New separated polynomial solutions to the Zernikesystem on the unit disk and interbasis expansionGEORGE S. POGOSYAN,1,2,3 KURT BERNARDO WOLF,4,* AND ALEXANDER YAKHNO1

1Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico2Yerevan State University, Yerevan, Armenia3Joint Institute for Nuclear Research, Dubna, Russia4Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico*Corresponding author: bwolf@fis.unam.mx

Received 23May 2017; revised 7 August 2017; accepted 22 August 2017; posted 23 August 2017 (Doc. ID 296673); published 19 September 2017

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on theunit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solutionbases involving Legendre and Gegenbauer polynomials in nonorthogonal coordinates, close to Cartesian ones.We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to beClebsch–Gordan coefficients. © 2017 Optical Society of America

OCIS codes: (000.3860) Mathematical methods in physics; (050.1220) Apertures; (050.5080) Phase shift; (080.1010) Aberrations

(global); (350.7420) Waves.

https://doi.org/10.1364/JOSAA.34.001844

1. INTRODUCTION: THE ZERNIKE SYSTEM

In 1934, Frits Zernike published a paper that gave rise to phase-contrast microscopy [1]. That paper presented a differentialequation of second degree to provide an orthogonal basis ofpolynomial solutions on the unit disk to describe wavefrontaberrations in circular pupils. This basis was also obtainedin Ref. [2] using the Schmidt orthogonalization process, asits authors noted that the reason to set up Zernike’s differentialequation had not been clearly justified. The two-dimensionaldifferential equation in r � �x; y� that Zernike solved is

bZΨ�r�≔ �∇2 − �r · ∇�2 − 2r · ∇�Ψ�r� � −EΨ�r�; (1)

on the unit disk D≔ fjrj ≤ 1g and Ψ�r� ∈ L2�D� (once twoparameters had been fixed by the condition of self-adjointness).The solutions found by Zernike are separable in polar coordi-nates �r;ϕ�, with Jacobi polynomials of degrees nr in the radiusr≔ jrj times trigonometric functions eimϕ in the angle ϕ. Thesolutions are thus classified by �nr ; m�, which add up to non-negative integers n � 2nr � jmj, providing the quantizedeigenvalues En � n�n� 2� for the operator in Eq. (1).

The spectrum �nr ; m� or �n; m� of the Zernike system is ex-actly that of the two-dimensional quantum harmonic oscillator.This evident analogy with the quantum oscillator spectrum hasbeen misleading, however. Two-term raising and lowering op-erators do not exist; only three-term recurrence relations havebeen found [3–7]. Beyond the rotational symmetry that ex-plains the multiplets in fmg, no Lie algebra has been shown

to explain the symmetry hidden in the equal spacing of nfamiliar from the oscillator model.

In Refs. [8,9] we have interpreted Zernike’s Eq. (1) as defin-ing a classical and a quantum system with a nonstandard“Hamiltonian” − 1

2 Z . This turns out to be interesting, becausein the classical system the trajectories turn out to be closedellipses, and in the quantum system, this Hamiltonian partakesin a cubic Higgs superintegrable algebra [10].

The key to solve the system was to perform a “vertical” mapfrom the disk D in r � �x; y� to a half-sphere in three-space~r � �x; y; z�, to be indicated as H� ≔ fj~rj � 1; z ≥ 0g. OnH� the issue of separability of solutions becomes clear: theorthogonal spherical coordinate system �ϑ;φ�, ϑ ∈ �0; 12 π�, φ ∈�−π; π� on H�, projects on the polar coordinates �r;ϕ� of D.But as shown in Fig. 1, the half-sphere can also be covered withother orthogonal and separated coordinate systems (i.e., thosewhose boundary coincides with one fixed coordinate): wherethe coordinate poles are along the x axis and the range of spheri-cal angles is ϑ 0 ∈ �0; π� and φ 0 ∈ �0; π�. Since the poles of thespherical coordinates can lie in any direction of the �x; y� planeand rotated around them, we take the x-axis orientation as rep-resenting the whole class of new solutions, which we identify bythe label II, to distinguish them from Zernike’s polar-separatedsolutions, which will be labeled I.

