$ and $m-q$ in the principal case when $r=pq$.

Theorem 2.1 suggests a natural construction of (p, q)-harmonic functions.

Proposition 2.2

Let $p,q\in {\mathbb {C}}$. Consider the function

$$\begin{aligned} u_m(z)=F(-p,m-q;m+1;|z|^2)z^m,\quad z\in {\mathbb {D}}, \end{aligned}$$

where $m\in {\mathbb {N}}$ and F is the hypergeometric function (0.5). Then $u_m$ is a (p, q)-harmonic function.

Proof

Clearly $u_m\in C^\infty ({\mathbb {D}})$. It is well-known that the hypergeometric function $y=F(a,b;c;\cdot )$ satisfies the hypergeometric equation $H_{a,b;c}y=0$ (see [3, Section 2.3]). The result now follows by Theorem 2.1. $\square$

Corollary 2.3

Let $p,q\in {\mathbb {C}}$. Consider the function

$$\begin{aligned} u_m(z)= F(-q,|m|-p;|m|+1;|z|^2){\bar{z}}^{|m|}, \quad z\in {\mathbb {D}}, \end{aligned}$$

where $m\in {\mathbb {Z}}^-$ and F is the hypergeometric function (0.5). Then $u_m$ is a (p, q)-harmonic function.

Proof

We consider the complex conjugate

$$\begin{aligned} \overline{u_m(z)}= F(-{\bar{q}},|m|-{\bar{p}};|m|+1;|z|^2) z^{|m|}, \quad z\in {\mathbb {D}}. \end{aligned}$$

By Proposition 2.2 we have that ${\bar{u}}_m$ is a $({\bar{q}},{\bar{p}})$-harmonic function. From hermitian symmetry we conclude that $u_m$ is a (p, q)-harmonic function. $\square$

Following earlier practice, 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 that

$$\begin{aligned} u(e^{i\theta }z)=e^{im\theta }u(z),\quad z\in {\mathbb {D}}{\setminus }\{0\}, \end{aligned}$$

for $e^{i\theta }\in {\mathbb {T}}$. Notice that every function u of the form (2.2) is homogeneous of order m with respect to rotations.

Theorem 2.4

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

$$\begin{aligned} u(z)=cF(-p,m-q;m+1;|z|^2)z^m,\quad z\in {\mathbb {D}}, \end{aligned}$$

(2.5)

for some $c\in {\mathbb {C}}$, where F is the hypergeometric function (0.5).

Proof

From Proposition 2.2 we know that every function u of the form (2.5) is (p, q)-harmonic.

Assume next that u is (p, q)-harmonic. Since u is homogeneous of order m, we can put u on the form (2.2) for some $f\in C^2(0,1)$. By Theorem 2.1 we have that

$$\begin{aligned} H_{-p,m-q,m+1}f(x)=0,\quad 0<x<1, \end{aligned}$$

where $H_{a,b;c}$ is as in (2.3). Below we shall check that

$$\begin{aligned} (m+1)f'(x)+p(m-q)f(x)=o(1/x^{m+1}) \end{aligned}$$

(2.6)

as $x\rightarrow 0$. Condition (2.6) allows us to apply [18, Proposition 1.3] to conclude that f is a constant multiple of the hypergeometric function $F(-p,m-q;m+1;\cdot )$. This will then complete the proof of the theorem.

We proceed to check (2.6). Recall formula (2.2). Since u is bounded near the origin we have that $f(x)={\mathcal {O}}(1/x^{m/2})$ as $x \rightarrow 0$. A differentiation of (2.2) gives that

$$\begin{aligned} \bar{\partial }u(z)=z^{m+1}f'(|z|^2),\quad z\in {\mathbb {D}}{\setminus }\{0\}. \end{aligned}$$

Since $\bar{\partial }u$ is bounded near the origin we have that $f'(x)={\mathcal {O}}(1/x^{(m+1)/2})$ as $x \rightarrow 0$. We have now checked (2.6). $\square$

Theorem 2.4 and its proof are modeled on [18, Theorem 2.1]. We have merely supplied some details.

We shall use the fact that the hypergeometric functions are analytic in ${\mathbb {D}}$. Let $H({\mathbb {D}})$ be the space of analytic functions in ${\mathbb {D}}$. The space $H({\mathbb {D}})$ is topologized in the usual manner using the semi-norms

$$\begin{aligned} \Vert f\Vert _K=\max _{z\in K}|f(z)|\end{aligned}$$

(2.7)

for $K\subset {\mathbb {D}}$ compact. Convergence in the space $H({\mathbb {D}})$ is usually referred to as normal convergence in ${\mathbb {D}}$.

Recall the terminology that a subset ${\mathcal {F}}$ of $H({\mathbb {D}})$ is called a normal family 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 ${\mathcal {F}}$. We refer to Conway [11, Chapter VII] for background.

Recall also the binomial series:

$$\begin{aligned} F(a,b;b;z)=\sum _{n=0}^\infty \frac{(a)_n}{n!}z^n=\frac{1}{(1-z)^a} \end{aligned}$$

(2.8)

for $z\in {\mathbb {D}}$.

Lemma 2.5

Let $p,q\in {\mathbb {C}}$ and consider the functions

$$\begin{aligned} f_m(z)=F(-p,m-q;m+1;z),\quad z\in {\mathbb {D}}, \end{aligned}$$

for $m\in {\mathbb {N}}$, where F is the hypergeometric function (0.5). Then ${\mathcal {F}}=\{f_m:\ m\in {\mathbb {N}}\}$ is a normal family of analytic functions in ${\mathbb {D}}$.

Proof

We shall prove that the functions in ${\mathcal {F}}$ are uniformly bounded on compact subsets of ${\mathbb {D}}$. The conclusion of the lemma then follows by a classical result of Montel (see Conway [11, Theorem VII.2.9]).

Let $K\subset {\mathbb {D}}$ be compact. Since $K\subset {\mathbb {D}}$ is compact there exists $0<r<1$ such that $\max _{z\in K}|z|<r$. Choose N such that

$$\begin{aligned} |1-(q+1)/(m+1)|\le 1/r \end{aligned}$$

for $m>N$. Observe that

$$\begin{aligned} \frac{(m-q)_n}{(m+1)_n}=\prod _{k=0}^{n-1}\frac{m-q+k}{m+1+k}= \prod _{k=0}^{n-1}\Big (1-\frac{q+1}{m+1+k}\Big ). \end{aligned}$$

Therefore $|(m-q)_n/(m+1)_n |\le 1/r^n$ for $n\in {\mathbb {N}}$ provided $m>N$.

