# Infinite Product Representation for the Szegö Kernel for an Annulus

Authors: Nuraddeen S. Gafai; Ali H. M. Murid; Nur H. A. A. Wahid
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The Szegö kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szegö kernel for an annulus can be expressed as a bilateral series and has a unique zero. In this paper, we show how to represent the Szegö kernel for an annulus as a basic bilateral series (also known asq-bilateral series). This leads to an infinite product representation through the application of Ramanujan’s sum. The infinite product clearly exhibits the unique zero of the Szegö kernel for an annulus. Its connection with the basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szegö kernel for general annulus and weighted Szegö kernel. Numerical comparisons on computing the Szegö kernel for an annulus based on the Kerzman-Stein integral equation, the bilateral series, and the infinite product are also presented.


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## Body

## 1. Introduction

The Ahlfors map is a branchingn-to-one map from an n-connected region onto the unit disk. It is intimately tied to the Szegö kernel of an n-connected region [1]. The boundary values of the Szegö kernel satisfy the Kerzman-Stein integral equation, which is a Fredholm integral equation of the second kind for a region with a smooth boundary [2]. The boundary values of the Alhfors map are completely determined from the boundary values of the Szegö kernel [1–3]. For an annulus region Ω, the Szegö kernel can be expressed as a bilateral series from which the zero can be determined analytically [4]. The Kerzman-Stein integral equation has been solved using the Adomian decomposition method in [5] to give another bilateral series form for the Szegö kernel for Ω that converges faster. There are various special functions in the form of bilateral and basic bilateral series [6–8]. For example, the bilateral basic hypergeometric series contain, as special cases, many interesting identities related to infinite products, theta functions, and Ramanujan's identities. It is therefore natural to ask if the bilateral series for the Szegö kernel for Ω can be summed as special functions or an infinite product that exhibits clearly its zero.In this paper, we show how to express the bilateral series for the Szegö kernel forΩ as a basic bilateral series (also known as q-bilateral series). Ramanujan’s sum is then applied to obtain the infinite product representation for the Szegö kernel for Ω. The product clearly exhibits the zero of the Szegö kernel for Ω, and its connection with the q-gamma function and the modified Jacobi theta function is shown. Using the symmetry of Ramanujan’s sum, we show how to easily transform the bilateral series for the Szegö kernel for Ω in [4] to the bilateral series in [5].The plan of the paper is as follows: After the presentation of some preliminaries in Section2, we derive the basic bilateral series and infinite product representations for the Szegö kernel for Ω in Section 3. We then derive a closed form of the Szegö for Ω in terms of q-gamma function and the modified Jacobi theta function. In Section 4, we show how to extend the representations in Section 3 to the general annulus using the transformation formula for the Szegö kernel under conformal mappings. Similar q-analysis for the weighted Szegö kernel for Ω is presented in Section 5. In Section 6, we give numerical comparisons for computing the Szegö kernel for Ω using bilateral series, infinite product, and integral equation formulations.


