# Fractional Zernike functions

Hajar Dkhissi a, Allal Ghanmi a and Safa Snounb CONTACT A. Ghanmi. Emails:
allal.ghanmi<EMAIL_ADDRESS>a Analysis, P.D.E $\&$ Spectral Geometry,
Lab M.I.A.-S.I., CeReMAR,
Department of Mathematics,P.O. Box 1014, Faculty of Sciences,
Mohammed V University in Rabat, Morocco  b Department of Mathematics, Faculty
of Sciences,
University of Gabes 6072, Tunisia

###### Abstract

We consider and provide an accurate study for the fractional Zernike functions
on the punctured unit disc, generalizing the classical Zernike polynomials and
their associated $\beta$-restricted Zernike functions. Mainly, we give the
spectral realization of the latter ones and show that they are orthogonal
$L^{2}$-eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian.
The algebraic and analytic properties for the fractional Zernike functions to
be established include the connection to special functions, their zeros, their
orthogonality property, as well as the differential equations, recurrence and
operational formulas they satisfy. Integral representations are also obtained.
Their regularity as poly-meromorphic functions is discussed and their
generating functions including a bilinear one of ”Hardy–Hille type” are
derived. Moreover, we prove that a truncated subclass defines a complete
orthogonal system in the underlying Hilbert space giving rise to a specific
Hilbertian orthogonal decomposition in terms of a second class of generalized
Bergman spaces.

###### keywords:

Zernike polynomials; $\beta$-restricted Zernike functions; fractional Zernike
functions; $\beta$-weighted Poly-Bergman spaces; Zeros set; Poly-meromorphy;
Generating functions; Orthogonality; Completeness.

###### Contents

  1. 1 Introduction
  2. 2 The $\beta$-restricted Zernike functions (spectral realization)
  3. 3 Fractional Zernike functions
    1. 3.1 Connection to special functions and explicit expression.
    2. 3.2 Poly-meromorphy
    3. 3.3 Differential equations.
    4. 3.4 Recurrence and operational formulas.
    5. 3.5 Generating and bilinear generating functions.
    6. 3.6 Integral representations
    7. 3.7 Completeness.

##  1 Introduction

The classical real Zernike polynomials are introduced in the framework of
optical problems, especially in order to analyze the figure of a circular
mirror. In Zernike”s paper on the knife edge test and the phase contrast
method [31], they are defined as eigenfunctions of a rotational invariant
second order partial differential equation. Next, they have been used in the
Nijboer’s works to develop the diffraction theory of optical aberrations.
Since then, they have been extensively employed to express the propagation of
a wavefront data in optical tests through imaging system [14, 15, 20], and to
represent the aberrations of optical systems (by atmospheric turbulence) [26,
29]. They are also used to study diffraction problems in the rotationally
symmetric system with circular pupils [24, 32] and pattern recognition [18,
28]. More recently, they are applied efficiently to characterize the shape of
any portion of molecular surfaces and to evaluate the shape complementarity of
protein-protein interfaces [23].

A generalized complex version (called Zernike or disc polynomials) defines
them as the orthogonal ones on the unit disc
$\mathbb{D}=\\{z\in\mathbb{C};\,|z|<1\\}$ with finite values at the boundary.
They are given by the Rodrigues type formula

$\displaystyle\mathcal{Z}^{\gamma}_{m,n}(z,\overline{z})$
$\displaystyle:=(-1)^{m+n}(1-|z|^{2})^{-\gamma}\frac{\partial^{m+n}}{\partial
z^{m}\overline{z}^{n}}\left(1-|z|^{2}\right)^{\gamma+m+n}$ (1.1)

for varying nonnegative integers $m,n$, and real $\gamma>-1$. This definition
agrees with the one provided by Koornwinder [19] as well as the one considered
by Dunkl [6]. Algebraic and analytic properties of
$\displaystyle\mathcal{Z}_{m,n}^{\gamma}(z,\bar{z})$ have been discussed in
many papers [1, 19, 30]. The corresponding Wiener and Paley type theorems have
been obtained by Kanjin in [16]. Recently, they have shown to be useful in the
concrete description of spectral properties of different types of Cauchy
transforms [7, 8].

In the present paper, we consider a specific generalization of the Zernike
polynomials in (1.1). Namely, we deal with the family of functions

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle:=(-1)^{m}z^{-\rho}(1-|z|^{2})^{-\kappa}\frac{\partial^{m}}{\partial
z^{m}}\left(z^{n+\rho}(1-|z|^{2})^{\kappa+m}\right)$ (1.2)

on the punctured unit disc $\mathbb{D}^{*}=\mathbb{D}\setminus\\{0\\}$, for
fixed real numbers $\rho,\kappa>-1$ and varying integers $m$ and $n$ such that
$m\geq 0$ and $n+\rho\geq 0$. Thus, for arbitrary nonnegative integer $\rho$,
they reduce further to the Zernike polynomials in (1.1) since for every
$\ell=0,1,\cdots$ we have

$\displaystyle z^{\ell}\mathcal{Z}^{\kappa,\ell}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{\mathcal{Z}_{m,n+\ell}^{\kappa}(z,\overline{z})}{(\kappa+m+1)_{n+\ell}}.$
(1.3)

Otherwise, they are no longer polynomials. Their study for arbitrary $\rho$
can be reduced to the subclass corresponding to $0\leq\rho<1$. More precisely,
we have

$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=z^{-[\rho]}\mathcal{Z}^{\kappa,\widetilde{\rho}}_{m,n+[\rho]}(z,\overline{z})$

whenever $n+[\rho]\geq 0$, where $[\rho]$ denotes the integer part of $\rho$
and $0\leq\widetilde{\rho}=\rho-[\rho]<1$. This justifies somehow the
following definition which can also be justified from being poly-meromorphic
(see Theorem 3.10).

######  Definition 1.1.

The functions $\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ in (1.2) are
referred to as fractional Zernike functions.

Contrary to the classical Zernike polynomials satisfying the symmetry
relationships
$\overline{\mathcal{Z}^{\kappa}_{m,n}(z,\overline{z})}=\mathcal{Z}^{\kappa}_{m,n}(\overline{z},z)=\mathcal{Z}^{\kappa}_{n,m}(z,\overline{z}),$
which play a crucial rule in their study, this relation is no longer valid for
the fractional Zernike functions
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ even if $\rho$ is a positive
integer. In fact, we have only
$\overline{\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})}=\mathcal{Z}^{\kappa,\rho}_{m,n}(\overline{z},z)$
for arbitrary $\rho$ and one gets from (1.3) the identity

$\displaystyle\overline{\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})}=\frac{(\kappa+1)_{m}}{(\kappa+1)_{n+\rho}}z^{\rho}\overline{z}^{-\rho}\mathcal{Z}^{\kappa,\rho}_{n+\rho,m-\rho}(z,\overline{z})$
(1.4)

valid for $\rho$ being a nonnegative integer. This reveals in particular that
the analytic and spectral properties of the functions
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ can not be directly
recovered from the Zernike polynomials, and the relevant properties may be
completely different from the classical ones, essentially when $\rho$ is non
integer. Thus, a concrete description of their algebraic and analytic
properties for fixed reals $\rho,\kappa>-1$ is desirable.

To this purpose we begin by considering the so-called $\beta$-restricted
Zernike functions $\psi^{\gamma,\eta}_{m,n}$. They are shown to be a special
class of polyanalytic excited states in the weighted Hilbert space
$L^{2,\alpha}_{\beta}(\mathbb{D})=L^{2}(\mathbb{D},d\mu_{\alpha,\beta})$ of
all complex-valued functions that are square integrable with respect to the
positive measure

$\displaystyle
d\mu_{\alpha,\beta}(z):=(1-|z|^{2})^{\alpha}|z|^{2\beta}dxdy;\quad
z=x+iy,\,\,\alpha,\beta>-1.$ (1.5)

The main results concerning the functions $\psi^{\gamma,\eta}_{m,n}$ are
summarized in Theorem 2.6. Namely, we prove that they form an orthogonal
system of eigenfunctions in $L^{2,\alpha}_{\beta}(\mathbb{D})$ for a perturbed
magnetic Laplacian, which is essentially the classical magnetic Schrödinger
operator on the hyperbolic disc perturbed by a particular potential (with zero
magnetic field) modeling the Aharonov–Bohm effect (see Remark 2.3). Moreover,
the $L^{2,\alpha}_{\beta}$-eigenspace of the considered Laplacian associated
with its lowest eigenvalue is shown to be the $\beta$-modified Bergman space
$\mathcal{A}^{2,\alpha}_{\beta}(\mathbb{D})$ on the punctured unit disc
$\mathbb{D}^{*}$ recently introduced and studied in [11, 12]. The other
$L^{2,\alpha}_{\beta}$-eigenspaces associated with the hyperbolic Landau
levels for the considered Laplacian can be seen as the polyanalytic analogs of
$\mathcal{A}^{2,\alpha}_{\beta}(\mathbb{D})$ (see Remark 2.7).

The motivation of considering $\psi^{\gamma,\eta}_{m,n}$ is that they can be
seen as the spectral side of fractional Zernike functions. For special values
of $\gamma$ and $\eta$ they are closely connected to by

$\displaystyle\mathcal{Z}^{\kappa_{m},\rho}_{m,n}(z,\overline{z})$
$\displaystyle=|z|^{2\eta}(1-|z|^{2})^{\frac{\alpha+1-\kappa_{m}}{2}}\psi^{\gamma,\eta}_{m,n}(z,\overline{z})$
(1.6)

for $m,n\geq 0$ with $\rho=\beta-2\eta$ and for $\kappa$ depending in $m$ and
given by $\kappa=\kappa_{m}=\alpha-2(\gamma+m)-1$. However, this last fact can
not be employed to recover the global properties of the fractional Zernike
functions $\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$. Only the local
ones for every fixed $m$, $n$ and $\rho$ with the specific $\kappa=\kappa_{m}$
can be derived.

For the concrete study of $\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ we
begin by establishing their explicit expressions, their different
hypergeometric representations, their expression in terms of the Jacobi
polynomials as well as their connection to the complex Zernike polynomials in
(1.1). Subsequently, the zero sets of
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ are described (Corollary
3.6) and shown to be the centered circles of radii being the zeros of the real
Jacobi polynomials. The orthogonality in the Hilbert space
$L^{2,\kappa}_{\rho}(\mathbb{D})=L^{2}(\mathbb{D},d\mu_{\kappa,\rho}$) is
discussed and the square norm is explicitly computed. The membership to a
specific class of poly-meromorphic functions in $\mathbb{D}$ is also
considered (Theorem 3.10). Moreover, we investigate the operational formulas
they satisfy including those of Burchnall type and discuss some recurrence
relations, the differential equations they obey (Theorems 3.15 and 3.12) and
so on. Certain associated generating functions are obtained such a bilinear
generating function analogous to the one Hardy–Hille generating function for
the generalized Laguerre polynomials. The latter one can be employed to derive
special integral representation for
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$. Another integral
representation of Cauchy-type is obtained as a special integral on the unit
circle. Finally, we show in Theorem 3.25 that the truncated fractional Zernike
functions

$\displaystyle\varUpsilon_{m,s}^{\kappa,\rho}(z,\overline{z}):=z^{s}|z|^{-s}\mathcal{Z}^{\kappa,\rho}_{m,m}(z,\overline{z}),\,s\in\mathbb{Z},\,m=0,1,\cdots,$
(1.7)

constitute an orthogonal basis of the Hilbert space
$L^{2,\kappa}_{\rho}(\mathbb{D})$. Accordingly, we define a second class of
poly-meromorphic Bergman spaces leading to a complete microlocal orthogonal
decomposition of the underlying Hilbert space
$L^{2,\kappa}_{\rho}(\mathbb{D})$. The obtained results will contribute
efficiently in the study of the associated isometric integral transforms (of
Bargmann type) on the configuration space on the positive half real line.

The remaining sections are organized as follows. Section 2 deals with the
spectral realization of the $\beta$-modified functions
$\psi^{\gamma,\eta}_{m,n}$ by Schrödinger’s factorization method. A proof of
$\psi^{\gamma,\eta}_{m,n}$ form an orthogonal system of eigenfuctions in
$L^{2,\alpha}_{\beta}(\mathbb{D})$ is also presented in this section. The
basic properties of the fractional functions as described above are stated and
proved in Section 3.

##  2 The $\beta$-restricted Zernike functions (spectral realization)

In this section we are concerned with the functions

$\displaystyle\psi^{\gamma,\eta}_{m,n}(z,\overline{z})$
$\displaystyle=(-1)^{m}z^{\eta-\beta}\overline{z}^{-\eta}(1-|z|^{2})^{\gamma-\alpha+m}\frac{\partial^{m}}{\partial
z^{m}}\big{(}z^{n+\beta-2\eta}(1-|z|^{2})^{\alpha-2\gamma-m-1}\big{)}$ (2.1)

for given reals $\alpha,\beta,\gamma,\eta$. They referred to as the
$\beta$-restricted Zernike functions (justified by Remark 2.7 below). We aim
to derive their basic properties and show that they form an orthogonal system
of $L^{2,\alpha}_{\beta}$-eigenfunctions for a perturbed magnetic Laplacian of
the form

$\displaystyle\Delta_{a,b}^{c,d}$
$\displaystyle=\Delta_{hyp}+(1-|z|^{2})\left(H_{a}^{b}(z)E-H_{c}^{d}(z)\overline{E}\right)+H_{a}^{b}(z)H_{c}^{d}(z)|z|^{2}$
(2.2)

acting on the weighted Hilbert space $L^{2,\alpha}_{\beta}(\mathbb{D})$,
$\alpha,\beta>-1$. Above $a$, $b$, $c$ and $d$ are given real numbers,
$\Delta_{hyp}=-(1-|z|^{2})^{2}{\partial^{2}}/{\partial z\partial{\bar{z}}}$ is
the Laplace–Beltrami operator on the hyperbolic disc,
$\overline{E}=\overline{z}{\partial}/{\partial\overline{z}}$ denotes the
complex conjugate of the complex Euler operator $E:=z{\partial}/{\partial z}$
and

$\displaystyle H_{a}^{b}(z):=a+b-\frac{b}{|z|^{2}}.$ (2.3)

It is worth noting that for particular values of $a,b,c,d$ one recovers the
magnetic Schrödinger operator on the hyperbolic unit disc representing the
Hamiltonian of a charged particle in motion under an external uniform magnetic
field [5, 3, 10, 13].