We now estimate the $f_m$’s with $m>N$. From (0.5) we have that

$$\begin{aligned} f_m(z)=\sum _{n=0}^\infty \frac{(-p)_n(m-q)_n}{(m+1)_n}\frac{z^n}{n!}, \quad z\in {\mathbb {D}}. \end{aligned}$$

Notice in this sum an appearance of the quotient $(m-q)_n/(m+1)_n$ considered in the previous paragraph. From the triangle inequality and the result of the previous paragraph we have that

$$\begin{aligned} |f_m(z)|\le \sum _{n=0}^\infty \frac{(|p|)_n}{n!}\frac{|z|^n}{r^n} =\frac{1}{(1-|z|/r)^{|p|}} \end{aligned}$$

for $|z|<r$ and $m>N$, where the last equality follows by the binomial series (2.8). This proves that the functions in ${\mathcal {F}}$ are uniformly bounded on K. $\square$

The following limit theorem will be much useful.

Theorem 2.6

Let $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} \lim _{m\rightarrow \infty } F(-p,m-q;m+1;z)=(1-z)^p,\quad z\in {\mathbb {D}}, \end{aligned}$$

with normal convergence, where F is the hypergeometric function (0.5).

Proof

Let the $f_m$’s be as in Lemma 2.5. From Lemma 2.5 we have that the set ${\mathcal {F}}=\{f_m:\ m\in {\mathbb {N}}\}$ is a normal family. From (0.5) we have that

$$\begin{aligned} f_m^{(n)}(0)=\frac{(-p)_n(m-q)_n}{(m+1)_n} \end{aligned}$$

for $m,n\in {\mathbb {N}}$. Recall that the Pochhammer symbol $(\cdot )_n$ is a monic polynomial of degree n. Passing to the limit we have that $\lim _{m\rightarrow \infty }f_m^{(n)}(0)=(-p)_n$ for $n\in {\mathbb {N}}$.

From (2.8) we have that $f^{(n)}(0)=(-p)_n$ for $n\in {\mathbb {N}}$, where

$$\begin{aligned} f(z)=(1-z)^p,\quad z\in {\mathbb {D}}. \end{aligned}$$

A standard argument now yields that $f_m\rightarrow f$ in $H({\mathbb {D}})$ as $m\rightarrow \infty$. Assume to reach a contradiction that there exists a compact set $K\subset {\mathbb {D}}$ such that $\{f_m\}_{m=0}^\infty$ does not converge uniformly to f on K. Passing to a subsequence we can assume that

$$\begin{aligned} \max _{z\in K}|f_{m_k}(z)-f(z)|\ge \delta >0 \end{aligned}$$

(2.9)

for $k=1,2,\dots$. Since the set ${\mathcal {F}}$ is a normal family, we can after passage to another subsequence if necessary, assume that $f_{m_k}\rightarrow g$ in $H({\mathbb {D}})$ as $k\rightarrow \infty$ for some $g\in H({\mathbb {D}})$. From the first paragraph of the proof we have $g^{(n)}(0)=(-p)_n$ for $n\in {\mathbb {N}}$, and a uniqueness argument gives that $g=f$ in ${\mathbb {D}}$. Thus $f_{m_k}\rightarrow f$ in $H({\mathbb {D}})$ as $k\rightarrow \infty$, which contradicts (2.9). $\square$

We emphasize that Theorem 2.6 appears much natural in view of the binomial series (2.8).

4 A generalized power series

From the product rule for differentiation we have that

$$\begin{aligned} (fg)^{(n)}=\sum _{k=0}^{n} \left( {\begin{array}{c}n\\ k\end{array}}\right) f^{(n-k)}g^{(k)} \end{aligned}$$

for, say, $f,g\in C^n[0,1)$.

Lemma 3.1

Let $j,k,m\in {\mathbb {N}}$ be such that $m \ge j$. Let u be a function of the form (2.2) for some $f \in C^{\infty }[0,1)$. Then

$$\begin{aligned} \partial ^{j} \bar{\partial }^{k}u(z)=z^{m+k-j} \sum _{l=0}^{j} \left( {\begin{array}{c}j\\ l\end{array}}\right) \frac{(m+k)!}{(m+k+l-j)!} |z|^{2l}f^{(k+l)}(|z|^2) \end{aligned}$$

for $z \in {\mathbb {D}}$.

Proof

Recall (2.2). Differentiating with respect to $\bar{z}$ we have that

$$\begin{aligned} \bar{\partial }^{k}u(z)=z^{m+k} f^{(k)}(|z|^2) \end{aligned}$$

for $z\in {\mathbb {D}}$. Another differentiation using the product rule for differentiation gives that

$$\begin{aligned} \partial ^{j} \bar{\partial }^{k}u(z)&=\sum _{l=0}^{j} \left( {\begin{array}{c}j\\ l\end{array}}\right) \frac{(m+k)!}{(m+k-(j-l))!} z^{m+k-(j-l)} f^{(k+l)}(|z|^2){\bar{z}}^l\\&=z^{m+k-j} \sum _{l=0}^{j} \left( {\begin{array}{c}j\\ l\end{array}}\right) \frac{(m+k)!}{(m+k+l-j)!} |z|^{2l} f^{(k+l)}(|z|^2) \end{aligned}$$

for $z\in {\mathbb {D}}$. This completes the proof of the lemma. $\square$

Let $C^{\infty }({\mathbb {D}})$ be the set of smooth complex-valued functions in the unit disc ${\mathbb {D}}$. The space $C^{\infty }({\mathbb {D}})$ is topologized by means of the family of semi-norms

$$\begin{aligned} \Vert u\Vert _{j,k;K}= \max _{z \in K}|\partial ^{j} \bar{\partial }^{k}u(z)|, \end{aligned}$$

(3.1)

where $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ is compact. Recall that $u_m\rightarrow u$ in $C^\infty ({\mathbb {D}})$ as $m\rightarrow \infty$ means that $\lim _{m\rightarrow \infty }\Vert u_m-u\Vert _{j,k;K}=0$ for all $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact.

A (formal) series $\sum _{m=0}^\infty u_m$ of functions $u_m\in C^\infty ({\mathbb {D}})$ for $m\in {\mathbb {N}}$ is said to be absolutely convergent in $C^\infty ({\mathbb {D}})$ if

$$\begin{aligned} \sum _{m=0}^\infty \Vert u_m\Vert _{j,k;K}<+\infty \end{aligned}$$

whenever $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ is compact. By completeness of $C^\infty ({\mathbb {D}})$ we have that every series absolutely convergent in $C^\infty ({\mathbb {D}})$ is convergent in $C^\infty ({\mathbb {D}})$.

Theorem 3.2

Let $\lbrace f_m \rbrace _{m=0}^\infty$ be a sequence in $C^{\infty }[0,1)$ such that

$$\begin{aligned} \limsup _{m \rightarrow \infty }\Big (\max _{0 \le x \le r}|f^{(n)}_{m}(x)|\Big )^{1/m} \le 1 \end{aligned}$$

for $n\in {\mathbb {N}}$ and $0< r < 1$. Set

$$\begin{aligned} u_m(z)=f_m(|z|^2)z^m, \quad z\in {\mathbb {D}}, \end{aligned}$$

for $m=0,1,2,\dots$. Then $\limsup _{m \rightarrow \infty } \Vert u_m\Vert ^{1/m}_{j,k;K}<1$ for all $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact, where $\Vert \cdot \Vert _{j,k;K}$ is as in (3.1).