## 2. Preliminaries

LetΩ=z:ρ<z<1 be an annulus with 0<ρ<1 and a point a∈Ω. The boundary Γ of Ω consists of two smooth Jordan curves with the outer curve Γ0 oriented counterclockwise and the inner curve Γ1 oriented clockwise. The positive direction of the contour Γ=Γ0∪Γ1 is usually that for which the region is on the left as one traces the boundary.Letφnzn=1∞ be an orthonormal basis for the Hardy spaces H2Γ. Since the Szegö kernel Sz,a is the reproducing kernel for H2Γ, it can be written as [4]
(1)Sz,a=∑n=0∞φnzφna¯,a∈Ω,with absolute and uniform convergence on compact subsets of Ω. An orthogonal basis for H2Γ is znn=−∞∞. Thus
(2)zn2=∫Γz2ndz=2π1+ρ2n+1,where dz is the arc length measure. Therefore, an orthonormal basis for H2Γ is [3, 4]
(3)zn2π1+ρ2n+1n=−∞∞.Using (1) and (3), the series representation for the Szegö kernel for Ω is given by [4]
(4)Sz,a=12π∑n=−∞∞za¯n1+ρ2n+1,a∈Ω,z∈Ω∪Γ.Series (4) is a bilateral series. It has a zero at z=−ρ/a¯ [4].Another bilateral series representation for the Szegö kernel forΩ is given by [5] (in an equivalent form)
(5)Sz,a=12π∑n=−∞∞−1nρnρ2n−za¯,z∈Ω∪Γ,a∈Ω,which is initially obtained by solving the Kerzman-Stein integral equation using the Adomian decomposition method. It is also shown in [5] how to derive (5) directly from (4) using geometric series. It is illustrated in [5] that series (5) converges faster than (4).More generally, ifΩ1 is any doubly connected region with the smooth boundary Γ1, and fz is a biholomorphic map of Ω1 onto Ω, then the Szegö kernel for Ω1 can be obtained via the transformation formula as [1]
(6)S1z,a=f′zSfz,faf′a¯=f′zf′a2π∑n=−∞∞fzfa¯n1+ρ2n+1,a∈Ω1,z∈Ω1∪Γ1,where ρ is unknown but can be computed.The Szegö kernelS1z,a can also be computed without using conformal mapping. The boundary values of the Szegö kernel S1z,a on Γ1 satisfy the Kerzman-Stein integral equation [2, 4],
(7)S1z,a+∫ΓAz,wS1w,adw=gz,z∈Γ1,where
(8)Az,w=12πTwz−w−Tz¯z¯−w¯,z≠w∈Γ1,0,z=w∈Γ1,gz=−12πiTz¯z¯−a¯,z∈Γ1,Tz=z′t∣z′t∣,z∈Γ1,and zt is a parametrization of Γ1. The function Az,w is known as the Kerzman-Stein kernel, and it is continuous on the boundary of Ω1 [9, 10]. In fact, the integral equation (7) is also valid for an n-connected region.Since bilateral series and basic bilateral series will be used throughout this paper, we recall some facts aboutq-series notations and results.Let0<q<1 and α∈ℂ. The q-shifted factorial is defined as [7]
(9)qα;qn=1,n=0,1−qα1−qα+1⋯1−qα+n−1,n=1,2,…,11−qα−11−qα−2⋯1−qα−n,n=−1,−2,….This notation yields the shifted factorial as a special case through(10)limq⟶1qα;qnq;qn=αα+1⋯α+n−1,n=1,2,….Ifα is written in place of qα, then (9) becomes
(11)α;qn=1,n=0,1−α1−αq⋯1−αqn−1,n=1,2,…,11−αq−11−αq−2⋯1−αq−n,n=−1,−2,….It can be shown that [7]
(12)1−α1−αqn=α;qnαq;qn,n=0,±1,±2,….Ifn⟶∞, it is standard to write
(13)α;q∞=∏n=0∞1−αqn,which is absolutely convergent for all finite values of α, real or complex, when q<1 [6]. This yields
(14)α;qn=α;q∞αqn;q∞.Observe thatα;q∞ would have zero as a factor if α=1. It would be zero also if α=q−1,q−2,q−3,…, but these are all outside the circle z=1 since q<1 [8].The bilateral basic hypergeometric series in baseq with one numerator and one denominator parameters is defined by [6–8]
(15)ψ11α;β;q;z=∑n=−∞∞α;qnβ;qnzn.The series is convergent forq<1 and β/α<z<1.The classical Ramanujan’sψ11 summation is given by [7, 8]
(16)ψ11α;β;q;z=αz;q∞q/αz;q∞β/α;q∞q;q∞z;q∞β/αz;q∞q/α;q∞β;q∞,β/α<z<1.The special caseβ=αq of Ramanujan’s 1ψ1 summation yields [8]
(17)∑n=−∞∞zn1−αqn=αz;q∞q/αz;q∞q;q∞2z;q∞q/z;q∞α;q∞q/α;q∞,also known as Cauchy’s formula. Due to symmetry in α and z on the right-hand side of (17), it implies [8]
(18)∑n=−∞∞zn1−αqn=∑n=−∞∞αn1−zqn.Theq-gamma function is defined as [7]
(19)Γqx=q;q∞qx;q∞1−q1−x,0<q<1,x=ℂ−0,−1,−2,….Another important special function that is used in this paper is the modified Jacobi theta function defined by [7]
(20)θx;q=x;q∞q/x;q∞,where x≠0 and q<1. For a more detailed discussion on q-series and historical perspectives, see, for example, [6–8] and the references therein.


