A series expansion for generalized harmonic functions

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.


Introduction
Let D be the open unit disc in the complex plane C and denote by ∂ z = ∂/∂z and ∂ z = ∂/∂z the usual complex partial derivatives. This work is concerned with second order partial differential operators of the form L p,q = (1 − |z| 2 )∂ z∂z + pz∂ z + qz∂ z − pq, z ∈ D, (0.1) where p, q ∈ C are complex parameters. Of particular interest are solutions of the associated homogeneous equation We say that a function u is ( p, q)-harmonic if u is twice continuously differentiable in D (in symbols u ∈ C 2 (D)) and L p,q u = 0 in D, where L p,q is as in (0. that a function u is ( p, q)-harmonic if and only if its complex conjugateū is (q,p)harmonic. Observe also that a (0, 0)-harmonic function is a harmonic function in D in the usual sense. An interesting example of a ( p, q)-harmonic function is the function (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 thatū p,q = uq ,p for p, q ∈ C.
Recall that elements of the unit circle T = ∂D act on the unit disc D as rotations. On a function level we consider rotation operators R e iθ u(z) = u(e iθ z), z ∈ D, (0. 3) for e iθ ∈ T acting on functions u in D. A basic observation concerning the differential operator L p,q is the commutativity relation L p,q R e iθ u = R e iθ L p,q u, u ∈ C 2 (D), for e iθ ∈ 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 Z be the set of integers. For a suitably smooth function u in D we define its m-th homogeneous part by the formula u m (z) = 1 2π T e −imθ u(e iθ z) dθ, z ∈ D, (0. 4) for m ∈ Z. Notice that the m-th homogeneous part u m of u is the m-th Fourier coefficient of the vector-valued function where R e iθ is as in (0.3). We set C ∞ (D) = ∞ n=0 C n (D), where C n (D) is the space of n-times continuously differentiable functions in D for n ∈ N = {0, 1, 2, . . . }. We topologize these spaces in the usual way using the semi-norms u j,k;K = max z∈K |∂ j∂ k u(z)|, where j, k ∈ N are non-negative integers and K ⊂ D is a compact subset of D.
Let us recall the classical hypergeometric function defined by for n = 1, 2, . . . 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 for some c m ∈ C when m ∈ 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 ∈ Z − = Z\N is a negative integer (see Corollary 4.4). Notice that 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 lim sup |m|→∞ u m 1/|m| j,k;K < 1 for all j, k ∈ N and K ⊂ D compact (see Theorem 4.6). This result enables us to use the classical root test to establish absolute convergence of the function series . A further analysis leads to a function series characterization of ( p, q)-harmonic functions. A function u in D is ( p, q)-harmonic if and only if it has the form 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 (see Theorem 6.3). This latter series expansion generalizes a well-known partial fraction decomposition formula for the classical Poisson kernel for 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: 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 ∈ C\Z − are such that Re( p)+ Re(q) > −1. In the final section of the paper we comment briefly on the connection to such theory.
The authors thank the referee for a careful reading of the manuscript.

The function u p,q is (p, q)-harmonic
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 ∈ C. Then L p,q u p,q = 0 in D, where L p,q is as in (0.1).
Proof Recall Lemma 1.2. Notice that the differential operator z∂ satisfies the product rule for differentiation. It is straightforward to check that whenever the formula makes sense. Differentiating using the product rule we have that for z ∈ D. From Lemma 1.1 we now have that for z ∈ D. Expanding the above product we see that for z ∈ D. Multiplying by a factor (1 − |z| 2 )/|z| 2 we see that for z ∈ D. Notice an appearance of the classical Poisson kernel in the rightmost term above. Using the partial fraction formula (1.1) we have that for z ∈ D. A simplification of terms now leads to the formula for z ∈ 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 D.

A sequence of hypergeometric functions
Let us first consider a second order partial differential operator of the form where p, q, r ∈ 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 ∈ 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 D\{0} of the form We introduce also the ordinary differential operator where a, b, c ∈ C are complex parameters. Notice that the famous hypergeometric ordinary differential equation 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 ∈ C. Let u be a function of the form (2. 2) for some f ∈ C 2 (0, 1) and m ∈ Z. Then where c = m + 1 and for z ∈ D\{0}, and similarly that for z ∈ D\{0}. Another differentiation gives that for z ∈ D\{0}. A calculation using these formulas gives that for z ∈ D\{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.
Equations (2.4) say that a and b are the zeros of the quadratic polynomial Notice that 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.
where m ∈ N and F is the hypergeometric function (0.5). Then u m is a ( p, q)-harmonic function.
The result now follows by Theorem 2.1.