Proof

Fix $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact. Set $r=\max _{z\in K}|z|<1$. From Lemma 3.1 we have that

$$\begin{aligned} \partial ^j \bar{\partial }^k u_m(z)=z^{m+k-j} \sum _{l=0}^{j} \left( {\begin{array}{c}j\\ l\end{array}}\right) \frac{(m+k)!}{(m+k+l-j)!} |z|^{2l}f_m^{(k+l)}(|z|^2) \end{aligned}$$

for $z \in {\mathbb {D}}$ provided $m\ge j$. We next apply the triangle inequality to see that

$$\begin{aligned} \Vert u_m\Vert _{j,k;K}&\le r^{m+k-j} \Big (\sum _{l=0}^{j} \left( {\begin{array}{c}j\\ l\end{array}}\right) \frac{(m+k)!}{(m+k+l-j)!} r^{2l}\Big ) \max _{k\le n\le j+k} \Vert f_m^{(n)}\Vert _{[0,r^2]}\nonumber \\&\le r^{m+k-j} (m+k)^j (1+r^2)^j \max _{k\le n\le j+k} \Vert f_m^{(n)}\Vert _{[0,r^2]} \end{aligned}$$

(3.2)

for $m\ge j$, where we have used the notation (2.7). We shall next apply the m-th root to (3.2) and pass to the limit as $m\rightarrow \infty$. In view of the assumption on the sequence $\{f_m\}_{m=0}^\infty$ we conclude from (3.2) that

$$\begin{aligned} \limsup _{m\rightarrow \infty } \Vert u_m\Vert _{j,k;K}^{1/m} \le r. \end{aligned}$$

Since $0<r<1$, this yields the conclusion of the theorem. $\square$

From the conclusion of Theorem 3.2 we have that $\sum _{m=0}^\infty \Vert u_m\Vert _{j,k;K}<+\infty$ whenever $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ is compact. Thus the series $\sum _{m=0}^\infty u_m$ is absolutely convergent in $C^{\infty }({\mathbb {D}})$.

We can think of a function series

$$\begin{aligned} \sum _{m=0}^\infty f_m(|z|^2)z^m \end{aligned}$$

(3.3)

as a vector-valued power series. The sequence of coefficients $\{f_m\}_{m=0}^\infty$ in (3.3) is now a sequence of functions in $C^{\infty }[0,1)$. From this point of view, Theorem 3.2 generalizes a well-known root criteria for power series (see for instance Conway [11, Theorem III.1.3]).

Of particular concern to us are sequences $\{f_m\}_{m=0}^\infty$ of the form

$$\begin{aligned} f_m(x)=F(-p,m-q;m+1;x),\quad 0\le x <1, \end{aligned}$$

for $m\in {\mathbb {N}}$, where $p,q\in {\mathbb {C}}$ and F is the hypergeometric function (0.5). We next observe that Theorem 2.6 guarantees that every such sequence $\{f_m\}_{k=0}^\infty$ satisfies the assumption of Theorem 3.2.

Proposition 3.3

Let $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} \limsup _{m \rightarrow \infty }\big (\max \limits _{0 \le x \le r} \big |F^{(n)}(-p,m-q;m+1;x)\big |\big )^{1/m} \le 1 \end{aligned}$$

for $n\in {\mathbb {N}}$ and $0< r < 1$, where F is the hypergeometric function (0.5).

Proof

Recall the fact that the complex derivative $f\mapsto f'$ is continuous in the topology of normal convergence of analytic functions (see Conway [11, Theorem VII.2.1]). Recall Montel’s theorem characterizing normal families of analytic functions (see Conway [11, Theorem VII.2.9]). In view of these two results, the proposition follows from Theorem 2.6. $\square$

5 Analysis of homogeneous parts

Notice that the rotation operators ${\mathbb {T}}\ni e^{i\theta }\mapsto R_{e^{i\theta }}$ from (0.3) have the group properties that $R_1=I$ is the identity and

$$\begin{aligned} R_{e^{i(\theta +\tau )}}=R_{e^{i\theta }}R_{e^{i\tau }} \end{aligned}$$

for $e^{i\theta },e^{i\tau }\in {\mathbb {T}}$.

Recall the notion of homogeneity with respect to rotations made precise in the paragraph before Theorem 2.4. Observe that a function u in ${\mathbb {D}}$ is homogeneous of order $m\in {\mathbb {Z}}$ with respect to rotations if and only if $R_{e^{i\theta }}u=e^{im\theta }u$ for $e^{i\theta }\in {\mathbb {T}}$.

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 $n\in {\mathbb {N}}\cup \{\infty \}$. Then $u_m\in C^n({\mathbb {D}})$ and $u_m$ is homogeneous of order m with respect to rotations.

Proof

Differentiations under the integral in (0.4) show that $u_m\in C^n({\mathbb {D}})$. From formula (0.4) we have that

$$\begin{aligned} u_m=\frac{1}{2\pi }\int _{\mathbb {T}}e^{-im\tau }R_{e^{i\tau }}u\, d\tau \end{aligned}$$

(4.1)

in a vector-valued sense. Let $e^{i\theta }\in {\mathbb {T}}$. From the group property of rotation operators we have that

$$\begin{aligned} R_{e^{i\theta }}u_m= \frac{1}{2\pi }\int _{\mathbb {T}}e^{-im\tau }R_{e^{i\theta }}R_{e^{i\tau }}u\, d\tau =\frac{1}{2\pi }\int _{\mathbb {T}}e^{-im\tau } R_{e^{i(\theta +\tau )}}u\, d\tau \end{aligned}$$

in a vector-valued sense. A change of variables now gives that

$$\begin{aligned} R_{e^{i\theta }}u_m=\frac{e^{im\theta }}{2\pi } \int _{\mathbb {T}}e^{-im\tau } R_{e^{i\tau }}u\, d\tau =e^{im\theta }u_m. \end{aligned}$$

Since $e^{i\theta }\in {\mathbb {T}}$ is arbitrary, this yields that $u_m$ is homogeneous of order m with respect to rotations. $\square$

Below we shall make use of the commutativity relation

$$\begin{aligned} L_{p,q}R_{e^{i\theta }}u=R_{e^{i\theta }}L_{p,q}u,\quad u\in C^2({\mathbb {D}}), \end{aligned}$$

(4.2)

for $e^{i\theta }\in {\mathbb {T}}$. In order to prove (4.2) it suffices to check that the differential operators $z\partial$ and ${\bar{z}}\bar{\partial }$ commute with the rotations $R_{e^{i\theta }}$ which is evident.