## 3. Szegö Kernel for an Annulus and Basic Bilateral Series

In this section, we express the bilateral series (4) as a basic bilateral series and derive the infinite product representation of the Szegö kernel for Ω. It is given in the following theorem.Theorem 1.
LetΩ be the annulus z:ρ<z<1 bounded by Γ. For a∈Ω, z∈Ω∪Γ, the Szegö kernel for Ω can be represented by
(21)Sz,a=12π1+ρ1ψ1−ρ;−ρ3;ρ2;a¯z,(22)=12π∏n=0∞1+a¯zρ2n+1a¯z+ρ2n+11−ρ2n+221−a¯zρ2na¯z−ρ2n+21+ρ2n+12.
The zero ofSz,a in Ω is the zero of the factor a¯z+ρ, that is, z=−ρ/a¯.Proof.
From (4), we have
(23)Sz,a=12π∑n=−∞∞a¯zn1+ρ2n+1=12π∑n=−∞∞a¯zn1−−ρρ2n.
Lettingα=−ρ and q=ρ2 yields
(24)Sz,a=12π∑n=−∞∞a¯zn1−αqn,(25)=12π1−α∑n=−∞∞1−α1−αqna¯zn.
Applying (12) and (15) gives
(26)Sz,a=12π1−α∑n=−∞∞α,qnαq,qna¯zn=12π1−α1ψ1α;αq;q;a¯z.
Note that the1ψ1 series above is convergent because q=ρ2<1 and β/α=αq/α=q=ρ2<a¯z<1. Substituting α=−ρ and q=ρ2 into (26) gives (21).
Applying Ramanujan’s sum (16) to (26), gives
(27)Sz,a=12π1−ααa¯z;q∞q/αa¯z;q∞q;q∞2a¯z;q∞q/a¯z;q∞q/α;q∞αq;q∞.
But from (14), with n=1, we have
(28)1−ααq;q∞=α;q∞.
Thus, (27) becomes
(29)Sz,a=12παa¯z;q∞q/αa¯z;q∞q;q∞2a¯z;q∞q/a¯z;q∞q/α;q∞α;q∞,(30)=12π∏n=0∞1−αa¯zqn1−qn+1/αa¯z1−qn+121−a¯zqn1−qn+1/a¯z1−qn+1/α1−αqn.
Substitutingα=−ρ and q=ρ2 into (30) gives (22).
The infinite product (22) would have poles if
(31)1−a¯zρ2n=0ora¯z−ρ2n+2=0,which implies
(32)z=1a¯ρ2norz=ρ2n+2a¯.
But(33)1aρ2n>1,ρ2n+2a¯<ρ2n+1<ρ.
Therefore, the poles are all outsideΩ.
The infinite product (22) would have zeros if
(34)1+a¯zρ2n+1=0ora¯z+ρ2n+1=0,which implies
(35)z=−1a¯ρ2n+1orz=−ρ2n+1a¯.
For the first case(36)1aρ2n+1>1ρ2n+1>1,which is outside Ω. For the second case, observe that
(37)ρ2n+1<ρ2n+1a¯=ρ2n+1∣a∣<ρ2n,which clearly has a zero inside Ω when n=0. Thus, the infinite product (22) for Sz,a has only one zero inside Ω at z=−ρ/a¯. This completes the proof.We note that the series representation (21) for Sz,a is valid only for ρ≤∣z∣≤1, while the infinite product representation (22) for Sz,a is meaningful for all z∈ℂ except for the infinitely many poles at z=0,ρ−2n/a¯,ρ2n+2/a¯.We next show that the Szegö kernel forΩ can also be expressed in terms of the basic gamma function and modified Jacobi theta function. By applying (20) to (29) and substituting α=−ρ and q=ρ2, we have
(38)Sz,a=12πθαa¯z;q∞q;q∞2θa¯z;q∞q/α;q∞α;q∞=12πθ−ρa¯z;ρ2∞ρ2;ρ2∞2θa¯z;ρ2∞−ρ;ρ2∞2.Applying (19) with q=ρ2, observe that
(39)ρ2;ρ2∞−ρ;ρ2∞=ρ2;ρ2∞ρ2x;ρ2∞=Γρ2x1−ρ21−x,where x satisfies ρ2x=−ρ. This equation may be written as
(40)e2x−1lnρ=eiπ,which yields a solution
(41)x=12+iπ2lnρ.Thus, (38) becomes
(42)Sz,a=Γρ2λ22π1−ρ221−λθ−ρa¯z;ρ2∞θa¯z;ρ2∞,λ=12+iπ2lnρ.This can be regarded as a closed-form expression for the Szegö kernel forΩ.In the following, we show how to easily transform series (4) to series (5) using (18). Letting α=−ρ and q=ρ2, (4) becomes
(43)Sz,a=12π∑n=−∞∞a¯zn1−αqn=12π∑n=−∞∞αn1−a¯zqn,where in the last step we have used (18). By replacing α=−ρ and q=ρ2, we get
(44)Sz,a=12π∑n=−∞∞−1nρn1−a¯zρ2n.Lettingn=−m yields
(45)Sz,a=12π∑m=−∞∞−1−mρ−m1−a¯zρ−2m=12π∑m=−∞∞−1mρmρ2m−a¯z,which is the same as (5).