To this end, we have to factorize the considered Laplacian in terms of some
first order differential operators (leading in particular to their Rodrigues
type formula). Thus, if we set $h_{\alpha,\beta}(z)=h(z)^{\alpha}|z|^{2\beta}$
with $h(z)=1-|z|^{2}$, we can consider the first order differential operator

$A_{\gamma,\eta}f(z):=h_{1-\gamma,-\eta}(z)\frac{\partial}{\partial\overline{z}}(h_{\gamma,\eta}f)(z)$

for given fixed reals $\gamma$ and $\eta$. Its explicit expression is given by

$A_{\gamma,\eta}f(z)=\left\\{(1-|z|^{2})\frac{\partial}{\partial\overline{z}}-H_{\gamma}^{\eta}(z)\right\\}f(z).$
(2.4)

The corresponding null space is closely connected to the set
$\text{Hol}(\mathbb{D}^{*})$ of holomorphic functions on the punctured unit
disc. Namely, we have
$\ker(A_{\gamma,\eta})=h_{-\gamma,-\eta}\text{Hol}(\mathbb{D}^{*})$. Moreover,
the formal adjoint operator $A^{*_{\alpha,\beta}}_{\gamma,\eta}$ of
$A_{\gamma,\eta}$ with respect to the inner scalar product

$\displaystyle{\left\langle{f,g}\right\rangle}_{\alpha,\beta}:=\int_{\mathbb{D}}f(z)\overline{g(z)}d\mu_{\alpha,\beta}(z)$
(2.5)

in $L^{2,\alpha}_{\beta}(\mathbb{D})$ is given by

$\displaystyle A^{*_{\alpha,\beta}}_{\gamma,\eta}f(z)$
$\displaystyle:=-h_{\gamma-\alpha,\eta-\beta}(z)\frac{\partial}{\partial
z}(h_{\alpha-\gamma+1,\beta-\eta}f)(z)$ (2.6)

in account of the conventional calculation. Accordingly, we perform

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}=A_{\gamma,\eta}A^{*_{\alpha,\beta}}_{\gamma,\eta}\quad
and\quad\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}=A^{*_{\alpha,\beta}}_{\gamma,\eta}A_{\gamma,\eta}.$
(2.7)

Straightforward computation leads to the explicit expression of these second
order differential operators in terms of $\Delta_{a,b}^{c,d}$ in (2.2) (we
omit the proof).

######  Lemma 2.1.

The expression of $\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ in the
$z$-coordinate is given by

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$
$\displaystyle=\Delta_{\gamma+1,\eta}^{\gamma-\alpha-1,\eta-\beta}+(\alpha-\gamma+1).$

Moreover, the operators $\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ and
$\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}$ satisfy

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}=\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,-}+(\alpha-2\gamma).$
(2.8)

######  Remark 2.2.

For $\alpha=-2$ and $\beta=0$ we have
$\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}=\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,-}-2(\gamma+1)$.
Also $H_{\alpha-\gamma+1}^{\beta-\eta}=-H_{\gamma+1}^{\eta}$ so that the
Laplacian $\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ reduces further to

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$
$\displaystyle=-h\left\\{h\frac{\partial^{2}}{\partial
z\partial\overline{z}}-H_{\gamma+1}^{\eta}(z)\left(E-\overline{E}\right)\right\\}+(H_{\gamma+1}^{\eta}(z))^{2}|z|^{2}-(\gamma+1).$
(2.9)

For the particular cases of $\gamma,\eta$ we recover the Landau-like
Hamiltonian on $\mathbb{D}$ (see e.g. [3, 10, 13]).

######  Remark 2.3.

The considered operators $\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ and
$\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}$ can be realized geometrically as
magnetic Schrödinger operators associated with a singular real differential
$1$-form (vector potential)
$\theta_{\alpha,\beta}=\theta_{\alpha}+\widetilde{\theta_{\beta}}$ with
$d\widetilde{\theta_{\beta}}=0$ and $d\theta_{\alpha}$ is the Khäler two form
on the hyperbolic unit disc up to a multiplicative constant. More precisely,
we have

$\displaystyle\theta_{\alpha,\beta}(z)$
$\displaystyle=\frac{i\alpha\left({\bar{z}}dz-
zd\bar{z}\right)}{1-|z|^{2}}-i\beta\left(\frac{dz}{z}-\frac{d\overline{z}}{\overline{z}}\right).$
(2.10)

Now, by means of the identity (2.8) we can establish the following (we omit
the proof).

######  Lemma 2.4.

The following commutation rules hold trues

$\displaystyle(i)$
$\displaystyle\qquad\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}A_{\gamma,\eta}=A_{\gamma,\eta}\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}\quad\mbox{and}\quad
A^{*_{\alpha,\beta}}_{\gamma,\eta}\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}=\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}A^{*_{\alpha,\beta}}_{\gamma,\eta}.$
$\displaystyle(ii)$
$\displaystyle\qquad\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}A^{*_{\alpha,\beta}}_{\gamma+1,\eta}=A^{*_{\alpha,\beta}}_{\gamma+1,\eta}\left(\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,+}+(\alpha-2\gamma)\right).$
$\displaystyle(iii)$ $\displaystyle\qquad
A_{\gamma+1,\eta}\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}=\left(\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,+}+(\alpha-2\gamma)\right)A_{\gamma+1,\eta}.$
$\displaystyle(iv)$
$\displaystyle\qquad\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,-}A_{\gamma,\eta}=A_{\gamma,\eta}\left(\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}-(\alpha-2\gamma)\right).$
$\displaystyle(v)$ $\displaystyle\qquad
A^{*_{\alpha,\beta}}_{\gamma,\eta}\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,-}=\left(\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,-}-(\alpha-2\gamma)\right)A^{*_{\alpha,\beta}}_{\gamma,\eta}.$

Lemma 2.4 is efficient in analyzing and studying the family of functions
$\psi^{\gamma,\eta}_{m,n}$ in (2.1). In fact, we show that they can be
obtained by successive application of $A^{*}_{\gamma+j,\eta}$;
$j=1,2,\cdots,m$, to the ground state functions. Namely, we consider the
differential operator

$\displaystyle A^{*,m}_{\gamma,\eta}(f)$
$\displaystyle:=A^{*_{\alpha,\beta}}_{\gamma+1,\eta}\circ
A^{*_{\alpha,\beta}}_{\gamma+2,\eta}\circ\cdots\circ
A^{*_{\alpha,\beta}}_{\gamma+m,\eta}(f).$

Then, we claim the following.

######  Lemma 2.5.

The closed expression of Rodrigues type for $A^{*,m}_{\gamma,\eta}$ is given
by

$\displaystyle A^{*,m}_{\gamma,\eta}(f)$
$\displaystyle=(-1)^{m}h_{\gamma-\alpha+m,\eta-\beta}\frac{\partial^{m}}{\partial
z^{m}}(h_{\alpha-\gamma,\beta-\eta}f).$ (2.11)

###### Proof.

Starting from the definition of $A^{*,m}_{\gamma,\eta}$ one gets

$\displaystyle A^{*,m}_{\gamma,\eta}(f)$
$\displaystyle=(-1)^{m}h_{\gamma-\alpha-1,\eta-\beta}\left(h^{2}\frac{\partial}{\partial
z}\right)^{m}(h_{\alpha-\gamma-m+1,\beta-\eta}f).$

Then (2.11) readily follows thanks to the fact
$(h^{2}\partial)^{m}(f)=h^{m+1}\partial^{m}(h^{m-1}f)$ in [9]. ∎

The main result in this section is the following.

######  Theorem 2.6.

Fix $\gamma$ such that $\alpha>2\gamma+1$. Then, for integers $m,n$ such that
$n>2\eta-\beta-1$ and $0\leq m<(\alpha-1-2\gamma)/2$, the following assertions
hold.

  * (i)

The function $\psi^{\gamma,\eta}_{m,n}$ is a
$L^{2,\alpha}_{\beta}$-eigenfunction of
$\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ with
$E^{\gamma,\alpha}_{m}=(m+1)(\alpha-2\gamma-m)$ as corresponding eigenvalue.

  * (ii)

The functions $\psi^{\gamma,\eta}_{m,n}$ form an orthogonal system in the
Hilbert space $L^{2,\alpha}_{\beta}(\mathbb{D})$ and their square norm
(induced from (2.5)) is given by

$\displaystyle{\left\|{\psi^{\gamma,\eta}_{m,n}}\right\|}_{\alpha,\beta}^{2}=\frac{\pi
m!}{(\alpha-2(\gamma+m)-1)}\frac{\Gamma(\alpha-2\gamma-m)\Gamma(n+\beta-2\eta+1)}{\Gamma(n+\alpha+\beta-2(\gamma+\eta+m))}.$
(2.12)

Here $\Gamma$ is the Gamma Euler function.

###### Proof.

In virtue of the algebraic identity $(iii)$ in Lemma 2.4 and the identity
(2.8) we can proceed by mathematical induction to get

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}A^{*,m}_{\gamma,\eta}$
$\displaystyle=A^{*,m}_{\gamma,\eta}\mathcal{L}_{\gamma+m,\eta}^{\alpha,\beta,+}+\sum_{j=0}^{m-1}(\alpha-2(\gamma+j))A^{*,m}_{\gamma,\eta}$
$\displaystyle=A^{*,m}_{\gamma,\eta}\left(\mathcal{L}_{\gamma+m+1,\eta}^{\alpha,\beta,-}+(\alpha-2(\gamma+m))\right)+\sum_{j=0}^{m-1}(\alpha-2(\gamma+j))A^{*,m}_{\gamma,\eta}$
$\displaystyle=A^{*,m}_{\gamma,\eta}\mathcal{L}_{\gamma+m+1,\eta}^{\alpha,\beta,-}+\sum_{j=0}^{m}(\alpha-2(\gamma+j))A^{*,m}_{\gamma,\eta}$
$\displaystyle=A^{*,m}_{\gamma,\eta}\mathcal{L}_{\gamma+m+1,\eta}^{\alpha,\beta,-}+(m+1)(\alpha-2\gamma-m)A^{*,m}_{\gamma,\eta}.$

Accordingly, it becomes clear that the functions
$A^{*,m}_{\gamma,\eta}(\varphi^{\gamma,\eta}_{m})$ are eigenfunctions of
$\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ whenever
$\varphi^{\gamma,\eta}_{m}$ belongs to the null space of
$A_{\gamma+m+1,\eta}$,

$\ker(A_{\gamma+m+1,\eta})=\\{f:\mathbb{D}^{*}\longrightarrow\mathbb{C};\,\,A_{\gamma+m+1,\eta}f=0\\}\subseteq\ker\left(\mathcal{L}_{\gamma+m+1,\eta}^{-}\right).$

This is the case when considering

$\displaystyle\varphi^{\gamma,\eta}_{m}(z)=\varphi^{\gamma,\eta}_{m,n}(z):=z^{n}(1-|z|^{2})^{-(\gamma+m+1)}|z|^{-2\eta};\
\ n\in\mathbb{Z}.$ (2.13)

More precisely, the functions
$A^{*,m}_{\gamma,\eta}(\varphi^{\gamma,\eta}_{m})=A^{*,m}_{\gamma,\eta}(z^{n}h_{-(\gamma+m+1),-\eta})$
are given by

$\displaystyle A^{*,m}_{\gamma,\eta}$
$\displaystyle(z^{n}h_{-(\gamma+m+1),-\eta})=(-1)^{m}h_{\gamma-\alpha+m,\eta-\beta}\frac{\partial^{m}}{\partial
z^{m}}(z^{n}h_{\alpha-2\gamma-m-1,\beta-2\eta})$ (2.14)
$\displaystyle=(-1)^{m}(1-|z|^{2})^{\gamma-\alpha+m}|z|^{2(\eta-\beta)}\frac{\partial^{m}}{\partial
z^{m}}\big{(}z^{n}|z|^{2(\beta-2\eta)}(1-|z|^{2})^{\alpha-2(\gamma+m)+m-1}\big{)}$

thanks to Lemma 2.5. The latter formula reduces further to the expression of
the $\beta$-restricted Zernike functions in (2.1). Moreover, they satisfy

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}\left(\psi^{\gamma,\eta}_{m,n}\right)=(m+1)(\alpha-2\gamma-m)\psi^{\gamma,n}_{m,n}=E^{\gamma,\alpha}_{m}\psi^{\gamma,\eta}_{m,n}.$
(2.15)

Now, for their orthogonality in $L^{2,\alpha}_{\beta}(\mathbb{D})$ one can use
their explicit expressions in terms of certain special functions (see for
example Remark 3.7 below). However, we present below another proof using the
factorization method. To this purpose, notice first that
$A^{*,m}_{\gamma,\eta}=A^{*_{\alpha,\beta}}_{\gamma+1,\eta}\circ
A^{*,m-1}_{\gamma+1,\eta}$ and that
$\varphi^{\gamma,\eta}_{m,n}=\varphi^{\gamma+1,\eta}_{m-1,n}$. It follows