We now return to (p, q)-harmonic functions.

Proposition 4.2

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

Proof

From Proposition 4.1 we have that $u_m\in C^2({\mathbb {D}})$ since $u\in C^2({\mathbb {D}})$. Recall formula (4.1). Applying the operator $L_{p,q}$ we have that

$$\begin{aligned} L_{p,q}u_m=\frac{1}{2\pi }\int _{\mathbb {T}}e^{-im\tau }L_{p,q}R_{e^{i\tau }}u\, d\tau \end{aligned}$$

in a vector-valued sense. We now use the commutativity relation (4.2) to conclude that

$$\begin{aligned} L_{p,q}u_m=\frac{1}{2\pi }\int _{\mathbb {T}}e^{-im\tau } R_{e^{i\tau }}L_{p,q}u\, d\tau =0, \end{aligned}$$

where the last equality is evident since u is (p, q)-harmonic. $\square$

Let us next calculate the homogeneous parts of a (p, q)-harmonic function.

Theorem 4.3

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

$$\begin{aligned} u_m(z)= c_mF(-p,m-q;m+1;|z|^2)z^m,\quad z\in {\mathbb {D}}, \end{aligned}$$

(4.3)

for some $c_m\in {\mathbb {C}}$, where F is the hypergeometric function (0.5).

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)-harmonic. The result now follows from Theorem 2.4. $\square$

Corollary 4.4

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

$$\begin{aligned} u_m(z)= c_mF(-q,|m|-p;|m|+1;|z|^2) {\bar{z}}^{|m|},\quad z\in {\mathbb {D}}, \end{aligned}$$

(4.4)

for some $c_m\in {\mathbb {C}}$, where F is the hypergeometric function (0.5).

Proof

The complex conjugate ${\bar{u}}_m$ is the $-m=|m|$-th homogeneous part of the function ${\bar{u}}$. By hermitian symmetry, the function ${\bar{u}}$ is $({\bar{q}},{\bar{p}})$-harmonic. From Theorem 4.3 we have that

$$\begin{aligned} \overline{u_m(z)}= a_m F(-{\bar{q}},|m|-{\bar{p}};|m|+1;|z|^2) z^{|m|},\quad z\in {\mathbb {D}}, \end{aligned}$$

for some $a_m\in {\mathbb {C}}$. A complex conjugation now yields (4.4) with $c_m={\bar{a}}_m$. $\square$

As a by-product from Theorem 4.3 and Corollary 4.4 we have that $u_m\in C^\infty ({\mathbb {D}})$ for $m\in {\mathbb {Z}}$ if u is (p, q)-harmonic.

Recall formula (0.4) for the homogeneous parts of a function u in ${\mathbb {D}}$. From the triangle inequality we have that

$$\begin{aligned} \sup _{|z|=r}|u_m(z)|\le \sup _{|z|=r}|u(z)|\end{aligned}$$

(4.5)

for $0<r<1$ and $m\in {\mathbb {Z}}$. Notice that (4.5) generalizes a well-known bound for Fourier coefficients.

We next estimate the constants $c_m$ appearing in Theorem 4.3 or Corollary 4.4.

Lemma 4.5

Let $p,q\in {\mathbb {C}}$. Let u be a (p, q)-harmonic function and consider its m-th homogeneous part $u_m$ for $m\in {\mathbb {Z}}$. Let $c_m$ be as in (4.4) or (4.4) depending on whether $m\in {\mathbb {N}}$ or $m\in {\mathbb {Z}}^-$. Then $\limsup _{|m|\rightarrow \infty } |c_m|^{1/|m|}\le 1$.

Proof

For simplicity we consider the case $m\in {\mathbb {N}}$. The case $m\in {\mathbb {Z}}^-$ is analogous or follows by hermitian symmetry. Let $0<r<1$. From (4.4) and (4.5) we have that

$$\begin{aligned} |c_m||F(-p,m-q;m+1;r^2) |r^m \le \max _{|z|=r}|u(z)|\end{aligned}$$

(4.6)

for $m\in {\mathbb {N}}$. From Theorem 2.6 we have that

$$\begin{aligned} \lim _{m\rightarrow \infty } F(-p,m-q;m+1;r^2)=(1-r^2)^p\ne 0. \end{aligned}$$

Therefore, a passage to the limit in (4.6) gives that

$$\begin{aligned} \limsup _{m\rightarrow \infty } |c_m|^{1/m}\le 1/r. \end{aligned}$$

Since $0<r<1$ is arbitrary we conclude that $\limsup _{m\rightarrow \infty } |c_m|^{1/m}\le 1$. This yields the conclusion of the lemma. $\square$

The following theorem is our main result about the asymptotic behavior of the homogeneous parts of a (p, q)-harmonic function.

Theorem 4.6

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

$$\begin{aligned} \limsup _{|m|\rightarrow \infty }\Vert u_m\Vert _{j,k;K}^{1/|m|}<1 \end{aligned}$$

for all $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact.

Proof

From Theorem 4.3 and Corollary 4.4 we have that $u_m\in C^\infty ({\mathbb {D}})$ for $m\in {\mathbb {Z}}$.

We consider first the case $m\in {\mathbb {N}}$. From Theorem 4.3 we have (4.4) for $m\in {\mathbb {N}}$, where $c_m\in {\mathbb {C}}$. Notice that the function $u_m$ in (4.4) is constructed from the function

$$\begin{aligned} f_m(x)=c_mF(-p,m-q;m+1;x),\quad 0\le x<1, \end{aligned}$$

as in Theorem 3.2. From Proposition 3.3 and Lemma 4.5 we have that the assumption of Theorem 3.2 is satisfied. An application of Theorem 3.2 yields the conclusion that $\limsup _{m\rightarrow \infty }\Vert u_m\Vert _{j,k;K}^{1/m}<1$ for $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact.

We next consider the case $m\in {\mathbb {Z}}^-$. The complex conjugate ${\bar{u}}_{m}$ of $u_m$ is the $(-m)$-th homogeneous part of the $({\bar{q}},{\bar{p}})$-harmonic function ${\bar{u}}$. From the result of the previous paragraph we thus have that

$$\begin{aligned} \limsup _{m\rightarrow -\infty }\Vert u_m\Vert _{j,k;K}^{1/|m|} =\limsup _{m\rightarrow \infty }\Vert ({\bar{u}})_m\Vert _{k,j;K}^{1/m}<1 \end{aligned}$$

for $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact. This completes the proof of the theorem. $\square$

We point out that Theorem 2.6 forms an integral part in the proof of Theorem 4.6.

Recall the notion of absolute convergence in $C^\infty ({\mathbb {D}})$, see the paragraph just before Theorem 3.2.