## 4. Szegö Kernel for General Annulus

Consider the general annulusΩ2=z:r2<z−z0<r1 with boundary denoted by Γ2. The region Ω2 reduces to Ω if z0=0, r2=ρ, and r1=1.Theorem 2.
Letz0∈ℂ, z∈Ω2∪Γ2, and a∈Ω2. The Szegö kernel for Ω2 can be represented by the bilateral series as
(46)S2z,a=12π∑n=−∞∞a¯−z0¯nr12n+1+r22n+1z−z0n,(47)=12π∑n=−∞∞−1nr1n+1r2nr22nr12−r12nz−z0a¯−z0¯. The zero of S2z,a in Ω2 is z=z0−r1r2/a¯−z0¯.Proof.
Observe that the functionfz=z−z0/r1 maps Ω2 onto Ω with ρ=r2/r1.
Applying the transformation formula (6) yields
(48)S2z,a=f′zSfz,faf′a¯=1r1Sz−z0r1,a−z0r11r1¯=1r1Sz−z0r1,a−z0r1.
Applying (4) to (48) with z and a replaced by z−z0/r1 and a−z0/r1, respectively, gives
(49)S2z,a=12πr1∑n=−∞∞z−z0a−z0¯/r12n1+r2/r12n+1,which simplifies to (46).
Applying (5) to (48) instead of z and a replaced by z−z0/r1 and a−z0/r1, respectively, gives
(50)S2z,a=12πr1∑n=−∞∞−1nr2/r1nr2/r12n−z−z0a−z0¯/r12,which simplifies to (47).
Using the fact thatSz,a has a zero at z=−ρ/a¯ for Ω, the zero of S2z,a for Ω2 is z−z0/r1=−ρ/a−z0¯/r1 which implies z=z0−ρr12/a¯−z0¯=z0−r1r2/a¯−z0¯. This completes the proof.
Similarly, the infinite product representation ofS2z,a for Ω2 can be obtained by applying (22) to (48) with z and a replaced by z−z0/r1 and a−z0/r1, respectively.