$\psi^{\gamma,\eta}_{m,n}=A^{*,m}_{\gamma,\eta}(\varphi^{\gamma,\eta}_{m,n})=A^{*_{\alpha,\beta}}_{\gamma+1,\eta}\circ
A^{*,m-1}_{\gamma+1,\eta}(\varphi^{\gamma,\eta}_{m,n})=A^{*_{\alpha,\beta}}_{\gamma+1,\eta}(\psi^{\gamma+1,\eta}_{m-1,n}).$

Accordingly, making use of (2.15) we obtain

$\displaystyle{\left\langle{\psi^{\gamma,\eta}_{m,n},\psi^{\gamma,\eta}_{j,k}}\right\rangle}$
$\displaystyle={\left\langle{\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,+}(\psi^{\gamma+1,\eta}_{m-1,n}),\psi^{\gamma+1,\eta}_{j-1,k}}\right\rangle}=E^{\gamma+1,\alpha}_{m-1}{\left\langle{\psi^{\gamma+1,\eta}_{m-1,n},\psi^{\gamma+1,\eta}_{j-1,k}}\right\rangle}.$

More generally, by induction we arrive at

$\displaystyle{\left\langle{\psi^{\gamma,\eta}_{m,n},\psi^{\gamma,\eta}_{j,k}}\right\rangle}$
$\displaystyle=\prod_{\ell=1}^{s}E^{\gamma+\ell,\alpha}_{m-\ell}{\left\langle{\psi^{\gamma+s,\eta}_{m-s,n},\psi^{\gamma+s,\eta}_{j-s,k}}\right\rangle};\,1\leq
s\leq m.$

Therefore, without lost of generality we can assume that $m\leq j$ and take
$s=m$ to get

$\displaystyle{\left\langle{\psi^{\gamma,\eta}_{m,n},\psi^{\gamma,\eta}_{j,k}}\right\rangle}$
$\displaystyle=\prod_{\ell=1}^{m}E^{\gamma+\ell,\alpha}_{m-\ell}{\left\langle{\psi^{\gamma+m,\eta}_{0,n},\psi^{\gamma+m,\eta}_{j-m,k}}\right\rangle}$
$\displaystyle=\prod_{\ell=1}^{m}E^{\gamma+\ell,\alpha}_{m-\ell}{\left\langle{\psi^{\gamma+m,\eta}_{0,n},A^{*}_{\gamma+m+1,\eta}\circ
A^{*}_{\gamma+m+2,\eta}\circ\cdots\circ
A^{*}_{\gamma+j,\eta}(\varphi^{\gamma+m,\eta}_{j-m,k})}\right\rangle}$
$\displaystyle=\prod_{\ell=1}^{m}E^{\gamma+\ell,\alpha}_{m-\ell}{\left\langle{A_{\gamma+j,\eta}\circ\cdots\circ
A_{\gamma+m+1,\eta}(\varphi^{\gamma+m,\eta}_{0,n}),\varphi^{\gamma+m,\eta}_{j-m,k}}\right\rangle}.$

The last identity holds by observing that
$\psi^{\gamma+s,\eta}_{0,n}=\varphi^{\gamma+s,\eta}_{0,n}$, which readily
follows from (2.11) or (2.14). Next, since $\varphi^{\gamma+m,\eta}_{0,n}$
belongs to $\ker(A_{\gamma+m+1,\eta})$ and then
$A_{\gamma+j,\eta}\circ\cdots\circ A_{\gamma+m+2,\eta}\circ
A_{\gamma+m+1,\eta}(\varphi^{\gamma+m,\eta}_{0,n})$ vanishes whenever $m<j$,
we obtain

$\displaystyle{\left\langle{\psi^{\gamma,\eta}_{m,n},\psi^{\gamma,\eta}_{j,k}}\right\rangle}$
$\displaystyle=\left(\prod_{\ell=1}^{m}E^{\gamma+\ell,\alpha}_{m-\ell}\right){\left\langle{\varphi^{\gamma+m,\eta}_{0,n},\varphi^{\gamma+m,\eta}_{0,k}}\right\rangle}\delta_{m,j}.$

To the computation of the quantity
${\left\langle{\varphi^{\gamma+m,\eta}_{0,n},\varphi^{\gamma+m,\eta}_{0,k}}\right\rangle}$
we make use of (2.13) giving the explicit expression of
$\varphi^{\gamma,\eta}_{m,n}$. This yields

$\displaystyle{\left\langle{\varphi^{\gamma+m,\eta}_{0,n},\varphi^{\gamma+m,\eta}_{0,k}}\right\rangle}$
$\displaystyle=\int_{\mathbb{D}}(1-|z|^{2})^{\alpha-2(\gamma+m+1)}|z|^{2(\beta-2\eta)}z^{n}\overline{z}^{k}d\lambda(z)$
$\displaystyle=\pi\left(\int_{0}^{1}(1-t)^{\alpha-2(\gamma+m+1)}t^{n+\beta-2\eta}dt\right)\delta_{n,k}$
$\displaystyle=\pi B(n+\beta-2\eta+1,\alpha-2(\gamma+m)-1)\delta_{n,k},$

where $B(a,b)$ denotes the classical beta function. The validity of the
previous formula requires that $n>2\eta-\beta-1$ and $\alpha-2\gamma-1>2m$
with $\alpha-2\gamma-1>0$. Finally, since

$\prod_{\ell=1}^{m}E^{\gamma+\ell,\alpha}_{m-\ell}=m!(\alpha-2(\gamma+m))_{m}=m!\frac{\Gamma(\alpha-2\gamma-m)}{\Gamma(\alpha-2(\gamma+m))},$

we arrive at

${\left\langle{\psi^{\gamma,\eta}_{m,n},\psi^{\gamma,\eta}_{j,k}}\right\rangle}=\frac{\pi
m!}{(\alpha-2(\gamma+m)-1)}\frac{\Gamma(\alpha-2\gamma-m)\Gamma(n+\beta-2\eta+1)}{\Gamma(n+\alpha+\beta-2(\gamma+\eta+m))}\delta_{m,j}\delta_{n,k}.$

This completes the proof. ∎

######  Remark 2.7.

The functions in (2.1) corresponding to $\gamma=-1$, $\eta=0$ and $m=0$ reduce
further to $\psi^{-1,0}_{0,n}(z,\overline{z})=z^{n}$, for varying integer
$n>-(\beta+1)$, whose square norm in $L^{2,\alpha}_{\beta}(\mathbb{D})$ is
given by

${\left\|{\psi^{-1,0}_{0,n}}\right\|}_{\alpha,\beta}^{2}=\pi\frac{\Gamma(\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+2)}.$

They form an orthogonal basis of the $\beta$-modified Bergman space
$\mathcal{A}^{2,\alpha}_{\beta}(\mathbb{D})$ defined as the closed subspace in
$L^{2,\alpha}_{\beta}(\mathbb{D})$ formed by the holomorphic functions on the
punctured disc $\mathbb{D}^{*}$ (see [11, 12] for details). In other words,
the $\beta$-modified Bergman space is the $L^{2}$-eigenspace of our magnetic
Laplacian $\mathcal{L}_{\gamma+1,\eta}^{\alpha,\beta,+}$ associated with its
lowest Landau level. For the particular case of $\beta=0$ we recover the
classical Bergman space on the unit disc with respect to the weight function
being of the generalized Gegenbauer form $(1-|z|^{2})^{\alpha}$.

######  Remark 2.8.

The functions $\psi^{\gamma,\eta}_{m,n}$ do not form a complete system in
$L^{2,\alpha}_{\beta}(\mathbb{D})$. However, for fixed $m$ such that $0\leq
m<(\alpha-1-2\gamma)/2$ and varying integer $n\geq 2\eta-\beta$ they span a
specific closed subspace $\mathcal{A}^{2,\alpha}_{\beta,m}(\mathbb{D})$ in
$L^{2,\alpha}_{\beta}(\mathbb{D})$. This gives rise to what can be called the
$m$-th generalized (or also poly-meromophic) $\beta$-modified Bergman space on
$\mathbb{D}^{*}$ and can be seen as the polyanalytic analog of the
$\beta$-modified Bergman space. Its reproducing kernel is given in Remark 3.20
below.

##  3 Fractional Zernike functions

In this section we provide an accurate theoretical study for the fractional
Zernike functions in (1.2). We discuss their connection to some special
functions, zeros, orthogonality in $L^{2,\kappa}_{\rho}(\mathbb{D})$,
regularity, differential equations, recurrence and operational formulas. Some
results concerning the generating functions, the integral representations and
completeness are also obtained.

###  3.1 Connection to special functions and explicit expression.

We begin by establishing the explicit expression of the fractional Zernike
functions $\mathcal{Z}^{\kappa,\rho}_{m,n}$ in terms of the classical Zernike
polynomials. Thus, for given real $b$ and nonnegative integer $m$ we define
the infected minimum $m\wedge^{*}b$ to be

$m\wedge^{*}b=\left\\{\begin{array}[]{ll}\min(m,b),&\,b=0,1,2,\cdots\\\
m,&b\in\mathbb{R},\,b\neq 0,1,2,\cdots.\\\ \end{array}\right.$

######  Proposition 3.1.

For every $\rho>-1$ we have

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{m!\Gamma(\rho+1)}{(\kappa+m+1)_{n}}\sum_{j=0}^{m\wedge^{*}\rho}\frac{(-1)^{j}}{j!(m-j)!\Gamma(\rho-j+1)}\frac{\left(1-|z|^{2}\right)^{j}}{z^{j}}\mathcal{Z}^{\kappa+j}_{m-j,n}(z,\overline{z}).$
(3.1)

###### Proof.

Using the facts (3.4) and

$z^{n}(1-|z|^{2})^{\kappa+m}=\frac{(-1)^{n}}{(\kappa+m+1)_{n}}\frac{\partial^{n}}{\partial\overline{z}^{n}}\left((1-|z|^{2})^{\kappa+m+n}\right)$

we get

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{(-1)^{m+n}}{(\kappa+m+1)_{n}}z^{-\rho}(1-|z|^{2})^{-\kappa}\frac{\partial^{m}}{\partial
z^{m}}\left(z^{\rho}\frac{\partial^{n}}{\partial\overline{z}^{n}}\left((1-|z|^{2})^{\kappa+m+n}\right)\right)$
$\displaystyle=\frac{(-1)^{m+n}m!}{(\kappa+m+1)_{n}}z^{-\rho}(1-|z|^{2})^{-\kappa}\sum_{j=0}^{m}\frac{(-\rho)_{j}}{j!(m-j)!}z^{\rho-j}\frac{\partial^{m-j+n}}{\partial
z^{m-j}\partial\overline{z}^{n}}\left((1-|z|^{2})^{\kappa+j+m-j+n}\right),$

which can be rewritten as (3.1). ∎

######  Remark 3.2.

For $\rho=0$ we recover the Zernike polynomials
$\mathcal{Z}_{m,n}^{\kappa}(z,\overline{z})$ up to the multiplicative constant
$1/(\kappa+m+1)_{n}$, while when $\rho=1$ we get

$\displaystyle\mathcal{Z}_{m,n+1}^{\kappa}(z,\overline{z})$
$\displaystyle=(\kappa+m+n+1)\left(z\mathcal{Z}^{\kappa}_{m,n}(z,\overline{z})+m\left(1-|z|^{2}\right)\mathcal{Z}^{\kappa+1}_{m-1,n}(z,\overline{z})\right),$

which is exactly the three terms recurrence formula for the Zernike
polynomials [1, p. 403, Eq. (5.1)]. This follows since that $(-\rho)_{j}=0$
for $j\geq\rho+1$ whenever $\rho=0,1,2,\cdots$. More generally, from (1.3)
with $\rho$ being a nonnegative integer we obtain new recurrence formula for
the classical complex Zernike polynomials

$\displaystyle\mathcal{Z}_{m,n+\rho}^{\kappa}(z,\overline{z})$
$\displaystyle=m!\Gamma(\rho+1)(\kappa+m+n+1)_{\rho}\sum_{j=0}^{m\wedge\rho}\frac{(-1)^{j}z^{\rho-j}\left(1-|z|^{2}\right)^{j}}{j!(m-j)!\Gamma(\rho-j+1)}\mathcal{Z}^{\kappa+j}_{m-j,n}(z,\overline{z}).$
(3.2)

The explicit expression of the few first terms of
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ can be computed easily from
the Rodrigues formula (1.2) or also using (3.1). Thus, those corresponding to
$m=0$ reduce to the monomials,
$\mathcal{Z}^{\kappa,\rho}_{0,n}(z,\overline{z})=z^{n}$. For $m=1$ and $m=2$
we get respectively

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{1,n}(z,\overline{z})$
$\displaystyle=(\kappa+n_{\rho}+1)\overline{z}z^{n}-n_{\rho}z^{n-1}$

and

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{2,n}(z,\overline{z})$
$\displaystyle=(\kappa+n_{\rho}+1)(\kappa+n_{\rho}+2)\overline{z}^{2}z^{n}-2n_{\rho}(\kappa+n_{\rho}+1)\overline{z}z^{n-1}+n_{\rho}(n_{\rho}-1)z^{n-2},$

where we have set $n_{\rho}=n+\rho$. A general formula for the explicit
expression of $\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ is given by
the following assertion.