Corollary 4.7

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

Proof

Theorem 4.6 allows us to apply the root test to conclude that

$$\begin{aligned} \sum _{m=-\infty }^\infty \Vert u_m\Vert _{j,k;K}<+\infty \end{aligned}$$

for $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact. $\square$

The following lemma is well-known but included for the sake of completeness.

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

$$\begin{aligned} u=\lim _{N\rightarrow +\infty }\sum _{m=-N}^N \Big (1-\frac{|m|}{N+1}\Big )u_m \end{aligned}$$

in $C^n({\mathbb {D}})$.

Proof

It is straightforward to check that

$$\begin{aligned} \sum _{m=-N}^N \Big (1-\frac{|m|}{N+1}\Big )u_m =\frac{1}{2\pi }\int _{\mathbb {T}}K_N(e^{i\theta }) R_{e^{i\theta }}u\, d\theta \end{aligned}$$

in a vector-valued sense, where

$$\begin{aligned} K_N(e^{i\theta })= \sum _{m=-N}^N\Big (1-\frac{|m|}{N+1}\Big )e^{im\theta }, \quad e^{i\theta }\in {\mathbb {T}}, \end{aligned}$$

is the Fejér kernel. It is well-known that the $K_N$’s are non-negative and form an approximate identity as $N\rightarrow \infty$. Since $u\in C^n({\mathbb {D}})$, it is well-known that the function

$$\begin{aligned} {\mathbb {T}}\ni e^{i\theta }\mapsto R_{e^{i\theta }}u\in C^n({\mathbb {D}}) \end{aligned}$$

is continuous from ${\mathbb {T}}$ into $C^n({\mathbb {D}})$. The proof is now completed by a standard argument. We refer to Katznelson [15, Section I.2] for details. $\square$

We think of Lemma 4.8 as a version of Fejér’s theorem adapted to our context.

We next deduce that (p, q)-harmonic functions belong to the space $C^\infty ({\mathbb {D}})$.

Corollary 4.9

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

Proof

From Lemma 4.8 we have that

$$\begin{aligned} u=\lim _{N\rightarrow +\infty }\sum _{m=-N}^N \Big (1-\frac{|m|}{N+1} \Big ) u_m \end{aligned}$$

(4.7)

in $C^2({\mathbb {D}})$ since u has such regularity. From Corollary 4.7 we know that the function series $\sum _{m=-\infty }^\infty u_m$ is absolutely convergent in $C^\infty ({\mathbb {D}})$. We can thus omit the convergence factors in (4.7) and deduce that $u=\sum _{m=-\infty }^\infty u_m$ in $C^\infty ({\mathbb {D}})$. By completeness we have that $u\in C^\infty ({\mathbb {D}})$. $\square$

6 A series expansion of harmonic functions

The analysis from Sect. 4 leads to a natural function series description of (p, q)-harmonic functions.

Theorem 5.1

Let $p,q\in {\mathbb {C}}$. Then u is a (p, q)-harmonic function if and only if it has the form

$$\begin{aligned} u(z)&=\sum _{m=0}^\infty c_mF(-p,m-q;m+1;|z|^2)z^m\nonumber \\&\quad \ +\sum _{m=1}^\infty c_{-m}F(-q,m-p;m+1;|z|^2){\bar{z}}^m, \quad z\in {\mathbb {D}}, \end{aligned}$$

(5.1)

for some sequence $\{c_m\}_{m=-\infty }^\infty$ of complex numbers such that

$$\begin{aligned} \limsup _{|m|\rightarrow \infty } |c_m|^{1/|m|}\le 1, \end{aligned}$$

(5.2)

where F is the hypergeometric function (0.5). Moreover, the sums in (5.1) are absolutely convergent in the space $C^\infty ({\mathbb {D}})$ when (5.2) holds.

Proof

Consider first a (formal) function series of the form (5.1) with coefficient sequence $\{c_m\}_{m=-\infty }^\infty$ satisfying (5.2). From Theorem 3.2 and Proposition 3.3 we have that

$$\begin{aligned} \limsup _{|m|\rightarrow \infty } \Vert u_m\Vert ^{1/|m|}_{j,k;K}<1 \end{aligned}$$

for $j,k\in {\mathbb {N}}$ and $K\subset {\mathbb {D}}$ compact, where

$$\begin{aligned} u_m(z)= c_m F(-p,m-q;m+1;|z|^2) z^{m},\quad z\in {\mathbb {D}}, \end{aligned}$$

for $m\in {\mathbb {N}}$ and

$$\begin{aligned} u_m(z)= c_m F(-q,|m|-p;|m|+1;|z|^2) {\bar{z}}^{|m|},\quad z\in {\mathbb {D}}, \end{aligned}$$

for $m\in {\mathbb {Z}}^-$. The root test now applies to show that the series $\sum _{m=-\infty }^\infty u_m$ in (5.1) is absolutely convergent in $C^\infty ({\mathbb {D}})$. This defines a function $u=\sum _{m=-\infty }^\infty u_m$ by (5.1). From Proposition 2.2 and Corollary 2.3 we have that each term $u_m$ in (5.1) is (p, q)-harmonic. Since the series expansion (5.1) converges in the space $C^\infty ({\mathbb {D}})$, we have that $u\in C^\infty ({\mathbb {D}})$. Applying the operator $L_{p,q}$ we have $L_{p,q}u=\sum _{m=-\infty }^\infty L_{p,q}u_m =0$ in $C^\infty ({\mathbb {D}})$, which shows that u is (p, q)-harmonic.

Consider next a (p, q)-harmonic function u. We proceed to derive (5.1). Denote by $u_m$ the m-th homogeneous part of u for $m\in {\mathbb {Z}}$. From Theorem 4.3 we have (4.4) for $m\in {\mathbb {N}}$ and from Corollary 4.4 we have (4.4) for $m\in {\mathbb {Z}}^-$, where $c_m\in {\mathbb {C}}$ for $m\in {\mathbb {Z}}$. From Corollary 4.9 we have (5.1) in the form that $u= \sum _{m=-\infty }^\infty u_m$ in $C^\infty ({\mathbb {D}})$. By Corollary 4.7 we have that the function series $\sum _{m=-\infty }^\infty u_m$ in (5.1) is absolutely convergent in $C^\infty ({\mathbb {D}})$. From Lemma 4.5 we have that (5.2) holds. $\square$

We emphasize that the function series (5.1) has an interpretation of homogeneous expansion of the (p, q)-harmonic function u.

We mention that Theorem 5.1 improves on a result by Ahern et al. [1, Theorem 2.1] when specialized to the present setting. Theorem 5.1 contains also some earlier results of Olofsson and Wittsten. Theorem 5.1 with $(p,q)=(0,\alpha )$, $\alpha \in {\mathbb {R}}$, yields [21, Theorem 1.2]. Theorem 5.1 with $(p,q)=(\alpha /2,\alpha /2)$, $\alpha \in {\mathbb {R}}$, yields [18, Theorem 2.2].