Corollary 2.3 Let p, q ∈ C. Consider the function
where m ∈ Z − and F is the hypergeometric function (0.5). Then u m is a ( p, q)harmonic function.
Proof We consider the complex conjugate By Proposition 2.2 we have thatū m is a (q,p)-harmonic function. From hermitian symmetry we conclude that u m is a ( p, q)-harmonic function.
Following earlier practice, a function u in D\{0} is said to be homogeneous of order m ∈ Z with respect to rotations if it has the property that for e iθ ∈ T. Notice that every function u of the form (2.2) is homogeneous of order m with respect to rotations.
for some c ∈ 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 ∈ C 2 (0, 1). By Theorem 2.1 we have that where H a,b;c is as in (2.3). Below we shall check that as x → 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; ·). 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 Since∂u is bounded near the origin we have that We have now checked (2.6).
is usually referred to as normal convergence in D.
Recall the terminology that a subset F of H (D) is called a normal family if every sequence of functions of F has a subsequence which converges in H (D). The limit function is not required to belong to F. We refer to Conway [11, Chapter VII] for background.
Recall also the binomial series: Lemma 2.5 Let p, q ∈ C and consider the functions for m ∈ N, where F is the hypergeometric function (0.5). Then F = { f m : m ∈ N} is a normal family of analytic functions in D.
Proof We shall prove that the functions in F are uniformly bounded on compact subsets of D. The conclusion of the lemma then follows by a classical result of Montel (see Conway [11,Theorem VII.2.9]).
We now estimate the f m 's with m > N . From (0. 5) we have that 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 for |z| < r and m > N , where the last equality follows by the binomial series (2.8). This proves that the functions in F are uniformly bounded on K .
The following limit theorem will be much useful.
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 F = { f m : m ∈ N} is a normal family. From (0.5) we have that for m, n ∈ N. Recall that the Pochhammer symbol (·) n is a monic polynomial of degree n. Passing to the limit we have that lim m→∞ f (n) A standard argument now yields that f m → f in H (D) as m → ∞. Assume to reach a contradiction that there exists a compact set K ⊂ D such that { f m } ∞ m=0 does not converge uniformly to f on K . Passing to a subsequence we can assume that for k = 1, 2, . . . . Since the set F is a normal family, we can after passage to another subsequence if necessary, assume that f m k → g in H (D) as k → ∞ for some g ∈ H (D). From the first paragraph of the proof we have g (n) (0) = (− p) n for n ∈ N, and a uniqueness argument gives that g = f in D. Thus f m k → f in H (D) as k → ∞, which contradicts (2.9).
We emphasize that Theorem 2.6 appears much natural in view of the binomial series (2.8).

A generalized power series
From the product rule for differentiation we have that for, say, f , g ∈ C n [0, 1).
for z ∈ D.
Proof Recall (2.2). Differentiating with respect toz we have that for z ∈ D. Another differentiation using the product rule for differentiation gives that for z ∈ D. This completes the proof of the lemma.
Let C ∞ (D) be the set of smooth complex-valued functions in the unit disc D. The space C ∞ (D) is topologized by means of the family of semi-norms where j, k ∈ N and K ⊂ D is compact. Recall that u m → u in C ∞ (D) as m → ∞ means that lim m→∞ u m − u j,k;K = 0 for all j, k ∈ N and K ⊂ D compact. A (formal) series ∞ m=0 u m of functions u m ∈ C ∞ (D) for m ∈ N is said to be absolutely convergent in C ∞ (D) if ∞ m=0 u m j,k;K < +∞ whenever j, k ∈ N and K ⊂ D is compact. By completeness of C ∞ (D) we have that every series absolutely convergent in C ∞ (D) is convergent in C ∞ (D).
Proof Fix j, k ∈ N and K ⊂ D compact. Set r = max z∈K |z| < 1. From Lemma 3.1 we have that for z ∈ D provided m ≥ j. We next apply the triangle inequality to see that for m ≥ 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 → ∞. In view of the assumption on the sequence { f m } ∞ m=0 we conclude from (3.2) that lim sup m→∞ u m 1/m j,k;K ≤ r .
Since 0 < r < 1, this yields the conclusion of the theorem.
From the conclusion of Theorem 3.2 we have that ∞ m=0 u m j,k;K < +∞ whenever j, k ∈ N and K ⊂ D is compact. Thus the series ∞ m=0 u m is absolutely convergent in C ∞ (D).
We can think of a function series Of particular concern to us are sequences { f m } ∞ m=0 of the form for m ∈ N, where p, q ∈ 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 satisfies the assumption of Theorem 3.2.