The coordinate system II is orthogonal on H� but projectson nonorthogonal ones on D; the new separated solutions con-sist of Legendre and Gegenbauer polynomials [9]. Of course,the spectrum fEng in Eq. (1) is the same as in the coordinate

1844 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article

1084-7529/17/101844-05 Journal © 2017 Optical Society of America

system I. Recall also that coordinates that separate a differentialequation lead to extra commuting operators and constants ofthe motion. In this paper, we proceed to find the I–II interbasisexpansions between the original Zernike and the newly foundsolution bases; its compact expression in terms of su(2)Clebsch–Gordan coefficients certainly indicates that some kindof deeper symmetry is at work.

The solutions of the Zernike system [1] in the new coordi-nate system, which we indicate by ϒI and ϒII on H�, and ΨI

and ΨII on D, are succinctly derived and written out inSection 2. In Section 3, we find the overlap between them,add some remarks in the concluding Section 4, and reservefor Appendix A some special-function developments.

2. TWO COORDINATE SYSTEMS, TWOFUNCTION BASES

The Zernike differential Eq. (1) in r � �x; y� on the diskD canbe “elevated” to a differential equation on the half-sphereH� inFig. 1 through first defining the coordinates ~ξ � �ξ1; ξ2; ξ3� by

ξ1 ≔ x; ξ2 ≔ y; ξ3 ≔ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − x2 − y2

q; (2)

then relating the measures of H� and D through

d2S�~ξ� � dξ1dξ2ξ3

� dxdyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − x2 − y2

p � d2rffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − jrj2

p (3)

and the partial derivatives by ∂x � ∂ξ1 − �ξ1∕ξ3�∂ξ3and ∂y � ∂ξ2 − �ξ2∕ξ3�∂ξ3 .A. Map Between H� and D Operators

Due to the change in measure (3), the Zernike operator on D,bZ in Eq. (1), must be subject to a similarity transformation bythe root of the factor between d2S�~ξ� and d2r; thus we definethe Zernike operator on the half-sphereH� and its solutions asbW ≔ �1−2�1∕4bZ �1 − jrj2�−1∕4;

ϒ�~ξ�≔ �1 − jrj2�1∕4Ψ�r�: (4)

In this way the inner products required for functions on thedisk and on the sphere are related by

�Ψ;Ψ 0�D ≔ZDd2rΨ�r��Ψ 0�r�

�ZH�

d2S�~ξ�ϒ�~ξ��ϒ 0�~ξ�≕ �ϒ;ϒ 0�H� : (5)

Perhaps rather surprisingly, the Zernike operator bW in Eq. (4)on ~ξ ∈ H� has the structure of (−2 times) a SchrödingerHamiltonian,

bWϒ�~ξ� ��ΔLB �

ξ21 � ξ224ξ23

� 1

�ϒ�~ξ� � −Eϒ�~ξ�; (6)

which is a sum of the Laplace–Beltrami operatorΔLB � L21 � L22 � L23, where

L1 ≔ ξ3∂ξ2 − ξ2∂ξ3 ; L2 ≔ ξ1∂ξ3 − ξ3∂ξ1 ; L3 ≔ ξ2∂ξ1 − ξ1∂ξ1(7)

are the generators of a formal so(3) Lie algebra. The secondsummand in Eq. (6) represents a radial potential

V W �r�≔ − r2∕8�1 − r2�, which has the form of a repulsiveoscillator constrained to �−1; 1�, whose rather delicateboundary conditions were addressed in Ref. [9].