We shall next turn to formulas for the coefficients $c_m$ appearing in (5.1).

Proposition 5.2

Let $p,q\in {\mathbb {C}}$. Let u be a (p, q)-harmonic function of the form (5.1) for some sequence $\{c_m\}_{m=-\infty }^\infty$ of complex numbers satisfying (5.2). Then

$$\begin{aligned} c_m=\lim _{r\rightarrow 0} \frac{1}{2\pi r^{|m|}} \int _{\mathbb {T}}u(re^{i\theta })e^{-im\theta }\, d\theta \end{aligned}$$

for $m\in {\mathbb {Z}}$.

Proof

We consider the case $m\in {\mathbb {N}}$. Using the expansion (5.1) it is straightforward to check that

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}}u(re^{i\theta })e^{-im\theta }\, d\theta =c_mF(-p,m-q;m+1;r^2)r^m \end{aligned}$$

for $0<r<1$. We next divide by $r^m$ and pass to the limit to see that

$$\begin{aligned} \lim _{r\rightarrow 0}\frac{1}{2\pi r^m} \int _{\mathbb {T}}u(re^{i\theta })e^{-im\theta }\, d\theta =c_mF(-p,m-q;m+1;0)=c_m. \end{aligned}$$

This yields the conclusion of the proposition when $m\in {\mathbb {N}}$.

The remaining case $m\in {\mathbb {Z}}{\setminus }{\mathbb {N}}$ is proved similarly or follows by hermitian symmetry. We omit the details. $\square$

The integral quantity

$$\begin{aligned} I_m(r)=\frac{1}{2\pi r^{|m|}} \int _{\mathbb {T}}u(re^{i\theta })e^{-im\theta }\, d\theta \end{aligned}$$

(5.3)

in Proposition 5.2 is naturally thought of as a line integral. In fact,

$$\begin{aligned} I_m(r)=\frac{1}{2\pi i}\oint _{|z|=r}\frac{u(z)}{z^{m+1}}\, dz \quad \text {and}\quad I_{-m}(r)=-\frac{1}{2\pi i}\oint _{|z|=r}\frac{u(z)}{{\bar{z}}^{m+1}}\, d{\bar{z}} \end{aligned}$$

for $m\in {\mathbb {N}}$ and $0<r<1$, where the circle of integration is traversed once in positive direction. These formulas are straightforward to check.

Theorem 5.3

Let $p,q\in {\mathbb {C}}$. Let u be a (p, q)-harmonic function of the form (5.1) for some sequence $\{c_m\}_{m=-\infty }^\infty$ of complex numbers satisfying (5.2). Then

$$\begin{aligned} c_m=\partial ^m u(0)/m! \quad \text {and}\quad c_{-m}=\bar{\partial }^m u(0)/m! \end{aligned}$$

for $m\in {\mathbb {N}}$.

Proof

We shall calculate the integral limit in Proposition 5.2. Let $I_m(r)$ be as in (5.3). Recall from Corollary 4.9 that $u\in C^\infty ({\mathbb {D}})$. Consider the Taylor expansion of u at the origin of degree $m\ge 0$:

$$\begin{aligned} u(z)=\sum _{\begin{array}{c} j,k\ge 0 \\ j+k\le m \end{array}} \frac{1}{j! k!} \partial ^j\bar{\partial }^k u(0)z^j{\bar{z}}^k +{\mathcal {O}}(|z|^{m+1}) \end{aligned}$$

as $z\rightarrow 0$. We have that

$$\begin{aligned} I_m(r)&= \frac{1}{2\pi r^m} \int _{\mathbb {T}}\Big ( \sum _{\begin{array}{c} j,k\ge 0 \\ j+k\le m \end{array}} \frac{1}{j! k!} \partial ^j\bar{\partial }^k u(0)r^{j+k}e^{i(j-k)\theta } \Big ) e^{-im\theta }\, d\theta +{\mathcal {O}}(r)\\&=\partial ^m u(0)/m! +{\mathcal {O}}(r) \end{aligned}$$

as $r\rightarrow 0$, where the last equality follows by cancellation. A passage to the limit as $r\rightarrow 0$ now yields that $c_m= \partial ^m u(0)/m!$.

We now turn to the formula for $c_{-m}$. A similar analysis as in the previous paragraph shows that

$$\begin{aligned} I_{-m}(r)= \bar{\partial }^m u(0)/m! +{\mathcal {O}}(r) \end{aligned}$$

as $r\rightarrow 0$. A passage to the limit as $r\rightarrow 0$ yields that $c_{-m}= \bar{\partial }^m u(0)/m!$. This completes the proof of the theorem. $\square$

The coefficient formulas in Theorem 5.3 leads to an addendum to Theorem 5.1.

Theorem 5.4

Let $p,q\in {\mathbb {C}}$. Let u be a (p, q)-harmonic function. Then

$$\begin{aligned} u(z)&=\sum _{m=0}^\infty \frac{\partial ^m u(0)}{m!} F(-p,m-q;m+1;|z|^2)z^m\\&\quad \ +\sum _{m=1}^\infty \frac{\bar{\partial }^m u(0)}{m!} F(-q,m-p;m+1;|z|^2){\bar{z}}^m \end{aligned}$$

for $z\in {\mathbb {D}}$, where F is the hypergeometric function (0.5).

Proof

Recall the series expansion (5.1) and (5.2) established in Theorem 5.1. By Theorem 5.3 we have that $c_m= \partial ^m u(0)/m!$ and $c_{-m}= \bar{\partial }^m u(0)/m!$ for $m\in {\mathbb {N}}$. This yields the conclusion of the theorem. $\square$

As a by-product of Theorem 5.4 we have a uniqueness result of classical type.

Corollary 5.5

Let $p,q\in {\mathbb {C}}$. Let u be a (p, q)-harmonic function. Assume that $u(0)=0$ and

$$\begin{aligned} \partial ^m u(0)=\bar{\partial }^mu(0)=0 \end{aligned}$$

for $m=1,2,\dots$. Then $u(z)=0$ for all $z\in {\mathbb {D}}$.

Proof

The result is evident by Theorem 5.4. $\square$

7 Further properties of the function $u_{p,q}$

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. In this section we derive the series expansion of $u_{p,q}$ (see Theorem 6.3). Of interest are also properties of integral means of $u_{p,q}$ (see Theorems 6.4 and 6.6).

Lemma 6.1

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Let $m\in {\mathbb {N}}$. Then

$$\begin{aligned} \partial ^m u_{p,q}(z)=\frac{(p+1)_m}{(1-z)^m}u_{p,q}(z)+{\bar{z}} g_m(z), \quad z\in {\mathbb {D}}, \end{aligned}$$

(6.1)

where $g_m\in C^\infty ({\mathbb {D}})$.