######  Proposition 3.3.

We have

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\sum_{j=0}^{m\wedge^{*}(n+\rho)}\frac{(-1)^{j}m!\Gamma(n+\rho+1)\Gamma(\kappa+m+1)}{j!(m-j)!\Gamma(n+\rho-j+1)\Gamma(\kappa+j+1)}\left(1-|z|^{2}\right)^{j}z^{n-j}\overline{z}^{m-j}.$
(3.3)

###### Proof.

Using the fact $(x)^{\underline{n}}=\Gamma(x+1)/\Gamma(x-n+1)$ for the
decreasing factorial $(x)^{\underline{n}}=x(x-1)\cdots(x-n+1)$, we obtain

$\frac{\partial^{m}}{\partial
z^{m}}\left(1-xz\right)^{a}=(-1)^{m}\frac{\Gamma(a+1)}{\Gamma(a+1-m)}x^{m}\left(1-xz\right)^{a-m}$

and

$\displaystyle\frac{\partial^{m}}{\partial
z^{m}}\left(z^{a}\right)=(-a)_{m}z^{a-m}=\varepsilon^{*}_{a,m}\frac{\Gamma(a+1)}{\Gamma(a+1-m)}z^{a-m},$
(3.4)

where for the nonnegative integer $m$ we have set

$\varepsilon^{*}_{a,m}=\left\\{\begin{array}[]{lll}1,&a\geq
m;\,a=0,1,\cdots\\\ 0,&a<m;\,a=0,1,\cdots\\\ 1,&a\in\mathbb{R},\,a\neq
0,1,\cdots.\\\ \end{array}\right.$

Thus, applying the Leibnitz formula for high order derivation of a product
yields

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=(-1)^{m}z^{-\rho}(1-|z|^{2})^{-\kappa}\frac{\partial^{m}}{\partial
z^{m}}\big{(}z^{n+\rho}(1-|z|^{2})^{\kappa+m}\big{)}$
$\displaystyle=\sum_{j=0}^{m}\varepsilon^{*}_{n+\rho,j}\frac{(-1)^{j}m!\Gamma(n+\rho+1)\Gamma(\kappa+m+1)}{j!(m-j)!\Gamma(n+\rho-j+1)\Gamma(\kappa+j+1)}\overline{z}^{m-j}z^{n-j}\left(1-|z|^{2}\right)^{j}.$

This gives rise to (3.3). ∎

Below, we present different hypergeometric representations of
$\mathcal{Z}^{\kappa,\rho}_{m,n}$ in terms of the Gauss hypergeometric
function defined on the open unit disc by power series

${{}_{2}F_{1}}\left(\begin{array}[]{c}a,b\\\
c\end{array}\bigg{|}z\right)=\sum_{n=0}^{\infty}{\frac{(a)_{n}(b)_{n}}{(c)_{n}}}{\frac{z^{n}}{n!}}$

provided that $c\neq 0,-1,-2,\cdots$.

######  Proposition 3.4.

The functions $\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ are given in
terms of the ${{}_{2}F_{1}}$ function by

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=(\kappa+1)_{m}z^{n}\overline{z}^{m}{{}_{2}F_{1}}\left(\begin{array}[]{c}-m,-n-\rho\\\
\kappa+1\end{array}\bigg{|}1-\frac{1}{|z|^{2}}\right).$ (3.7)

###### Proof.

By means of $(a)^{\underline{n}}=(-1)^{n}(-a)_{n}=\Gamma(a+1)/\Gamma(a-n+1)$
combined with $(-1)^{j}(-m)_{j}(m-j)!=m!$, we can rewrite (3.3) as

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=(\kappa+1)_{m}z^{n}\overline{z}^{m}\sum_{j=0}^{m\wedge^{*}(n+\rho)}\frac{(-m)_{j}(-n-\rho)_{j}}{(\kappa+1)_{j}j!}\left(1-\frac{1}{|z|^{2}}\right)^{j}$
$\displaystyle=(\kappa+1)_{m}z^{n}\overline{z}^{m}{{}_{2}F_{1}}\left(\begin{array}[]{c}-m,-n-\rho\\\
\kappa+1\end{array}\bigg{|}1-\frac{1}{|z|^{2}}\right).$

∎

There are several equivalent expressions for $\mathcal{Z}^{\kappa,\rho}_{m,n}$
in terms of the Gauss hypergeometric functions which follow from the well-
known linear transformations for ${{}_{2}F_{1}}$. Thus, from the second and
the third ones in [22, $\S$ 2.4., p. 47], it follows

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=(\kappa+1)_{m}z^{n-m}{{}_{2}F_{1}}\left(\begin{array}[]{c}-m,n+\kappa+\rho+1\\\
\kappa+1\end{array}\bigg{|}1-|z|^{2}\right)$ (3.10)

and

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=(\kappa+1)_{m}z^{-\rho}\overline{z}^{m-n-\rho}{{}_{2}F_{1}}\left(\begin{array}[]{c}-n-\rho,\kappa+m+1\\\
\kappa+1\end{array}\bigg{|}1-|z|^{2}\right).$ (3.13)

However, starting from (3.7) and applying the linear transformation [25, Eq.
(15.8.7)] for the limiting case one obtains

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=(\kappa+\rho+n+1)_{m}z^{n}\overline{z}^{m}{{}_{2}F_{1}}\left({-m,-n-\rho\atop-\kappa-\rho-
n-m}\bigg{|}\frac{1}{|z|^{2}}\right).$ (3.14)

The same transformation applied to (3.10) gives rise to

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=(-n-\rho)_{m}z^{n-m}{{}_{2}F_{1}}\left({-m,\kappa+n+\rho+1\atop
n+\rho-m+1}\bigg{|}|z|^{2}\right).$ (3.15)

The latter one remains valid for $\rho$ being integer and $m\leq n+\rho$.

The next result is concerned with the expression of
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ in terms of the the real
Jacobi polynomials.

######  Proposition 3.5.

For $\rho=0,1,\cdots$, we have

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{(\kappa+1)_{m\vee(n+\rho)}(m\wedge(n+\rho))!}{(\kappa+1)_{n+\rho}}\frac{z^{n}\overline{z}^{m}}{|z|^{2(m\wedge(n+\rho))}}P^{(\kappa,|m-n-\rho|)}_{m\wedge(n+\rho)}(2|z|^{2}-1),$
(3.16)

while when $\rho>-1$ is non-integer or $m=m\wedge(n+\rho)$ we have

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=m!z^{n-m}P_{m}^{(\kappa,n-m+\rho)}(2|z|^{2}-1).$ (3.17)

###### Proof.

Consider first the case of $\rho=0,1,\cdots$. Starting from (3.10) and the
assumption $m\leq n+\rho$ one can make use of the facts
$(-n-\rho)_{m}=(-1)^{m}(n+\rho-m+1)_{m}$,
$P^{(a,b)}_{m}(x)=(-1)^{m}P^{(b,a)}_{m}(-x)$ and

$\displaystyle{{}_{2}F_{1}}\left(\begin{array}[]{c}-m,m+a+b+1\\\
a+1\end{array}\bigg{|}x\right)=\frac{m!}{(a+1)_{m}}P^{(a,b)}_{m}(1-2x)$ (3.20)

to get

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=m!z^{n-m}P^{(\kappa,n+\rho-m)}_{m}(2|z|^{2}-1).$ (3.21)

The result corresponding to the case $m\geq n+\rho$ is immediate from the
previous one using the like-symmetry relationship (1.4) (it is also immediate
from (3.13)). In fact, we have

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{(\kappa+1)_{m}(n+\rho)!}{(\kappa+1)_{n+\rho}}z^{-\rho}\overline{z}^{m-n-\rho}P^{(\kappa,m-n-\rho)}_{n+\rho}(2|z|^{2}-1).$
(3.22)

To conclude one observes that both expressions can rewritten in the unified
form

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{(\kappa+1)_{m\vee(n+\rho)}(m\wedge(n+\rho))!}{(\kappa+1)_{n+\rho}}|z|^{|m-n-\rho|-\rho}e^{i(n-m)\arg
z}P^{(\kappa,|m-n-\rho|)}_{m\wedge(n+\rho)}(2|z|^{2}-1),$

which clearly leads to (3.16). This can also be deduced from the functions
$z^{\rho}\mathcal{Z}^{\kappa,\rho}_{m,n}$ being the classical Zernike
functions in (1.1) up to multiplicative constant and next making appeal to the
one in [9].

Finally, the formula (3.17) for $\rho$ being non-integer is exactly (3.16)
(when $\rho$ integer and $m\leq n+\rho$). It can be obtained by means of
(3.10) and (3.20) valid whenever $m=m\wedge^{*}(n+\rho)$, which corresponds to
$\rho>-1$ being non-integer or also to $m=m\wedge(n+\rho)$. A direct proof can
be handled starting from the derivation formula [4, p. 1, 1.1.2 Eq. 2] that we
can rewrite as

$\displaystyle\frac{d^{m}}{dz^{m}}\left(z^{a}(1-xz)^{n}\right)=m!z^{a-m}(1-xz)^{b-m}P_{m}^{(a-m,b-m)}(1-2xz).$
(3.23)

Therefore, with the specification $a=n+\rho$, $b=\kappa+m$ and
$x=\overline{z}$, the expression of $\mathcal{Z}^{\kappa,\rho}_{m,n}$ in (1.2)
becomes (3.17). ∎

The next result concerning the zeros of $\mathcal{Z}^{\kappa,\rho}_{m,n}$ is
an immediate consequence of Proposition 3.5.

######  Corollary 3.6.

The point $0$ is a zero of $\mathcal{Z}^{\kappa,\rho}_{m,n}$ when
$\rho=0,1,2,\cdots$ if and only if $n>m$ or $m>n$ with $\rho=0$. However, when
$\rho$ is non-integer, it is a zero of $\mathcal{Z}^{\kappa,\rho}_{m,n}$ only
when $n>m$. The other zeros are the circles centered at the origin with radii
$(1+r_{m,n}^{\kappa,\rho})/2$, where $r_{m,n}^{\kappa,\rho}$ are the zeros,
located at the segment $(0,1)$, of the real Jacobi polynomials
$P_{m}^{(\kappa,n-m+\rho)}(x)$ when $\rho$ is non-integer and
$P_{m\wedge(n+\rho)}^{(\kappa,|n-m+\rho|)}(x)$ when $\rho$ is a non-negative
integer.

######  Remark 3.7.

According to Proposition 3.5, the expression of the $\beta$-restricted Zernike
functions $\psi^{\gamma,\eta}_{m,n}$ in terms of the Jacobi polynomials reads

$\displaystyle\psi^{\gamma,\eta}_{m,n}(z,\overline{z})$
$\displaystyle=(-1)^{m}m!\frac{z^{n-m}}{|z|^{2\eta}}(1-|z|^{2})^{\frac{\kappa-\alpha-1}{2}}P_{m}^{(n-m+\rho,\kappa)}(1-2|z|^{2}),$
(3.24)

where $\rho=\beta-2\eta>-1$ is non-integer and $\kappa=\alpha-2(\gamma+m)-1$.

We conclude this subsection by discussing the orthogonality of the considered
functions.

######  Corollary 3.8.

The functions $\mathcal{Z}^{\kappa,\rho}_{m,n}$ form an orthogonal system in
$L^{2,\kappa}_{\rho}(\mathbb{D})$ with square norm given by

$\displaystyle{\left\|{\mathcal{Z}^{\kappa,\rho}_{m,n}}\right\|}_{L^{2,\kappa}_{\rho}(\mathbb{D})}^{2}=\frac{\pi
m!(n+\rho)!\Gamma(m+\kappa+1)}{(m+n+\rho+\kappa+1)\Gamma(n+\rho+\kappa+1)}=:\frac{1}{\gamma^{\kappa,\rho}_{m,n}}$
(3.25)

###### Proof.

We provide explicit computation only when $\rho$ being integer. For the case
of $\rho$ non-integer one can proceeds as for $m=m\wedge(n+\rho)$ and $\rho$
is integer. Thus, let $\rho$ a fixed integer and set

$d_{m,n}^{\rho,\kappa}:=\frac{(\kappa+1)_{m\vee(n+\rho)}(m\wedge(n+\rho))!}{(\kappa+1)_{n+\rho}}.$

Then, from Proposition 3.5 and the use of the polar coordinates
$z=\sqrt{t}e^{i\theta}$; $0\leq t<1$, $0\leq\theta<2\pi$, we get

$\displaystyle I_{m,n,j,k}^{\rho,\kappa}$
$\displaystyle:=\int_{D}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})\overline{\mathcal{Z}^{\kappa,\rho}_{j,k}(z,\overline{z})}|z|^{2\rho}(1-|z|^{2})^{\kappa}d\Lambda(z)$
$\displaystyle=\pi
d_{m,n}^{\rho,\kappa}d_{j,k}^{\rho,\kappa}\left(\int_{0}^{1}t^{|m-n-\rho|}(1-t)^{\kappa}P^{(\kappa,|m-n-\rho|)}_{m\wedge(n+\rho)}(2t-1)P^{(\kappa,|m-n-\rho|)}_{j\wedge(k+\rho)}(2t-1)dt\right)\delta_{n-m,k-j}$
$\displaystyle=\frac{\pi
d_{m,n}^{\rho,\kappa}d_{j,k}^{\rho,\kappa}}{2^{|m-n-\rho|+\kappa+1}}\left(\int_{-1}^{1}(1+x)^{|m-n-\rho|}(1-x)^{\kappa}P^{(\kappa,|m-n-\rho|)}_{m\wedge(n+\rho)}(x)P^{(\kappa,|m-n-\rho|)}_{j\wedge(k+\rho)}(x)dx\right)\delta_{n-m,k-j}.$

Now, by the orthogonal property for the classical Jacobi polynomials [22,
p.212], it follows

$\displaystyle I_{m,n,j,k}^{\rho,\kappa}$ $\displaystyle=\frac{\pi
d_{m,n}^{\rho,\kappa}d_{j,k}^{\rho,\kappa}}{2^{|m-n-\rho|+\kappa+1}}{\left\|{P^{(\kappa,|m-n-\rho|)}_{m\wedge(n+\rho)}}\right\|}^{2}\delta_{n-m,k-j}\delta_{m\wedge(n+\rho),j\wedge(k+\rho)}$
$\displaystyle=\frac{\pi
m!(n+\rho)!\Gamma(m+\kappa+1)}{(m+n+\rho+\kappa+1)\Gamma(n+\rho+\kappa+1)}\delta_{m,j}\delta_{n,k}.$

This proves the orthogonality of $\mathcal{Z}^{\kappa,\rho}_{m,n}$ in the
Hilbert space $L^{2,\kappa}_{\rho}(\mathbb{D})$. ∎

###  3.2 Poly-meromorphy

In analogy with the definition of the polyanalytic functions defined as those
satisfying the Cauchy–Riemann equation
$\partial^{n}/\partial\overline{z}^{n}=0$ one has to suggest the following or
fthe poly-meromorpy [2, p 199].