Analysis of homogeneous parts
Notice that the rotation operators T e iθ → R e iθ from (0.3) have the group properties that R 1 = I is the identity and
Recall the notion of homogeneity with respect to rotations made precise in the paragraph before Theorem 2.4. Observe that a function u in D is homogeneous of order m ∈ Z with respect to rotations if and only if R e iθ u = e imθ u for e iθ ∈ T.  Since e iθ ∈ T is arbitrary, this yields that u m is homogeneous of order m with respect to rotations.
Below we shall make use of the commutativity relation for e iθ ∈ T. In order to prove (4.2) it suffices to check that the differential operators z∂ andz∂ commute with the rotations R e iθ which is evident. We now return to ( p, q)-harmonic functions.

Proposition 4.2 Let p, q ∈ C. Let u be a ( p, q)-harmonic function and denote by u m its m-th homogeneous part for some m ∈ Z. Then u m is ( p, q)-harmonic.
Proof From Proposition 4.1 we have that u m ∈ C 2 (D) since u ∈ C 2 (D). Recall formula (4.1). Applying the operator L p,q we have that in a vector-valued sense. We now use the commutativity relation (4.2) to conclude that where the last equality is evident since u is ( p, q)-harmonic.
Let us next calculate the homogeneous parts of a ( p, q)-harmonic function. for some c m ∈ C, where F is the hypergeometric function (0.5).
for some c m ∈ C, where F is the hypergeometric function (0.5).
Proof The complex conjugateū m is the −m = |m|-th homogeneous part of the functionū. By hermitian symmetry, the functionū is (q,p)-harmonic. From Theorem 4.3 we have that for some a m ∈ C. A complex conjugation now yields (4.4) with c m =ā m .

As a by-product from Theorem 4.3 and Corollary 4.4 we have that
Recall formula (0.4) for the homogeneous parts of a function u in D. From the triangle inequality we have that Therefore, a passage to the limit in (4.6) gives that lim sup m→∞ |c m | 1/m ≤ 1/r .
Since 0 < r < 1 is arbitrary we conclude that lim sup m→∞ |c m | 1/m ≤ 1. This yields the conclusion of the lemma.
The following theorem is our main result about the asymptotic behavior of the homogeneous parts of a ( p, q)-harmonic function.  Proof Theorem 4.6 allows us to apply the root test to conclude that ∞ m=−∞ u m j,k;K < +∞ for j, k ∈ N and K ⊂ D compact.
The following lemma is well-known but included for the sake of completeness. Proof It is straightforward to check that in a vector-valued sense, where is the Fejér kernel. It is well-known that the K N 's are non-negative and form an approximate identity as N → ∞. Since u ∈ C n (D), it is well-known that the function T e iθ → R e iθ u ∈ C n (D) is continuous from T into C n (D). The proof is now completed by a standard argument. We refer to Katznelson [15, Section I.2] for details.
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 ∞ (D). in C 2 (D) since u has such regularity. From Corollary 4.7 we know that the function series ∞ m=−∞ u m is absolutely convergent in C ∞ (D). We can thus omit the convergence factors in (4.7) and deduce that u = ∞ m=−∞ u m in C ∞ (D). By completeness we have that u ∈ C ∞ (D).

A series expansion of harmonic functions
The analysis from Sect. 4 leads to a natural function series description of ( p, q)harmonic functions.
Proof We consider the case m ∈ N. Using the expansion (5.1) it is straightforward to check that for 0 < r < 1. We next divide by r m and pass to the limit to see that This yields the conclusion of the proposition when m ∈ N. The remaining case m ∈ Z\N is proved similarly or follows by hermitian symmetry. We omit the details.  Proof We shall calculate the integral limit in Proposition 5.2. Let I m (r ) be as in (5.3).

The integral quantity
Recall from Corollary 4.9 that u ∈ C ∞ (D). Consider the Taylor expansion of u at the origin of degree m ≥ 0: as z → 0. We have that as r → 0, where the last equality follows by cancellation. A passage to the limit as r → 0 now yields that c m = ∂ m u(0)/m!. We now turn to the formula for c −m . A similar analysis as in the previous paragraph shows that as r → 0. A passage to the limit as r → 0 yields that c −m =∂ m u(0)/m!. This completes the proof of the theorem.
The coefficient formulas in Theorem 5.3 leads to an addendum to Theorem 5.1. 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 for integrable functions f ∈ L 1 (T) on T.
Let C(T) be the space of continuous functions on T and fix ϕ ∈ C(T). By the ( p, q)-harmonic Dirichlet problem for ϕ we understand the problem of finding a ( p, q)-harmonic function u such that lim r →1 u r = ϕ in C(T), where u r (e iθ ) = u(re iθ ), e iθ ∈ T, for 0 < r < 1. Following usual practice, we formulate this latter Dirichlet problem as L p,q u = 0 in D, u = ϕ on T, (7.1) where L p,q is as in (0.1). 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].
Funding Open access funding provided by Lund University.

Conflict of interest
The authors declare that they have no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.