Proof

We prove the lemma by induction on $m\in {\mathbb {N}}$. Clearly (6.1) holds for $m=0$ with $g_0=0$. By Lemma 1.1 we have that

$$\begin{aligned} \partial u_{p,q}(z)=\frac{p+1}{1-z}u_{p,q}(z)+{\bar{z}} g_1(z), \quad z\in {\mathbb {D}}, \end{aligned}$$

(6.2)

with

$$\begin{aligned} g_1(z)=-\frac{p+q+1}{1-|z|^2} u_{p,q}(z), \quad z\in {\mathbb {D}}. \end{aligned}$$

This proves (6.1) for $m=1$.

Assume next that (6.1) holds for some $m=n\ge 1$, that is,

$$\begin{aligned} \partial ^n u_{p,q}(z)=\frac{(p+1)_n}{(1-z)^n}u_{p,q}(z)+{\bar{z}} g_n(z), \quad z\in {\mathbb {D}}, \end{aligned}$$

where $g_n\in C^\infty ({\mathbb {D}})$. Differentiating we have that

$$\begin{aligned} \partial ^{n+1} u_{p,q}(z)=(p+1)_n\frac{n}{(1-z)^{n+1}}u_{p,q}(z) +\frac{(p+1)_n}{(1-z)^n}\partial u_{p,q}(z)+ {\bar{z}} \partial g_n(z) \end{aligned}$$

for $z\in {\mathbb {D}}$. We next use (6.2) to see that

$$\begin{aligned} \partial ^{n+1} u_{p,q}(z)&=(p+1)_n\frac{n}{(1-z)^{n+1}}u_{p,q}(z) +\frac{(p+1)_n}{(1-z)^n}\Big (\frac{p+1}{1-z}u_{p,q}(z)+{\bar{z}} g_1(z)\Big ) \\&\quad \ + {\bar{z}} \partial g_n(z)=\frac{(p+1)_{n+1}}{(1-z)^{n+1}}u_{p,q}(z) + {\bar{z}} g_{n+1}(z) \end{aligned}$$

for $z\in {\mathbb {D}}$, where

$$\begin{aligned} g_{n+1}(z)= \frac{(p+1)_{n}}{(1-z)^{n}} g_1(z)+\partial g_n(z), \quad z\in {\mathbb {D}}. \end{aligned}$$

This proves (6.1) for $m=n+1$. The conclusion of the lemma now follows by the principle of induction. $\square$

Notice that formula (6.1) interprets naturally as a congruence in $C^\infty ({\mathbb {D}})$ modulo ${\bar{z}} C^\infty ({\mathbb {D}})$.

Theorem 6.2

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} \partial ^m u_{p,q}(0)=(p+1)_m \quad \text {and}\quad \bar{\partial }^m u_{p,q}(0)=(q+1)_m \end{aligned}$$

for $m\in {\mathbb {N}}$.

Proof

We shall use Lemma 6.1. A point evaluation at the origin in (6.1) shows that $\partial ^m u_{p,q}(0)=(p+1)_m$ for $m\in {\mathbb {N}}$. Moreover, by hermitian symmetry ${\bar{u}}_{p,q}=u_{{\bar{q}},{\bar{p}}}$ we have that

$$\begin{aligned} \bar{\partial }^m u_{p,q}(0)=\bar{\partial }^m {\bar{u}}_{{\bar{q}},{\bar{p}}}(0) =\overline{ \partial ^m u_{{\bar{q}},{\bar{p}}}(0)}=\overline{ ({\bar{q}}+1)_m} = (q+1)_m \end{aligned}$$

for $m\in {\mathbb {N}}$. $\square$

We can now derive the series expansion of the function $u_{p,q}$.

Theorem 6.3

Let $p,q\in {\mathbb {C}}$. Then

$$\begin{aligned} \frac{(1-|z|^2)^{p+q+1}}{(1-z)^{p+1} (1-{\bar{z}})^{q+1}}&=\sum _{m=0}^\infty \frac{(p+1)_m}{m!}F(-p,m-q;m+1;|z|^2)z^m\\&\quad \ +\sum _{m=1}^\infty \frac{(q+1)_m}{m!}F(-q,m-p;m+1;|z|^2){\bar{z}}^m \end{aligned}$$

for $z\in {\mathbb {D}}$, where F is the hypergeometric function (0.5).

Proof

Let $u_{p,q}$ be as in (0.2). From Theorem 1.4 we know that the function $u_{p,q}$ is (p, q)-harmonic. By Theorem 5.4 it has the function series expansion

$$\begin{aligned} u_{p,q}(z)&=\sum _{m=0}^\infty \frac{\partial ^m u_{p,q}(0)}{m!} F(-p,m-q;m+1;|z|^2)z^m\\&\quad \ +\sum _{m=1}^\infty \frac{\bar{\partial }^m u_{p,q}(0)}{m!} F(-q,m-p;m+1;|z|^2){\bar{z}}^m \end{aligned}$$

for $z\in {\mathbb {D}}$. The result now follows from Theorem 6.2. $\square$

We point out that Theorem 6.3 generalizes a well-known partial fraction decomposition formula for the classical Poisson kernel for ${\mathbb {D}}$, see formula (1.1). Suitably specialized Theorem 6.3 yields [21, Theorem 2.5] and [18, Theorem 3.2].

Notice that the series expansion in Theorem 6.3 is absolutely convergent in $C^\infty ({\mathbb {D}})$.

We denote by ${\text {Re}}(z)$ and ${\text {Im}}(z)$ the real and imaginary parts of a complex number z, respectively. The standard Gamma function is defined by

$$\begin{aligned} \Gamma (z)=\int _0^\infty t^{z-1}e^{-t}\, dt \end{aligned}$$

for ${\text {Re}}(z)>0$. It is well-known that $\Gamma$ continues to a meromorphic function in ${\mathbb {C}}$ with simple poles at the points $z=0,-1,-2,\dots$.

We next turn to $L^1$ means of $u_{p,q}$.

Theorem 6.4

Let $u_{p,q}$ be as in (0.2) for some $p,q\in {\mathbb {C}}$ such that ${\text {Re}}(p)+{\text {Re}}(q)>-1$. Then

$$\begin{aligned} \frac{1}{2\pi } \int _{{\mathbb {T}}}|u_{p,q}(re^{i\theta })|\, d\theta \le e^{\frac{\pi }{2}|{\text {Im}}(p)-{\text {Im}}(q)|} \frac{\Gamma ({\text {Re}}(p)+{\text {Re}}(q)+1)}{\Gamma (\frac{{\text {Re}}(p)+{\text {Re}}(q)}{2}+1)^2} \end{aligned}$$

for $0\le r<1$, where $\Gamma$ is the Gamma function.