## 5. The Weighted Szegö Kernel for an Annulus and Basic Bilateral Series

The weighted Szegö kernel is defined in [11] as
(51)K^qtz,w=12π∑n=−∞∞w¯zn1+tq2n,t>0,q<z,w<1.To adopt the notations used in this paper, we changeq to ρ, w to a, and K^qtz,w to Sρtz,a in (51), which gives
(52)Sρtz,a=12π∑n=−∞∞a¯zn1+tρ2n,t>0,ρ<z,a<1.Note thatSρρz,a is exactly the kernel Sz,a for Ω discussed in Section 1. The zeros of the kernel Stρz,a are not discussed in [11] but have expressed interest on the effect of the weight on the location of its zeros. In the following theorem, we express the weighted Szegö kernel Sρtz,a as a basic bilateral series and derive its associated infinite product representation as well as its zeros.Theorem 3.
LetΩ be the annulus z:ρ<z<1 bounded by Γ. For a∈Ω, z∈Ω∪Γ, and t>0, the weighted Szegö kernel Sρtz,a for Ω can be represented by
(53)Stρz,a=12π1+t1ψ1−t;−tρ2;ρ2;a¯z,(54)=12π∏n=0∞1+ta¯zρ2na¯z+ρ2n+2/t1−ρ2n+221−a¯zρ2na¯z−ρ2n+21+ρ2n+2/t1+tρ2n.
The kernelSρtz,a has a zero in Ω only if t takes the form t=ρ±2m+1, m=0,1,2,⋯. In both cases, the zero is z=−ρ/a¯.Proof.
Observe that(55)Sρtz,a=12π∑n=−∞∞a¯zn1−−tρ2n.
Lettingα=−t and q=ρ2, the above equation becomes
(56)Sρtz,a=12π∑n=−∞∞a¯zn1−αqn,which is exactly the same form as (24). Applying the result (26) with α=−t, the above equation becomes
(57)Sρtz,a=12π1+t1ψ1−t;−tq;q;a¯z.
Series (57) is convergent because q=ρ2<1 and β/α=−tq/−t=q<ρ2<a¯z<1. Substituting q=ρ2 gives (41).
Applying the result (29) with α=−t to (57) yields
(58)Sρtz,a=12π−ta¯z;q∞q/−ta¯z;q∞q;q∞2a¯z;q∞q/a¯z;q∞q/−t;q∞−t;q∞.
Replacingq=ρ2 and applying (13) give (54).
In the proof of Theorem1, we have shown that the factors 1−a¯zρ2na¯z−ρ2n+2 have no zeros in Ω. The factors 1+ρ2n+2/t1+tρ2n would have zeros if
(59)ρ2n+2/t=−1ortρ2n=−1.
Sincet>0, we conclude that the kernel Sρtz,a has no poles in Ω for any t>0. The factors 1+ta¯zρ2na¯z+ρ2n+2/t would have zeros if
(60)1+ta¯zρ2n=0ora¯z+ρ2n+2/t=0,which implies
(61)z=−1ta¯ρ2norz=−ρ2n+2ta¯.
For the first case, observe that(62)1tρ2n<1ta¯ρ2n<1tρ2n+1.
To have a zero inΩ, we must have the condition
(63)ρ≤1tρ2n<1ta¯ρ2n<1tρ2n+1≤1,which means
(64)t≤1ρ2n+1andt≥1ρ2n+1.
Hence, we must havet=ρ−2n+1. In this case, the zero of Sρtz,a in Ω is z=−ρ/a¯.
For the second case, observe that(65)ρ2n+2t<ρ2n+2ta¯<ρ2n+1t.
To have a zero inΩ, we must have the condition
(66)ρ≤ρ2n+2t<ρ2n+2ta¯<ρ2n+1t≤1,which means
(67)t≤ρ2n+1andt≥ρ2n+1.
Hence, we must havet=ρ2n+1. In this case, the zero of Sρtz,a in Ω is also z=−ρ/a¯. This completes the proof.The weighted Szegö kernel can also be expressed in terms of the basic gamma function and the modified Jacobi theta function. By applying (20) to (58) with q=ρ2, we have
(68)Sρtz,a=12πθ−ta¯z;ρ2∞ρ2;ρ2∞2θa¯z;ρ2∞ρ2/−t;ρ2∞−t;ρ2∞.Observe that(69)ρ2;ρ2∞−t;ρ2∞=ρ2;ρ2∞ρ2x;ρ2∞=Γρ2x1−ρ21−x,where x satisfies ρ2x=−t. This equation may be written as
(70)2xlnρ=ln−t=ln−t+iarg−t=lnt+iπ,which yields a solution
(71)x=lnt+iπ2lnρ.Observe also that(72)ρ2;ρ2∞−ρ2/t;ρ2∞=ρ2;ρ2∞ρ2y;ρ2∞=Γρ2y1−ρ21−y,where y satisfies ρ2y=−ρ2/t. This equation may be written as
(73)2y−2lnρ=ln−1t=ln−1t+iarg−1t=−lnt+iπ,which yields a solution
(74)y=1+−lnt+iπ2lnρ.Thus, (68) becomes
(75)Sρtz,a=Γρ2μΓρ2νθ−ta¯z;ρ2∞2π1−ρ22−μ−νθa¯z;ρ2∞,μ=lnt+iπ2lnρ,ν=1+−lnt+iπ2lnρ.This can be regarded as a closed-form expression for the weighted Szegö kernel for an annulusΩ. Observe that (75) reduces to (42) when t=ρ.