######  Definition 3.9.

A complex-valued function $f$ on an open set $U$ in the complex plane is said
to be poly-meromorphic of order $n$ (of first kind) if there exist certain
meromorphic functions $\psi_{k}$; $k=0,1,\cdots,n-1$ on $U$ such that

$f(z)=\psi_{0}(z)+\overline{z}\psi_{1}(z)+\cdots+\overline{z}^{n-1}\psi_{n-1}(z).$

The main result in this subsection discusses the regularity of the considered
fractional Zernike functions.

######  Theorem 3.10.

The fractional Zernike functions $\mathcal{Z}^{\kappa,\rho}_{m,n}$ are
polynomials in $z$ and $\overline{z}$ if and only if $\rho=0$ or $m\leq n$.
Alternatively, they are poly-meromorphic functions of order $m$ with $0$ as
unique pole. Its order of multiplicity is given by
$Ord^{\kappa,\rho}_{m,n}=\rho$ for $m>n+\rho$ when $\rho=1,2,\cdots$, and by
$Ord^{\kappa,\rho}_{m,n}=m-n$ for $m>n$ when $\rho>-1$ is non-integer, or when
$n<m\leq n+\rho$ with $\rho=1,2,\cdots$.

###### Proof.

Set

$\displaystyle c^{\kappa,\rho}_{m,n,j}$
$\displaystyle:=\frac{(-1)^{j}m!\Gamma(n+\rho+1)\Gamma(\kappa+m+1)}{j!(m-j)!\Gamma(n+\rho-j+1)\Gamma(\kappa+j+1)}$

and for $p<q\leq m\wedge^{*}(n+\rho)$ consider the quantities

$S^{\kappa,\rho,m,n}_{q,p}:=R^{\kappa,\rho,m,n}_{q}-X^{p-q}R^{\kappa,\rho,m,n}_{p},$

where $R^{\kappa,\rho,m,n}_{p}$ is the polynomial of degree less or equal to
$p$ given by

$\displaystyle R^{\kappa,\rho,m,n}_{p}(X)$
$\displaystyle=\sum_{k=0}^{p}\left(\sum_{j=0}^{p-k}(-1)^{j}\frac{(j+k)!}{j!k!}c^{\kappa,\rho}_{m,n,j+k}\right)X^{p-k}.$
(3.26)

Notice for instance that its constant coefficients is given by
$S^{\kappa,\rho,m,n}_{q,p}(0)=R^{\kappa,\rho,m,n}_{q}(0)=c^{\kappa,\rho}_{m,n,q}.$
Thus, starting from (3.3) we can rewrite
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$ as

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=z^{n-[m\wedge^{*}(n+\rho)]}\overline{z}^{m-[m\wedge^{*}(n+\rho)]}R^{\kappa,\rho,m,n}_{m\wedge^{*}(n+\rho)}(|z|^{2}).$
(3.27)

But, since $c^{\kappa,\rho}_{m,n,p}\neq 0$, it becomes clear from (3.27) that
the functions $\mathcal{Z}^{\kappa,\rho}_{m,n}$ are polynomials if and only if
$\rho=0$ or $m\leq n$ independently of $\rho>-1$ being integer or not. This
assertion is also immediate from Proposition 3.5. Next, using the fact that
$R^{\kappa,\rho,m,n}_{q}=X^{q-p}R^{\kappa,\rho,m,n}_{p}+S^{\kappa,\rho,m,n}_{q,p}$
we obtain

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\overline{z}^{m-n}R^{\kappa,\rho,m,n}_{n}(|z|^{2})+z^{n-[m\wedge^{*}(n+\rho)]}\overline{z}^{m-[m\wedge^{*}(n+\rho)]}S^{\kappa,\rho,m,n}_{m\wedge^{*}(n+\rho),n}(|z|^{2}).$
(3.28)

A meticulous study of the different possible cases of $m$ compared to $n$ and
$n+\rho$ for given $\rho>-1$ leads to

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=\left\\{\begin{array}[]{lllll}\displaystyle
z^{n-m}R^{\kappa,\rho,m,n}_{m}(|z|^{2})&\mbox{if }m\leq n;\,\rho>-1\\\
\displaystyle\overline{z}^{m-n}R^{\kappa,\rho,m,n}_{n}(|z|^{2})&\mbox{if
}m>n;\,\rho=0\\\
\displaystyle\overline{z}^{m-n}R^{\kappa,\rho,m}_{n}(|z|^{2})+\frac{1}{z^{m-n}}S^{\kappa,\rho,m,n}_{m}(|z|^{2})&\mbox{if
}m>n;\,\rho\mbox{ non-integer}\\\ &\mbox{or }n<m\leq
n+\rho;\,\rho=1,2,\cdots\\\
\displaystyle\overline{z}^{m-n}R^{\kappa,\rho,m,n}_{n}(|z|^{2})+\frac{\overline{z}^{m-(n+\rho)}}{z^{\rho}}S^{\kappa,\rho,m,n}_{n+\rho}(|z|^{2})&\mbox{if
}m>n+\rho;\,\rho=1,2,\cdots.\end{array}\right.$

Moreover, this reveals that the regular part of
$\mathcal{Z}^{\kappa,\rho}_{m,n}$ is always a polyanalytic function of order
$m$ and anti-polyanalytic of order $n$. It is given by

$\displaystyle R^{\kappa,\rho}_{m,n}(|z|^{2})$
$\displaystyle=\left\\{\begin{array}[]{ll}z^{n-m}R^{\kappa,\rho,m,n}_{m}(|z|^{2}),&m\leq
n\\\ \overline{z}^{m-n}R^{\kappa,\rho,m,n}_{n}(|z|^{2}),&m\geq
n\end{array}\right.=z^{n-m\wedge n}\overline{z}^{m-m\wedge
n}R^{\kappa,\rho,m,n}_{m\wedge n}(|z|^{2}).$ (3.31)

However, in general $\mathcal{Z}^{\kappa,\rho}_{m,n}$ are poly-meremorphic
with $0$ as the unique pole for $\rho\neq 0$ and $m>n$. Thus, for $m>n+\rho$
with $\rho=1,2,\cdots$ the singular part is clearly given by
$z^{-\rho}\overline{z}^{m-(n+\rho)}S^{\kappa,\rho,m,n}_{n+\rho}(|z|^{2}).$ It
is given by $z^{n-m}S^{\kappa,\rho,m,n}_{m}(|z|^{2})$ whenever $n<m$ and
$\rho>-1$ non-integer or $n<m\leq n+\rho$ when $\rho=1,2,\cdots$. Therefore,
it becomes clear that the multiplicity of the singularity is given by

$\displaystyle Ord^{\kappa,\rho}_{m,n}$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\rho&\mbox{if
}m>n+\rho;\,\rho=1,2,\cdots\\\ \displaystyle{m-n}&\mbox{if }n<m;\,\rho\mbox{
non-integer}\\\ &\mbox{or }n<m\leq
n+\rho;\,\rho=1,2,\cdots.\end{array}\right.$

This completes the proof. ∎

######  Remark 3.11.

The obtained result can justifies somehow the appellation of
$\mathcal{Z}^{\kappa_{m},\rho}_{m,n}$ by fractional Zernike functions.

###  3.3 Differential equations.

In this subsection we are concerned with some second order differential
equations satisfied by the fractional Zernike functions.

######  Theorem 3.12.

Let $m_{\rho}=m+\rho$. Then, the function $\mathcal{Z}^{\kappa,\rho}_{m,n}$ is
solution of

$\displaystyle z^{2}(1-|z|^{2})\frac{\partial^{2}}{\partial
z^{2}}+\left((m_{\rho}-n+1)-(\kappa+m_{\rho}-n+2)|z|^{2}\right)z\frac{\partial}{\partial
z}+n(\kappa+m_{\rho}+1)|z|^{2}=\rho(n-m).$

###### Proof.

Consider the first order differential operator

$\displaystyle\nabla^{\kappa,\rho}(f)(z)=-\left((1-|z|^{2})\frac{\partial}{\partial
z}-\mathcal{Z}^{\kappa,\rho}_{1,0}(z,\overline{z})\right)(f)(z).$ (3.32)

Also, for varying $j=1,2,\cdots,m$ we set

$\displaystyle\nabla^{\kappa,\rho}_{j}(f)$
$\displaystyle:=-z^{-\rho}(1-|z|^{2})^{-\kappa-j+1}\frac{\partial}{\partial
z}\big{(}z^{\rho}(1-|z|^{2})^{\kappa+j}f\big{)},$ (3.33)

so that $\nabla^{\kappa,\rho}(f)(z)=\nabla^{\kappa,\rho}_{1}(f)(z)$.
Successive application of $\nabla^{\kappa,\rho}_{j}$ leads to the operator
$\widetilde{\nabla}^{\kappa,\rho}_{m}:=\nabla^{\kappa,\rho}_{1}\circ\nabla^{\kappa,\rho}_{2}\circ\cdots\circ\nabla^{\kappa,\rho}_{m}$
satisfying
$\widetilde{\nabla}^{\kappa,\rho}_{m+1}=\nabla^{\kappa,\rho}_{1}\circ\widetilde{\nabla}^{\kappa+1,\rho}_{m}$
since $\nabla^{\kappa,\rho}_{j+1}=\nabla^{\kappa+1,\rho}_{j}$. It is
explicitly given by

$\widetilde{\nabla}^{\kappa,\rho}_{m}(f)(z)=(-1)^{m}z^{-\rho}(1-|z|^{2})^{-\kappa}\frac{\partial^{m}}{\partial
z^{m}}\big{(}z^{\rho}(1-|z|^{2})^{\kappa+m}f\big{)}.$ (3.34)

Thus, in view of (1.2) it is clear that
$\widetilde{\nabla}^{\kappa,\rho}_{m}(e_{n})(z)=\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
for $e_{n}(z):=z^{n}$, and therefore

$\displaystyle\nabla^{\kappa,\rho}(\mathcal{Z}^{\kappa+1,\rho}_{m,n}(z,\overline{z}))=\nabla^{\kappa,\rho}_{1}\left(\nabla^{\kappa+1,\rho}_{1}\circ\nabla^{\kappa+1,\rho}_{2}\circ\cdots\circ\nabla^{\kappa+1,\rho}_{m}\right)(e_{n})=\mathcal{Z}^{\kappa,\rho}_{m+1,n}(z,\overline{z}).$
(3.35)

Now associated with the Euler differential operator
$E_{z}=z{\partial}/{\partial z}$ and the constant
$c^{\kappa,\rho}_{m,n}=m(n+\rho+\kappa+1)$, we define the first order
differential operator

$D^{\kappa,\rho}_{m,n}=\frac{1}{c^{\kappa,\rho}_{m,n}\overline{z}}\left(E_{z}-(n-m).\right)=\frac{1}{c^{\kappa,\rho}_{m,n}}\left(\frac{z}{\overline{z}}\frac{\partial}{\partial{z}}-\frac{(n-m)}{\overline{z}}\right).$

Hence making use of (3.17) combined with the differentiation formula of Jacobi
polynomials in [22, p.213 ] we obtain the identity

$D^{\kappa,\rho}_{m,n}(\mathcal{Z}^{\kappa,\rho}_{m,n})=\mathcal{Z}^{\kappa+1,\rho}_{m-1,n}.$
(3.36)

Therefore, from (3.35) and (3.36) it is immediate that
$\nabla^{\kappa,\rho}\circ
D^{\kappa,\rho}_{m,n}(\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z}))=\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$.
A direct computation shows that
$-c^{\kappa,\rho}_{m,n}\overline{z}\nabla^{\kappa,\rho}\circ
D^{\kappa,\rho}_{m,n}$ is given by

$\displaystyle z(1-|z|^{2})\frac{\partial^{2}}{\partial
z^{2}}+\left([m_{\rho}-n+1]-[\kappa+m_{\rho}-n+2]|z|^{2}\right)\frac{\partial}{\partial
z}+(m-n)\left(\frac{\rho}{z}-[\kappa+\rho+1]\overline{z}\right)$

This shows that the fractional Zernike function
$\mathcal{Z}^{\kappa,\rho}_{m,n}$ satisfy the desired differential equation. ∎

######  Remark 3.13.

In view of (3.35) and (3.36) the considered operators $\nabla^{\kappa,\rho}$
and $D^{\kappa,\rho}_{m,n}$ appear as creation and annihilation operators for
the fractional Zernike functions.

######  Remark 3.14.

Let $(P_{j}f)(z)=z^{j}f(z)$. Then, the commutation relation
$P_{j}\circ\nabla^{\kappa,\rho}_{m}\circ
P_{j}^{-1}=\nabla^{\kappa,\rho-j}_{m}$ holds for all $z\in\mathbb{D}^{*}$.
This follows by observing that from (1.2), we have

$\displaystyle
z^{j}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=\mathcal{Z}^{\kappa,\rho-j}_{m,n+j}(z,\overline{z}).$
(3.37)

The following can be proved using the close connection of
$\mathcal{Z}^{\kappa_{m},\rho}_{m,n}$ to the $\beta$-restricted Zernike
functions studied in Section 2.