Proof

From definition of powers we have that

$$\begin{aligned} u_{p,q}(z)=e^{(p+q+1)\log (1-|z|^2)} e^{-(p+q+2)\log |1-z|} e^{-i(p-q)\arg (1-z)} \end{aligned}$$

for $z\in {\mathbb {D}}$. Notice that the complex number $1-z$ lies in the open right half-plane when $z\in {\mathbb {D}}$. Passing to absolute values we have that

$$\begin{aligned} |u_{p,q}(z)|\le e^{\frac{\pi }{2}|{\text {Im}}(p)-{\text {Im}}(q)|} \frac{(1-|z|^2)^{{\text {Re}}(p)+{\text {Re}}(q)+1}}{|1-z|^{{\text {Re}}(p)+ {\text {Re}}(q)+2}} \end{aligned}$$

for $z\in {\mathbb {D}}$. We next apply [18, Theorem 3.1] with $\alpha ={\text {Re}}(p)+{\text {Re}}(q)$ to conclude that

$$\begin{aligned} \frac{1}{2\pi } \int _{{\mathbb {T}}}|u_{p,q}(re^{i\theta })|\, d\theta \le e^{\frac{\pi }{2}|{\text {Im}}(p)-{\text {Im}}(q)|} \frac{\Gamma (\alpha +1)}{\Gamma (\alpha /2+1)^2} \end{aligned}$$

for $0\le r<1$. This yields the conclusion of the theorem. $\square$

Remark 6.5

A similar analysis as in the proof of Theorem 6.4 shows that

$$\begin{aligned} \lim _{r\rightarrow 1} \frac{1}{2\pi } \int _{{\mathbb {T}}}|u_{p,q}(re^{i\theta })|\, d\theta =+\infty \end{aligned}$$

if ${\text {Re}}(p)+{\text {Re}}(q)\le -1$. We omit the details.

A classical result known as Gauss summation formula says that

$$\begin{aligned} \lim _{x\rightarrow 1}F(a,b;c;x)=\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)} \end{aligned}$$

(6.3)

when $a,b\in {\mathbb {C}}$ and $c\in {\mathbb {C}}{\setminus }\{0,-1,-2,\dots \}$ are such that ${\text {Re}}(c)>{\text {Re}}(a)+{\text {Re}}(b)$ (see [3, Theorem 2.2.2]).

Theorem 6.6

Let $p,q\in {\mathbb {C}}$ be such that ${\text {Re}}(p)+{\text {Re}}(q)>-1$. Then

$$\begin{aligned} \lim _{r\rightarrow 1}\frac{1}{2\pi } \int _{{\mathbb {T}}} u_{p,q}(re^{i\theta })\, d\theta = \frac{\Gamma (p+q+1)}{\Gamma (p+1)\Gamma (q+1)}, \end{aligned}$$

where $\Gamma$ is the Gamma function.

Proof

Recall the series expansion of the function $u_{p,q}$ established in Theorem 6.3. By cancellation of terms we have that

$$\begin{aligned} \frac{1}{2\pi } \int _{{\mathbb {T}}} u_{p,q}(re^{i\theta })\, d\theta = F(-p,-q;1;r^2) \end{aligned}$$

for $0\le r<1$. Since ${\text {Re}}(p)+{\text {Re}}(q)>-1$, we can apply Gauss summation formula (6.3) to calculate the limit of the above quantity as $r\rightarrow 1$. This yields the conclusion of the theorem. $\square$

8 The Dirichlet problem: concluding remarks

In a restricted range of parameters, the class of (p, q)-harmonic functions can be analyzed in terms of their boundary values. We shall discuss in this section some rudiments of such theory.

Let $p,q\in {\mathbb {C}}{\setminus }{\mathbb {Z}}^-$ be such that ${\text {Re}}(p)+{\text {Re}}(q)>-1$. The (p, q)-harmonic Poisson kernel is defined by

$$\begin{aligned} K_{p,q}(z)= c_{p,q}\frac{(1-|z|^2)^{p+q+1}}{(1-z)^{p+1} (1-{\bar{z}})^{q+1}}, \quad z\in {\mathbb {D}}, \end{aligned}$$

where

$$\begin{aligned} c_{p,q}= \frac{\Gamma (p+1)\Gamma (q+1)}{\Gamma (p+q+1)}. \end{aligned}$$

Notice that the constant $c_{p,q}$ is a non-zero complex number in this range of parameters. By Theorem 1.4 the function $K_{p,q}$ is (p, q)-harmonic. Theorems 6.4 and 6.6 ensure that the function $K_{p,q}$ satisfies some standard properties for approximate identities.

The (p, q)-harmonic Poisson integral is defined by

$$\begin{aligned} K_{p,q}[f](z)=\frac{1}{2\pi } \int _{{\mathbb {T}}} K_{p,q}(ze^{-i\theta })f(e^{i\theta })\, d\theta , \quad z\in {\mathbb {D}}, \end{aligned}$$

for integrable functions $f\in L^1({\mathbb {T}})$ on ${\mathbb {T}}$.

Let $C({\mathbb {T}})$ be the space of continuous functions on ${\mathbb {T}}$ and fix $\varphi \in C({\mathbb {T}})$. By the (p, q)-harmonic Dirichlet problem for $\varphi$ we understand the problem of finding a (p, q)-harmonic function u such that $\lim _{r\rightarrow 1}u_r=\varphi$ in $C({\mathbb {T}})$, where

$$\begin{aligned} u_r(e^{i\theta })=u(re^{i\theta }),\quad e^{i\theta }\in {\mathbb {T}}, \end{aligned}$$

for $0<r<1$. Following usual practice, we formulate this latter Dirichlet problem as

$$\begin{aligned} {\left\{ \begin{array}{ll} L_{p,q} u =0\quad \text {in}\ {\mathbb {D}}, \\ \quad \ \ u =\varphi \quad \text {on}\ {\mathbb {T}}, \end{array}\right. } \end{aligned}$$

(7.1)

where $L_{p,q}$ is as in (0.1).

Theorem 7.1

Let $p,q\in {\mathbb {C}}{\setminus }{\mathbb {Z}}^-$ be such that ${\text {Re}}(p)+{\text {Re}}(q)>-1$. Let $\varphi \in C({\mathbb {T}})$. Then a function u in ${\mathbb {D}}$ satisfies (7.1) if and only if it has the form

$$\begin{aligned} u(z)=K_{p,q}[\varphi ](z),\quad z\in {\mathbb {D}}. \end{aligned}$$

The proof of Theorem 7.1 follows a standard scheme for such results and is therefore omitted, see [18, 21] for details.

The present paper suggests a finer study of (p, q)-harmonic functions. Earlier results of such type concern Poisson integral representations, pointwise boundary limits, Green functions and Lipschitz continuity of generalized harmonic functions, see [18,19,20,21]. We mention here also work of Ahern and collaborators [1, 2].