## 6. Numerical Computation of the Szegö Kernel for an Annulus

In this section, we compare the speed of convergence of the three formulas for computing the Szegö kernel forΩ based on the two bilateral series (4) and (5) and the infinite product (22).To approximate (4) numerically, we calculate
(76)Sz,a≈S10z,a=12π∑k=−1010za¯k1+ρ2k+1,and S50 and S100.To approximate (5) numerically, we calculate
(77)Sz,a≈S10∗z,a=12π∑k=−1010−1kρkρ2k−za¯,and S50∗.To approximate (22) numerically, we compute
(78)Sz,a≈S15∗∗z,a=12π∏k=0151+a¯zρ2k+1za¯+ρ2k+11−ρ2k+221−za¯ρ2kza¯−ρ2k+21+ρ2k+12,and S20∗∗ and S25∗∗.The approximations are then compared with the numerical solution of the Kerzman-Stein Equation (7). To solve (7), we used the Nyström method [5] with the trapezoidal rule with n selected nodes on each boundary component Γ0 and Γ1. The approximate solution is represented by S~n where n is the number of nodes. All the computations were done using MATHEMATICA 12.3. Four numerical examples are given for different values of a and ρ. The results for the error norms are presented for each example.We consider an annulusΩ bounded by
(79)Γ0:z0t=eit,Γ1:z1t=ρe−it,with 0≤t≤2π.Example 1.
We consider an annulusΩ with a=0.7i and ρ=0.5. The results for the error norms are presented in Tables 1–3.Table 1
Error norms betweenS10 and S~n, S50 and S~n, and S100 and S~n.
nS10−S~n∞S50−S~n∞S100−S~n∞162.4536 (-02)2.97754 (-03)2.97758 (-03)322.75019 (-02)1.15906 (-05)1.16299 (-05)642.75136 (-02)3.91113 (-08)1.88349 (-10)1282.75136 (-02)3.92996 (-08)2.28878 (-15)Table 2
Error norms betweenS10∗ and S~n and S50∗ and S~n.
nS10∗−S~n∞S50∗−S~n∞162.94797 (-03)2.97758 (-03)321.78995 (-02)1.16299 (-05)641.77628 (-04)1.88351 (-10)1281.77628 (-04)1.81497 (-15)Table 3
Error norms betweenS15∗∗ and S~n, S20∗∗ and S~n, and S25∗∗ and S~n.
nS15∗∗−S~n∞S20∗∗−S~n∞S25∗∗−S~n∞162.97758 (-03)2.97758 (-03)2.97758 (-03)321.16296 (-05)1.16299 (-05)1.16299 (-05)641.44308 (-10)1.88038 (-10)1.8835 (-10)1283.1999 (-10)3.1275 (-13)1.82618 (-15)Example 2.
We consider an annulusΩ with a=−0.4−0.6i and ρ=0.3. The results for the error norms are presented in Tables 4–6.Table 4
Error norms betweenS10 and S~n, S50 and S~n, and S100 and S~n.
nS10−S~n∞S50−S~n∞S100−S~n∞161.29695 (-02)1.46732 (-03)1.46732 (-03)321.56432 (-02)7.88666 (-06)7.88666 (-06)641.5646 (-02)3.26124 (-08)2.2539 (-10)1281.5646 (-02)3.26942 (-08)2.85127 (-15)Table 5
Error norms betweenS10∗ and S~n and S50∗ and S~n.
nS10∗−S~n∞S50∗−S~n∞161.46686 (-03)1.46732 (-03)328.4009 (-06)7.88666 (-06)641.02367 (-06)2.2539 (-10)1281.02367 (-06)1.25883 (-15)Table 6
Error norms betweenS5∗∗ and S~n, S10∗∗ and S~n, and S15∗∗ and S~n.
nS5∗∗−S~n∞S10∗∗−S~n∞S15∗∗−S~n∞161.4675 (-03)1.46732 (-03)1.46732 (-03)327.70793 (-06)7.88666 (-06)7.88666 (-06)643.72977 (-07)2.2434 (-10)2.2539 (-10)1283.73107 (-07)2.2023 (-12)1.41308 (-15)Example 3.
We consider an annulusΩ with a=−0.8 and ρ=0.4. The results for the error norms are presented in Tables 7–9.Table 7
Error norms betweenS10 and S~n, S50 and S~n, and S100 and S~n.
nS10−S~n∞S50−S~n∞S100−S~n∞166.45804 (-02)8.28061 (-03)8.28061 (-03)326.82534 (-02)2.2673 (-04)2.2673 (-04)646.83565 (-02)9.0045 (-06)1.79491 (-07)1286.83565 (-02)9.08614 (-06)1.29631 (-10)Table 8
Error norms betweenS10∗ and S~n and S50∗ and S~n.
nS10∗−S~n∞S50∗−S~n∞168.28737 (-03)8.28061 (-03)322.33562 (-04)2.2673 (-04)641.79806 (-05)1.79491 (-07)1281.78806 (-05)1.1287 (-15)Table 9
Error norms betweenS5∗∗ and S~n, S10∗∗ and S~n, and S15∗∗ and S~n.
nS5∗∗−S~n∞S10∗∗−S~n∞S15∗∗−S~n∞168.27577 (-03)8.28061 (-03)8.28061 (-03)322.2189 (-04)2.26729 (-04)2.2673 (-04)641.13437 (-05)1.78984 (-07)1.79491 (-07)1281.14253 (-05)1.19798 (-09)7.90864 (-14)Example 4.
We consider an annulusΩ with a=−0.4−0.5i and ρ=0.1. The results for the error norms are presented in Tables 10–12.Table 10
Error norms betweenS10 and S~n, S50 and S~n, and S100 and S~n.
nS10−S~n∞S50−S~n∞S100−S~n∞163.15879 (-03)2.61429 (-04)2.61429 (-04)323.22447 (-03)2.08805 (-07)2.08805 (-07)643.28124 (-03)5.91022 (-11)1.33153 (-13)1283.28124 (-03)5.91168 (-11)1.33233 (-15)Table 11
Error norms betweenS10∗ and S~n and S50∗ and S~n.
nS10∗−S~n∞S50∗−S~n∞162.61429 (-04)2.61429 (-04)322.0879 (-07)2.08805 (-07)641.68217 (-11)1.33183 (-13)1281.67281 (-11)1.16606 (-15)Table 12
Error norms betweenS5∗∗ and S~n, S10∗∗ and S~n, and S15∗∗ and S~n.
nS5∗∗−S~n∞S10∗∗−S~n∞S15∗∗−S~n∞162.61429 (-04)2.61429 (-04)2.61429 (-04)322.08805 (-07)2.08805 (-07)2.08805 (-07)646.46416 (-13)1.3313 (-13)1.33121 (-13)1286.77069 (-13)1.49882 (-15)1.55654 (-15)The numerical results presented in Tables1–12 show that computations using the infinite product formula (22) converge faster than the bilateral series formulas (4) and (5).