######  Theorem 3.15.

For given fixed nonnegative integer $m$ and reals $\alpha>-1$ and $\gamma$
such that $\kappa_{m}=\alpha-2(\gamma+m)-1$, the fractional Zernike functions
$\mathcal{Z}^{\kappa_{m},\rho}_{m,n}$ for varying $n$ are eigenfunctions of

$\displaystyle-(1-|z|^{2})^{2}\partial\overline{\partial}-(1-|z|^{2})\left(mE-
H_{\kappa_{m}+m+1}^{\rho}(z)\overline{E}\right)+mH_{\kappa_{m}+m+1}^{\rho}(z)|z|^{2}$
(3.38)

with $m(\kappa_{m}+m+1)$ as corresponding eigenvalue.

###### Proof.

For the proof observe that the fractional Zernike functions
$\mathcal{Z}^{\kappa_{m},\rho}_{m,n}$ are closely connected to the
$\beta$-restricted Zernike functions
$\psi^{\gamma,\eta}_{m,n}(z,\overline{z})$ by (2.1) for every fixed
nonnegative integer $m$. The latter ones are eigenfunctions of the hamiltonian
$\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ in (2.7) with
$E^{\gamma,\alpha}_{m}=(m+1)(\alpha-2\gamma-m)$ as corresponding eigenvalue
(see $(i)$ in Theorem 2.6). The key observation to conclude is that the
operators $\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}$ and
$\mathcal{L}_{\gamma+a,\eta+b}^{\alpha+2a,\beta+2b,+}$ are unitary equivalent
for arbitrary reals $a$ and $b$. More precisely, since
$A^{*_{\alpha,\beta}}_{\gamma,\eta}(h_{a,b}f)=h_{a,b}A^{*_{\alpha+2a,\beta+2b}}_{\gamma+a,\eta+b}(f)$
and $A_{\gamma,\eta}(h_{a,b}f)=h_{a,b}A_{\gamma+a,\eta+b}(f)$, we obtains

$\displaystyle\mathcal{L}_{\gamma,\eta}^{\alpha,\beta,+}(h_{a,b}f)$
$\displaystyle=A_{\gamma,\eta}A^{*_{\alpha,\beta}}_{\gamma,\eta}(h_{a,b}f)=h_{a,b}A_{\gamma+a,\eta+b}A^{*_{\alpha+2a,\beta+2b}}_{\gamma+a,\eta+b}(f)=h_{a,b}\mathcal{L}_{\gamma+a,\eta+b}^{\alpha+2a,\beta+2b,+}(f).$

Subsequently, for $\kappa_{m}=\alpha-2(\gamma+m)-1$, $\rho=\beta-2\eta$,
$b=-\eta$ and $a={(\kappa_{m}-\alpha-1)}/{2}=-(\gamma+m+1)$, the fractional
Zernike function $\mathcal{Z}^{\kappa_{m},\rho}_{m,n}$ satisfies

$\mathcal{L}_{-m-1,0}^{\kappa_{m}-1,\rho,+}\mathcal{Z}^{\kappa_{m},\rho}_{m,n}=E^{\gamma,\alpha}_{m}\mathcal{Z}^{\kappa_{m},\rho}_{m,n}.$

But from Lemma 2.1, it is clear that the second order partial differential
equation in (3.38) is exactly
$\mathcal{L}_{-m-1,0}^{\kappa_{m}-1,\rho,+}-(\kappa_{m}+m+1)$. ∎

######  Corollary 3.16.

The Zernike polynomials $\mathcal{Z}^{(-1)}_{m,n}$, corresponding to the limit
case of $\kappa_{m}=-1$ and fixed $m$, are harmonic functions for the
Laplacian

$\left\\{(1-|z|^{2})\partial\overline{\partial}+m\left(E-\overline{E}\right)-m^{2}\right\\}\mathcal{Z}^{-1}_{m,n}.$

###### Proof.

This readily follows by specifying $\rho=0$ in Theorem 3.15 and choosing
$\alpha$ and $\gamma$ such that $m=(\alpha/2)-\gamma$. Indeed, in this case we
have $\kappa_{m}+m+1=m$ and the left hand side of (3.38) reduces further to
the Landau Hamiltonian
$(1-|z|^{2})\left\\{(1-|z|^{2})\partial\overline{\partial}+m\left(E-\overline{E}\right)\right\\}+m^{2}|z|^{2}$
with quantized constant magnetic field of magnitude $m$. ∎

###  3.4 Recurrence and operational formulas.

From the three terms recurrence formula in [22, p: 213] for the Jacobi
polynomials one can deduces

$\displaystyle
A_{m,b}z^{2}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})+B_{m,b}z\mathcal{Z}^{\kappa,\rho-1}_{m-1,n}(z,\overline{z})+C_{m,b}\mathcal{Z}^{\kappa,\rho-2}_{m-2,n}(z,\overline{z})=0$

valid for all $m=2,3,4,\cdots$, where we have set $A_{m,b}:=(b-m)(b+2)$,
$B_{m,b}:=b(b-1)(b-\kappa-1)$ and $C_{m,b}:=b(m-1)(\kappa+m-1)(b-\kappa-m)$
with $b=\kappa+n+m+\rho$. However, starting from the Rodriguez formula for the
fractional Zernike functions by rewriting it in the form

$\mathcal{Z}^{\kappa,\rho}_{m,n}=(-1)^{m}z^{-\rho}(1-|z|^{2})^{-\kappa}\partial_{z}^{m-1}\left(\partial_{z}(z^{n+\rho}(1-|z|^{2})^{\kappa+m})\right),$

one derives the recurrence formula

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=(\kappa+m)\overline{z}\mathcal{Z}^{\kappa,\rho}_{m-1,n}(z,\overline{z})-(n+\rho)(1-|z|^{2})\mathcal{Z}^{\kappa+1,\rho}_{m-1,n-1}(z,\overline{z}).$
(3.39)

But, by means of (3.37) we can rewrite the recurrence formula (3.39) as

$\displaystyle
z\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=(\kappa+m)\overline{z}\mathcal{Z}^{\kappa,\rho-1}_{m-1,n+1}(z,\overline{z})-(n+\rho)(1-|z|^{2})\mathcal{Z}^{\kappa+1,\rho-1}_{m-1,n}(z,\overline{z}).$
(3.40)

Moreover, we can prove the following

$\displaystyle\mathcal{Z}^{\kappa-1,\rho-1}_{m+1,n+1}$
$\displaystyle=[\kappa|z|^{2}+(m-n-\rho)(1-|z|^{2})]\mathcal{Z}^{\kappa,\rho}_{m,n}-m(n+\rho+\kappa+1)\overline{z}(1-|z|^{2})\mathcal{Z}^{\kappa+1,\rho}_{m-1,n}.$
(3.41)

Indeed, it follows from the use (3.36), leading to

$z\frac{\partial}{\partial_{z}}(\mathcal{Z}^{\kappa,\rho}_{m,n})-(n-m)\mathcal{Z}^{\kappa,\rho}_{m,n}=\overline{z}m(n+\rho+\kappa+1)\mathcal{Z}^{\kappa+1,\rho}_{m-1,n},$

combined with the derivation formula

$(1-|z|^{2})\frac{\partial}{\partial_{z}}(\mathcal{Z}^{\kappa,\rho}_{m,n})=-\rho(1-|z|^{2})\mathcal{Z}^{\kappa,\rho+1}_{m,n-1}+\kappa\overline{z}\mathcal{Z}^{\kappa,\rho}_{m,n}-\mathcal{Z}^{\kappa-1,\rho}_{m+1,n}.$

The latter one follows from the Rodrigues Formula.(1.2) In the sequel, we
obtain non-trivial recurrence formulas of Nielsen type for the fractional
Zernike functions. This follows as specific cases of the so-called Burchnall
representation type formulas for the fractional Zernike functions. To the
exact statement we let $\mathbb{a}^{\kappa,\rho,m,n}_{q,j,\ell,k}$ and
$\mathbb{b}^{\kappa,\rho}_{m,n,j,k}$ respectively, be the constants given by

$\displaystyle\mathbb{a}^{\kappa,\rho,m,n}_{q,j,\ell,k}:=\varepsilon^{*}_{\rho,m-j}\frac{(-1)^{m+j+\ell}m!q!\Gamma(\rho+1)\Gamma(\kappa+m+1)}{\ell!(m-j)!(j-\ell)!(q-\ell)!\Gamma(\rho-m+j+1)\Gamma(\kappa+m+n+1)}$
(3.42)

and

$\displaystyle\mathbb{b}^{\kappa,\rho,m,n}_{j,k}:=\frac{(-1)^{j+k}m!n!\Gamma(\kappa+m+n+1)}{j!k!(m-j)!(n-k)!\Gamma(\kappa+m+k+1)}.$
(3.43)

######  Proposition 3.17.

Let $\kappa,\rho,m$ and $n$ be as above. Let $p$ be a nonnegative integer and
$u$ a real such that $u\geq\max(-\kappa,-1)$. Then, we have

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n+q}(z,\overline{z})$
$\displaystyle=z^{q}\sum_{j=0}^{m}\sum_{\ell=0}^{j\wedge
q}\mathbb{a}^{\kappa,\rho,m,n}_{q,j,\ell,k}\left(\frac{1-|z|^{2}}{z}\right)^{m+\ell-j}\mathcal{Z}^{\kappa+m+\ell-j}_{j-\ell,n}(z,\overline{z}),$
(3.44) $\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n+q}(z,\overline{z})$
$\displaystyle=\frac{m!q!\Gamma(\kappa+m+1)}{\Gamma(\kappa+m+n+1)}z^{q}\sum_{j=0}^{m\wedge
q}\sum_{k=0}^{n}\frac{(-1)^{j}}{j!(m-j)!(q-j)!}\left(\frac{1-|z|^{2}}{z}\right)^{j}\mathcal{Z}^{\kappa+j,\rho}_{m-j,n}(z,\overline{z})$
(3.45)

and

$\displaystyle\mathcal{Z}^{\kappa+u,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\frac{\Gamma(\kappa+u+m+1)}{\Gamma(\kappa+u+m+n+1)}\sum_{j=0}^{m}\sum_{k=0}^{n}(-1)^{j+k}\mathbb{b}^{\kappa,\rho,m,n}_{j,k}\mathcal{Z}^{u-j-k}_{j,k}(z,\overline{z}).$
(3.46)

###### Proof.

Considering the operator

$\displaystyle B^{\kappa,\rho}_{m,n}(f)$
$\displaystyle=\frac{\partial^{m}}{\partial{z}^{m}}\left(z^{\rho}\frac{\partial^{n}}{\partial{\overline{z}}^{n}}\left((1-|z|^{2})^{\kappa+m+n}f\right)\right),$
(3.47)

for every sufficiently differentiable function $f$. Making use of the Leibnitz
formula applied from outside to inside we arrive at the Burchnall type formula

$\displaystyle B^{\kappa,\rho}_{m,n}(f)$
$\displaystyle=\sum_{j=0}^{m}\binom{m}{j}\frac{\partial^{m-j}}{\partial{z^{m-j}}}(z^{\rho})\frac{\partial^{j}}{\partial{z^{j}}}\left(\frac{\partial^{n}}{\partial{\overline{z}^{n}}}[(1-|z|^{2})^{\kappa+m+n}f]\right)$
(3.48)
$\displaystyle=z^{\rho}(1-|z|^{2})^{\kappa+m}\sum_{j=0}^{m}\sum_{\ell=0}^{j}\sum_{k=0}^{n}\mathbb{a}^{\kappa,\rho}_{m,n,j,\ell,k}z^{j-m}(1-|z|^{2})^{k+\ell-j}\mathcal{Z}^{\kappa+m+k+\ell-j}_{j-\ell,n-k}(z,\overline{z})\frac{\partial^{\ell+k}}{\partial
z^{\ell}\partial\overline{z}^{k}}(f),$

where the involved constant is given by

$\mathbb{a}^{\kappa,\rho}_{m,n,j,\ell,k}:=(-1)^{m+n+k}\frac{n!(q-\ell)!\Gamma(\kappa+m+n+1)}{q!k!(n-k)!\Gamma(\kappa+m+1)}\mathbb{a}^{\kappa,\rho,m,n}_{q,j,\ell}.$

Similarly we get (by Leibnitz formula from inside to outside)

$\displaystyle B^{\kappa,\rho}_{m,n}(f)$
$\displaystyle=\frac{\partial^{m}}{\partial{z^{m}}}\left(z^{\rho}\left[\sum_{k=0}^{n}\binom{n}{k}\frac{\partial^{n-k}}{\partial{\overline{z}^{n-k}}}\left((1-|z|^{2})^{\kappa+m+n}\right)\frac{\partial^{k}}{\partial{\overline{z}^{k}}}(f)\right]\right)$
$\displaystyle=(-1)^{m+n}z^{\rho}\sum_{j=0}^{m}\sum_{k=0}^{n}\mathbb{b}^{\kappa,\rho}_{m,n,j,k}(1-|z|^{2})^{\kappa+j+k}\mathcal{Z}^{\kappa+j+k,\rho}_{m-j,n-k}(z,\overline{z})\frac{\partial^{j+k}}{\partial
z^{j}\partial\overline{z}^{k}}(f).$ (3.49)

Therefore, since the action of $B^{\kappa,\rho}_{m,n}$ on the specific case of
$f=e_{q}=z^{q}$ reduces to

$\displaystyle B^{\kappa,\rho}_{m,n}(z^{q})$
$\displaystyle=(-1)^{m+n}(\kappa+m+1)_{n}z^{\rho}(1-|z|^{2})^{\kappa}\mathcal{Z}^{\kappa,\rho}_{m,n+q}(z,\overline{z}),$

we obtain (3.44) (resp. (3.45)) from (3.48) (resp. (3.49)). The identity
(3.46) follows from (3.49) by considering the particular case of
$f(z)=(1-|z|^{2})^{u}$ and observing that
$B^{\kappa,\rho}_{m,n}((1-|z|^{2})^{u}g)=B^{\kappa+u,\rho}_{m,n}(g)$. ∎

######  Remark 3.18.

The Burchnall representation in (3.49) for $f$ being an holomorphic function
simply reads

$\displaystyle B^{\kappa,\rho}_{m,n}(f)$
$\displaystyle=(-1)^{m+n}z^{\rho}h^{\kappa}\sum_{j=0}^{m}\mathbb{b}^{\kappa,\rho}_{m,n,j,0}(1-|z|^{2})^{j}\mathcal{Z}^{\kappa+j,\rho}_{m-j,n}(z,\overline{z})\frac{\partial^{j}(f)}{\partial
z^{j}}.$ (3.50)

Analog representation for arbitrary $f$ (not necessary holomorphic) can be
developed using the differential operator

$A^{\rho,\kappa}_{m,n}(f)=\frac{\partial^{m}}{\partial{z}^{m}}\left(z^{n+\rho}(1-|z|^{2})^{\kappa+m}f\right).$
(3.51)

More precisely, one obtains

$\displaystyle A^{\rho,\kappa}_{m,n}(f)$
$\displaystyle=(-1)^{m}m!z^{\rho}h^{\kappa}\sum_{j=0}^{m}\frac{(-1)^{j}(1-|z|^{2})^{j}}{j!(m-j)!}\mathcal{Z}^{\kappa+j,\rho}_{m-j,n}(z,\overline{z})\frac{\partial^{j}f}{\partial{z^{j}}}.$
(3.52)

###  3.5 Generating and bilinear generating functions.

The aim here is to obtain some generating and bilinear generating functions
for the fractional Zernike functions. First, it is worth noting that from
Proposition 3.5 and making use of the generating function for the Jacobi
polynomials in [22, p: 213] we obtain the generating function

$\sum_{m=0}^{+\infty}\frac{u^{m}}{m!}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})=2^{n}z^{2n+1-m+\rho+\kappa}\frac{(z-u+R(u,z))^{m-n-\rho}(z+u+R(u,z))^{-\kappa}}{R(u,z)}.$

Here $R(u,z)=1$ for $u=0$ and
$R(u,z)=(z^{2}+2uz(1-2|z|^{2})+z^{2}(1-2|z|^{2})^{2})^{1/2}$ when $u\neq 0$.

The next result gives the expression of special bilinear generating functions
as derivative of the confluent and Gauss hypergeometric functions by means of
the partial differential operator

$\displaystyle R_{m}^{\kappa,\rho}f(z)$
$\displaystyle=\frac{1}{(z\overline{w})^{\rho}(1-|z|^{2})^{\kappa}(1-|w|^{2})^{\kappa}}\frac{\partial^{2m}}{\partial
z^{m}\partial\overline{w}^{m}}\left((z\overline{w})^{\rho}(1-|z|^{2})^{\kappa+m}(1-|w|^{2})^{\kappa+m}f\right)(z)$

for sufficiently differential function $f$.

######  Proposition 3.19.

We have

$\displaystyle\sum_{n=0}^{+\infty}\frac{(a)_{n}}{n!(c)_{n}}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})\overline{\mathcal{Z}^{\kappa,\rho}_{m,n}(w,\overline{w})}$
$\displaystyle=R_{m}^{\kappa,\rho}\left({{}_{1}F_{1}}\left(\begin{array}[]{c}a\\\
c\end{array}\bigg{|}z\overline{w}\right)\right)$ (3.55)

and

$\displaystyle\sum_{n=0}^{+\infty}\frac{(a)_{n}(b)_{n}}{n!(c)_{n}}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})\overline{\mathcal{Z}^{\kappa,\rho}_{m,n}(w,\overline{w})}$
$\displaystyle=R_{m}^{\kappa,\rho}\left({{}_{2}F_{1}}\left(\begin{array}[]{c}a,b\\\
c\end{array}\bigg{|}z\overline{w}\right)\right).$ (3.58)

###### Proof.

This readily follows by means of the Rodrigues’ formula for
$\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$. Indeed we can rewrite the
left-hand side in (3.55) as

$\displaystyle\frac{1}{(z\overline{w})^{\rho}(1-|z|^{2})^{\kappa}(1-|w|^{2})^{\kappa}}\frac{\partial^{2m}}{\partial
z^{m}\partial\overline{w}^{m}}\left((z\overline{w})^{\rho}[(1-|z|^{2})(1-|w|^{2})]^{\kappa+m}\sum_{n=0}^{+\infty}\frac{(a)_{n}}{(c)_{n}}\frac{z^{n}}{n!}\right),$

which reduces further to (3.55). The formula (3.58) follows in a similar way.
∎

######  Remark 3.20.

For the special values of $a=1$, $b=\kappa+\rho+1$ and $c=\rho+1$ with
$\rho=\beta-2\eta$ and $\kappa=\kappa_{m}=\alpha-2(\gamma+m)-1$, the quantity
${n!(c)_{n}}/{(a)_{n}(b)_{n}}$ reduces to be the square norm of
$\psi^{\gamma,\eta}_{m,n}$ in (2.12) up to a multiplicative constant
$d^{\kappa,\rho}_{m}$ independent of $n$. Thus, formula in (3.58) leads to the
reproducing kernel

$\displaystyle K_{\gamma,\eta,m}^{\alpha,\beta}(z,w)$
$\displaystyle=\sum_{n=0}^{+\infty}\frac{\psi^{\gamma,\eta}_{m,n}(z,\overline{z})\overline{\psi^{\gamma,\eta}_{m,n}(w,\overline{w})}}{{\left\|{\psi^{\gamma,\eta}_{m,n}}\right\|}_{\alpha,\beta}^{2}}$

of the $m$-th generalized $\beta$-modified Bergman space introduced in Remark
2.8. In fact, we have

$\displaystyle K_{\gamma,\eta,m}^{\alpha,\beta}(z,w)$
$\displaystyle=d^{\kappa,\rho}_{m}\frac{[(1-|z|^{2})(1-|w|^{2})]^{\gamma+m}}{|zw|^{2\eta}}R_{m}^{\kappa,\rho}\left({{}_{2}F_{1}}\left(\begin{array}[]{c}a,b\\\
c\end{array}\bigg{|}z\overline{w}\right)\right).$

A closed formula for $K_{\gamma,\eta,m}^{\alpha,\beta}(z,w)$ needs further
investigation.

Below, we prove a bilinear generating function for the fractional Zernike
function that looks like the Hardy–Hille formula for the generalized Laguerre
polynomials. Thus, we deal with

$\displaystyle G^{\kappa,\rho}_{n}(z,w|t)$
$\displaystyle:=\sum_{m=0}^{+\infty}\frac{t^{m}\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})\mathcal{Z}^{\kappa,\rho}_{m,n}(\overline{w},w)}{m!(\kappa+1)_{m}}.$
(3.59)

######  Proposition 3.21.

For every sufficiently small $t$, the closed expression of
$G^{\kappa,\rho}_{n}(z,w|t)$ is given

$\displaystyle G^{\kappa,\rho}_{n}(z,w|t)$
$\displaystyle=\frac{(z+tw)^{n+\rho}(\overline{w}+t\overline{z})^{n+\rho}}{(z\overline{w})^{\rho}(1+t\overline{z}w)^{2(n+\rho)+\kappa+1)}}{{}_{2}F_{1}}\left(\begin{array}[]{c}-n-\rho,-n-\rho\\\
\kappa+1\end{array}\bigg{|}-\frac{t(1-|z|^{2})(1-|w|^{2})}{(z+tw)(\overline{w}+t\overline{z})}\right).$

###### Proof.

Using the hypergeometric representation (Proposition 3.5) we can rewrite
$G^{\kappa,\rho}_{n}(z,w|t)$ as

$\displaystyle G^{\kappa,\rho}_{n}(z,w|t)$
$\displaystyle=(z\overline{w})^{n}\sum_{m=0}^{+\infty}\frac{(\kappa+1)_{m}}{m!}(t\overline{z}w)^{m}$
$\displaystyle{{}_{2}F_{1}}\left(\begin{array}[]{c}-m,-n-\rho\\\
\kappa+1\end{array}\bigg{|}1-\frac{1}{|z|^{2}}\right){{}_{2}F_{1}}\left(\begin{array}[]{c}-m,-n-\rho\\\
\kappa+1\end{array}\bigg{|}1-\frac{1}{|w|^{2}}\right).$

Thus, one concludes for the result in Proposition 3.21 by making use of the
Meixner bilinear relation in [21, Eq. (12), p. 85]. ∎

###  3.6 Integral representations

By means of the classical integral representations for the Gauss
hypergeometric functions in the right hand side of (3.10) and (3.15) or for
the Jacobi polynomials in (3.17), we can derive different integral
representations for $\mathcal{Z}_{m,n}^{\rho,\kappa}(z,\bar{z})$. However, we
give below some non-trivial ones. The first one is based on the Cauchy
integral formula for holomorphic functions and following in spirit Kazantsev
and Bukhgeimwe idea [17].

######  Theorem 3.22.

The fractional Zernike functions $\mathcal{Z}^{\kappa,\rho}_{m,n}$ admits the
following integral representation

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=\dfrac{(-1)^{m}m!}{2\pi
i}z^{-\rho}(1-|z|^{2})^{-\kappa}\oint_{\mid
t\mid=1}t^{n+m+\rho+\kappa}\dfrac{(\overline{t}-\overline{z})^{\kappa+m}}{(t-z)^{m+1}}dt.$
(3.60)

###### Proof.

Make use of the ordinary binomial expansion with the factorial function [27,
p. 56],

$\displaystyle(1-\xi)^{-a}=\sum\limits_{j=0}^{+\infty}{(a)_{j}}\frac{\xi^{j}}{j!},$

to expand the factor $(1-|z|^{2})^{j}$ in the explicit expression of
$\mathcal{Z}^{\kappa_{m},\rho}_{m,n}$ given by (3.3). Also, we need to the
fact that

$\dfrac{\partial^{m}}{\partial z^{m}}(z^{j+n+\rho})=\dfrac{m!}{2\pi
i}\oint_{\mid t\mid=1}\dfrac{t^{j+n+\rho}}{(t-z)^{m+1}}dt,$

which follows from the Cauchy integral formula applied to the function
$\varphi_{z}(t)=t^{m+j+\rho}/(t-z)$. Thus, we obtain

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(z,\overline{z})$
$\displaystyle=(-1)^{m}z^{-\rho}(1-|z|^{2})^{-\kappa}\sum\limits_{j=0}^{+\infty}\dfrac{(-\kappa-m)_{j}}{j!}\overline{z}^{j}\frac{\partial^{m}}{\partial
z^{m}}\left(z^{n+\rho+j}\right)$ $\displaystyle=\dfrac{(-1)^{m}m!}{2\pi
i}z^{-\rho}(1-|z|^{2})^{-\kappa}\oint_{\mid
t\mid=1}\dfrac{t^{n+\rho}}{(t-z)^{m+1}}\left(\sum\limits_{j=0}^{+\infty}(-\kappa-m)_{j}\dfrac{(t\overline{z})^{j}}{j!}\right)dt$
$\displaystyle=\dfrac{(-1)^{m}m!}{2\pi
i}z^{-\rho}(1-|z|^{2})^{-\kappa}\oint_{\mid
t\mid=1}\dfrac{t^{n+\rho}(1-t\overline{z})^{\kappa+m}}{(t-z)^{m+1}}dt$
$\displaystyle=\dfrac{(-1)^{m}m!}{2\pi
i}z^{-\rho}(1-|z|^{2})^{-\kappa}\oint_{\mid
t\mid=1}t^{n+m+\rho+\kappa}\dfrac{(\overline{t}-\overline{z})^{\kappa+m}}{(t-z)^{m+1}}dt.$

This proves (3.60). ∎

The next integral representation for the fractional Zernike functions appears
as corollary of the bilinear generating function in Proposition 3.21.

######  Proposition 3.23.

Let $\gamma^{\kappa,\rho}_{m,n}$ be as in (3.25). The fractional Zernike
functions have the integral representation

$\displaystyle\mathcal{Z}^{\kappa,\rho}_{m,n}(w,\overline{w})=\frac{m!(\kappa+1)_{m}\gamma^{\kappa,\rho}_{m,n}}{w^{\rho}t^{m}}$
$\displaystyle\int_{D}\Xi^{\kappa,\rho}_{m,n}(z,w|t){{}_{2}F_{1}}\left(\begin{array}[]{c}-m,n+\kappa+\rho+1\\\
\kappa+1\end{array}\bigg{|}1-|z|^{2}\right)$ (3.63)
$\displaystyle\times{{}_{2}F_{1}}\left(\begin{array}[]{c}-n-\rho,-n-\rho\\\
\kappa+1\end{array}\bigg{|}-\frac{t(1-|z|^{2})(1-|w|^{2})}{(w+tz)(\overline{z}+t\overline{w})}\right)d\lambda(z);$
(3.66)

where

$\Xi^{\kappa,\rho}_{m,n}(z,w|t):=\frac{\overline{z}^{m-n-\rho}(w+tz)^{n+\rho}(\overline{z}+t\overline{w})^{n+\rho}}{(1+tz\overline{w})^{2(n+\rho)+\kappa+1}}(1-|z|^{2})^{\kappa}.$

###### Proof.

Starting from the expansion (3.59) one gets

$\mathcal{Z}^{\kappa,\rho}_{m,n}(w,\overline{w})=\frac{m!(\kappa+1)_{m}\gamma^{\kappa,\rho}_{m,n}}{t^{m}}\overline{{\left\langle{G^{\kappa,\rho}_{n}(\cdot,w|t),\mathcal{Z}^{\kappa,\rho}_{m,n}}\right\rangle}_{L^{2,\kappa}_{\rho}(\mathbb{D})}}.$

Next, by the explicit expression of the kernel function
$G^{\kappa,\rho}_{n}(z,w|t)$ given in Proposition 3.21 combined with the
hypergeometric representation in (3.10) we derive the integral representation
(3.63). ∎

######  Remark 3.24.

The integral in the right hand side of (3.63) is a rigid integral on complex
domain $\mathbb{D}$ in the sense that it is nontrivial and can not be reduced
to classical integral on real domains.