## 7. Conclusion

This paper has shown that the bilateral series for the Szegö kernel forΩ is a disguised bilateral basic hypergeometric series ψ11. Ramanujan’s sum for ψ11 is then applied to obtain the infinite product representation for the Szegö kernel for Ω. The product clearly exhibits the zero of the Szegö kernel for an Ω. The Szegö kernel can also be expressed as a closed form in terms of the q-gamma function and the modified Jacobi theta function. Similar q-analysis has also been conducted for the Szegó kernel for general Ω and for the weighted Szegö kernel for Ω. The numerical comparisons have shown that the infinite product method converges faster than the bilateral series methods for computing the Szegö kernel for Ω.For future work, it is natural to devote further investigation on the infinite product representation for the Szegö kernel for doubly connected regions via the transformation formula (6) and Theorem 1. This however requires knowledge of conformal mapping of doubly connected regions to annulus [12–15]. For some ideas on numerical methods for computing the zero of the Szegö kernel for doubly connected regions, see [16]. Alternatively, perhaps some computational intelligence algorithms can also be considered to compute the zero, like the monarch butterfly optimization (MBO) [17], earthworm optimization algorithm (EWA) [18], elephant herding optimization (EHO) [19], moth search (MS) algorithm [20], slime mould algorithm (SMA) [21], and Harris hawks optimization (HHO) [22].



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