###  3.7 Completeness.

Here we discuss the completeness of the fractional Zernike functions in
$L^{2,\kappa}_{\rho}(\mathbb{D})$. Notice for instance that for
$\rho=0,1,2,\cdots$ the functions $z^{\rho}\mathcal{Z}^{\kappa,\rho}_{m,n}$
for varying $m,n+\rho=0,1,2,\cdots$ constitute an orthogonal basis of
$L^{2,\kappa}_{\rho}(\mathbb{D})$ since they are closely connected to the
complex Zernike polynomials $\mathcal{Z}^{\kappa}_{m,n+\rho}$ by (1.3). The
latter ones are known to form an orthogonal basis for the Hilbert space
$L^{2,\kappa}_{0}(\mathbb{D})$ (see e.g., [6, 16]). A direct proof starts from
the observation that each $(\mathcal{Z}^{\kappa,\ell}_{m,n})_{m,n}$ is a
polynomial of exact degrees $m$ in $\overline{z}$ and $n$ in $z$, so that any
$z^{n}\overline{z}^{m}$ can be rewritten as

$\overline{z}^{m}z^{n}=\sum_{j=0}^{m}\sum_{k=0}^{n}a_{j,k}^{m,n}\mathcal{Z}^{\kappa,\ell}_{j,k}(z,\overline{z}).$

The coefficients $a_{j,k}^{m,n}$ are explicit and can be computed by the
formula

$\displaystyle a_{j,k}^{m,n}$
$\displaystyle=\gamma^{\kappa,\rho}_{m,n}\int_{\mathbb{D}}\overline{z}^{m}z^{n}\overline{\mathcal{Z}^{\kappa,\ell}_{j,k}}(z,\overline{z})|z|^{2\rho}(1-|z|^{2})^{\kappa}d\lambda(z),$

where $\gamma^{\kappa,\rho}_{m,n}$ is as in (3.25). In the sequel, we consider
only the case of $\rho$ non-integer $\rho>-1$.

######  Theorem 3.25.

The functions
$\varUpsilon_{m,s}^{\kappa,\rho}:=\overline{z}^{-s/2}\mathcal{Z}_{m,m+s/2}^{\kappa,\rho-s/2}$,
for varying nonnegative integer $m$ and varying integer $s$, form an
orthogonal complete system in $L^{2,\kappa}_{\rho}(\mathbb{D})$.

###### Proof.

The key observation is that for $z=\sqrt{(1+x)/2}\,e^{i\theta}$ with
$x\in[-1,1[$ and $\theta\in[0,2\pi[$ we have

$\varUpsilon_{m,s}^{\kappa,\rho}(z,\overline{z})=\frac{z^{s}}{|z|^{s}}\mathcal{Z}_{m,m}^{\kappa,\rho}(z,\overline{z})=m!e^{is\theta}P_{m}^{(\kappa,\rho)}(x).$

Subsequently, their orthogonality in $L^{2,\kappa}_{\rho}(\mathbb{D})$ readily
follows using the orthogonality of the Jacobi polynomials
$P_{m}^{(\kappa,\rho)}$ in the Hilbert $L^{2}_{\kappa,\rho}$ of square
integrable functions on $[-1,1[$ with respect to the measure
$(1-x)^{\kappa}(1+x)^{\rho}dx$. More exactly we have

$\displaystyle\int_{D}\varUpsilon_{m,s}^{\kappa,\rho}(z,\overline{z})\overline{\varUpsilon_{n,r}^{\kappa,\rho}(z,\overline{z})}|z|^{2\rho}(1-|z|^{2})^{\kappa}d\lambda(z)$
$\displaystyle=\frac{m!n!\pi}{2^{\kappa+\rho+1}}{\left\|{P_{m}^{(\kappa,\rho)}}\right\|}_{L^{2}_{\kappa,\rho}}^{2}\delta_{m,n}\delta_{s,r}.$

For their completeness, let $f\in F^{\perp}$, the orthogonal of
$F=Span\\{\varUpsilon_{m,s}^{\kappa,\rho};m=0,1,2,\cdots,s\in\mathbb{Z}\\}$ in
$L^{2,\kappa}_{\rho}(\mathbb{D})$. Thus, by Proposition 3.5, the assumption
that $f\in F^{\perp}$ becomes equivalent to

$\displaystyle{\left\langle{f,\varUpsilon_{m,s}^{\kappa,\rho}}\right\rangle}_{L^{2,\kappa}_{\rho}}=$
$\displaystyle\frac{m!}{2^{\kappa+\rho+2}}\int_{-1}^{1}\left(1-x\right)^{\kappa/2}\left(1+x\right)^{\rho/2}P_{m}^{(\kappa,\rho)}(x)\hat{f}^{\kappa,\rho}_{s}(x)dx=0$

for every integer $s$ and $m=0,1,2,\cdots$. The involved function is defined
by

$\hat{f}^{\kappa,\rho}_{s}(x):=\left(1-x\right)^{\kappa/2}\left(1+x\right)^{\rho/2}\hat{f}_{s}(x),$

where $\hat{f}_{s}(x)$ denotes the $s$-th Fourier coefficient of the function
$f_{x}:=\theta\longmapsto f(\sqrt{(1+x)/2}\,e^{i\theta})$ for every fixed
$x\in[-1,1[$. Clearly $\hat{f}^{\kappa,\rho}_{s}$ belongs to
$L^{2}([-1,1[;dt)$ since by means of the Cauchy-Schwartz inequality and the
Fubini’s theorem one gets

$\displaystyle\int_{-1}^{1}|\hat{f}^{\kappa,\rho}_{s}(x)|^{2}dx$
$\displaystyle\leq
2\pi\int_{-1}^{1}\left(1-x\right)^{\kappa}\left(1+x\right)^{\rho}\left(\int_{0}^{2\pi}\left|f\left(\sqrt{\frac{1+x}{2}}e^{i\theta}\right)\right|^{2}d\theta\right)dx=2^{\kappa+\rho+3}\pi{\left\|{f}\right\|}_{L^{2,\kappa}_{\rho}}^{2}.$

Therefore, $\hat{f}^{\kappa,\rho}_{s}=0$ a.e on $[-1,1[$ for every
$s\in\mathbb{Z}$ for the functions
$\left(1-x\right)^{\kappa/2}\left(1+x\right)^{\rho/2}P_{m}^{(\kappa,\rho)}$,
$m=0,1,2,\cdots$, being an orthogonal basis of $L^{2}\left([-1,1[,dt\right)$.
This implies in particular that the Fourier transform of $f_{x}\in
L^{2}([0,2\pi[,d\theta)$ satisfies
$\mathcal{F}(f_{x})(\ell)=\hat{f}_{-s}(x)=0$ for every $s\in\mathbb{Z}$ and
every fixed $x\in[-1,1[\setminus N$, where we have set
$N:=\cup_{s}\\{x\in[-1,1[;\,\hat{f}_{s}(x)\neq 0\\}$. This proves that the
function $f_{x}=0$ a.e. on $[0,2\pi[$ for almost every $x\in[-1,1[$.
Therefore, $f$ is a vanishing function almost every $t\in[-1,1[$. This
completes the proof. ∎

######  Corollary 3.26.

The Hilbert spaces
$A^{\kappa,\rho}_{s}(\mathbb{D}):=\overline{Span\\{\overline{z}^{-s/2}\mathcal{Z}_{m,m+s/2}^{\kappa,\rho-s/2};m=0,1,2,\cdots\\}}^{L^{2,\kappa}_{\rho}};\,\,s\in\mathbb{Z}$
defines a Hilbertian orthogonal decomposition of
$L^{2,\kappa}_{\rho}(\mathbb{D})$. Namely, we have

$L^{2,\kappa}_{\rho}(\mathbb{D})=\bigoplus_{s\in\mathbb{Z}}A^{\kappa,\rho}_{s}(\mathbb{D}).$

######  Definition 3.27.

The closed subspace $A^{\kappa,\rho}_{s}(\mathbb{D})$ are called generalized
(poly-meromorphic) Bergman spaces of second kind.

Acknowledgement: The assistance of the members of ”Ahmed Intissar” and
”Analysis, P.D.E. $\&$ Spectral Geometry” seminars is gratefully acknowledged.

Data availability statement: All data generated or analyzed during this study
are included in this article.

Conflict of interest: The authors declare that they have no conflict of
interest. .

## References

  * [1] Aharmim B., El Hamyani A., El Wassouli F., Ghanmi A., Generalized Zernike polynomials: operational formulae and generating functions. Integral Transforms Spec. Funct. 26, no. 6 (2015) 395–410.
  * [2] Balk M.B., Polyanalytic functions and their generalizations, in Complex analysis I: entire and meromorphic functions polyanalytic functions, A.A. Gonchar, V.P. Havin, N.K. Nikolski (eds.), Springer-Verlag Berlin Heidelberg 1997
  * [3] Boussejra A., Intissar A., $L^{2}$-Concrete spectral analysis of the invariant Laplacian $\Delta_{\alpha\beta}$ in the Unit complex ball $B^{n}$. J Func. Anal. 160 (1998) 115-140.
  * [4] Brychkov Y.A., Handbook of special functions. Derivatives, integrals, series and other formulas. CRC Press, Boca Raton, FL, 2008.
  * [5] Comtet A., On the Landau levels on the hyperbolic plane, Ann. Phys. 173 (1987) 185-209.
  * [6] Dunkl C.F., The Poisson kernel for Heisenberg polynomials on the disc. Math. Z. 187 (4) (1984) 527–547.
  * [7] El Harti R., Elkachkouri A., Ghanmi A., Solid Cauchy transform on weighted poly-Bergman spaces. To appear in FILOMAT.
  * [8] El Hamyani A., Ghanmi A., Intissar A., Generalized Zernike polynomials: Integral representation and Cauchy transform. Preprint 2016, arXiv:1605.00281.
  * [9] Ghanmi A., On $L^{2}$-eigenfunctions of twisted Laplacian on curved surfaces and suggested orthogonal polynomials. Oper. Matrices 4, no. 4, (2010) 533–540.
  * [10] Ghanmi A., Intissar A., Asymptotic of complex hyperbolic geometry and $L^{2}$-spectral analysis of Landau-like Hamiltonian. J. Math. Phys. 46, no. 3 (2005) 032107, 26 pp.
  * [11] Ghanmi A., Snoun S., Integral representations of Bargmann type for $\beta$-modified Bergman space on punctured unit. disc. Bull. Malays. Math. Sci. Soc., 45, no. 3, (2022) 1367–1381.
  * [12] Ghiloufi N., Safa S., Zeros of news Bergman kernels. J. Korean Math. Soc., 59, no. 3, (2022) 449–468.
  * [13] Graham C.R., The Dirichlet problem for the Bergman Laplacian I, Comm. Partial Differ. Equations 8 (1983) 433-476.
  * [14] Gray R.W., Dunn C., Thompson K.P., Rolland J.P., An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems. Opt. Express 20 (15) (2012) 16436–16449.
  * [15] Iskander D.R., Collins M.J., Davis B., Optimal modeling of corneal surfaces with Zernike polynomials. IEEE Trans. Biomed. Eng. 48 (1) (2001) 87–95.
  * [16] Kanjin Y., Laguerre and disk polynomial expansions with nonnegative coefficients. J. Fourier Anal. Appl. 3 (2013) 495–513.
  * [17] Kazantsev S.G., Bukhgeim A.A., Singular value decomposition for the 2D fan-beam Radon transform of tensor fields. J. Inverse Ill-Posed Probl. 12, no. 3, (2004) 245–278.
  * [18] Khotanzad A., Hong Y. H., Invariant image recognition by Zernike moments. IEEE Trans. Pattern Anal. Machine Intell. 12 (5), (1990) 489–497.
  * [19] Koornwinder T.H., Two-variable analogues of the classical orthogonal polynomials. Theory and application of special functions, R.A. Askey (ed.), Academic Press, New York, (1975) 435–495.
  * [20] Lakshminarayanan V., Fleck A., Zernike polynomials: a guide. J. Modern Optics, 58 (7) (2011) 545–561
  * [21] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. I. McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.
  * [22] Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical physics. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Volume 52. Springer 1966.
  * [23] Milanetti E., M. Miotto M., Di Rienzo L., Monti M., Gosti G., Ruocco G., 2D Zernike polynomial expansion: Finding the protein-protein binding regions. Computational and Structural Biotechnology Journal, 19, (2021) 29-36
  * [24] Nijboer B. R. A., The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations, Physica 13 (10) (1947) 605–620.
  * [25] NIST digital library of mathematical functions.
  * [26] Noll R. J., Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am. 66 (3) (1976) 207–211.
  * [27] Rainville E.D, Special functions. Chelsea Publishing Co., Bronx, NY, 1971.
  * [28] Wang L., Healey G., Using Zernike moments for the illumination and geometry invariant classification of multispectral texture. IEEE Trans. on Image Process. 7 (2) (1998) 196–203.
  * [29] Winker D. M., Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence. J. Opt. Soc. Am. A 8 (10) (1991) 1568–1574.
  * [30] Wünsche A., Generalized Zernike or disc polynomials. J. Comput. Appl. Math. 174 (1) (2005) 135–163.
  * [31] Zernike F., Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode. Physica, 1, 7-12, (1934) 689–704.
  * [32] Zernike F., Brinkman H.C., Hypersphärishe funktionen und die in sphärischen Bereichen orthogonalen polynome. Proc. Kon. Akad. v. Wet., Amsterdam 38 (1935) 161–170.