The coordinates ~ξ can be now expressed in terms of the twomutually orthogonal systems of coordinates on the sphere [11],as shown in Fig. 1:

System I∶ ξ1 � sin ϑ cos φ; ξ2 � sin ϑ sin φ;

ξ3 � cos ϑ; ϑjπ∕20 ; φjπ−π ; (8)

System II∶ ξ1 � cos ϑ 0; ξ2 � sin ϑ 0 cos φ 0;

ξ3 � sin ϑ 0 sin φ 0; ϑ 0jπ0 ; φ 0jπ0 : (9)

In the following, we succinctly give the normalized solutionsfor the differential Eq. (6) in terms of the angles for H� inthe coordinate systems I and II, and their projection as wave-fronts on the disk D of the optical pupil. The spectrum ofquantum numbers that classify each eigenbasis, �n; m� and�n1; n2�, will indeed be formally identical with that of thetwo-dimensional quantum harmonic oscillator in polar andCartesian coordinates, respectively.

B. Solutions in System I [Eq. (8)]

Zernike’s differential Eq. (1) is clearly invariant under rotationsaround the center of the disk, corresponding to rotations of bWin Eq. (6) around the ξ3 axis of the coordinate system (8) on thesphere. Written out in those coordinates, it has the form of aSchrödinger equation,

1

sin ϑ

∂∂ϑ

sin ϑ∂ϒI�ϑ;φ�

∂ϑ� 1

sin2θ

∂2ϒI�ϑ;φ�∂φ2

��E � 1

4tan2ϑ� 1

�ϒI�ϑ;φ� � 0; (10)

Fig. 1. Top row: orthogonal coordinate systems that separate on thehalf-sphere H�. Bottom row: their vertical projection on the disk D.Left: spherical coordinates with their pole at the �z axis; separatedsolutions will be marked by I. Right: spherical coordinates with theirpole along the�x axis, whose solutions are identified by II. The lattermaps on nonorthogonal coordinates on the disk that also separatesolutions of the Zernike equation.

Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A 1845

with a potential V W �ϑ� � − 18 tan

2 ϑ. Clearly this will separateinto a differential equation in φ with a separating constant m2,where m ∈ Z≔ f0;1;2;…g and solutions ∼eimφ. Thisseparation constant then enters into a differential equationin ϑ that also has the form of a one-dimensionalSchrödinger equation with an effective potential of thePöschl–Teller type V I

eff �ϑ� � �m2 − 14�csc2 ϑ − 1

4 sec2ϑ, whose

solutions with the proper boundary conditions at ϑ � 12 π

are Jacobi polynomials.On the half-sphere, the solutions to Eq. (6) are thus

ϒIn;m�ϑ;φ�≔

ffiffiffiffiffiffiffiffiffiffiffin� 1

π

r�sin ϑ�jmj�cos ϑ�1∕2

× P�jmj;0�12�n−jmj�

�cos 2ϑ�eimφ; (11)

where n ∈ Z�0 ≔ f0; 1; 2;…g is the principal quantum number

corresponding to En � n�n� 2� in Eq. (1). The index of theJacobi polynomial is the radial quantum number that countsthe number of radial nodes, nr ≔ 1

2 �n − jmj� ∈ Z�0 . Thus, in

each level n, the range of angular momenta arem ∈ f−n; −n� 2;…; ng. These solutions are orthonormal overthe half-sphere H� under the measure d2SI�ϑ;φ� �sin ϑdϑdφ with the range of the angles �ϑ;φ� given in Eq. (8).

Projected on the disk D in polar coordinates r � �r;ϕ�, theoriginal solutions of Zernike, orthonormal under the innerproduct in Eq. (5), are

ΨIn;m�r;ϕ�≔ �−1�nr

ffiffiffiffiffiffiffiffiffiffiffin� 1

π

rr jmjP�jmj;0�

nr �1 − 2r2�eimϕ; (12)

which are shown in Fig. 2 (top).

C. Solutions in System II [Eq. (9)]

The Zernike differential equation in the form of Eq. (6), afterreplacement of the second coordinate system �ϑ 0;φ 0� in Eq. (9)on the half-sphere, acting on functions separated as

ϒII�ϑ 0;φ 0� � 1ffiffiffiffiffiffiffiffiffiffiffisin ϑ 0p S�ϑ 0�T �φ 0�; (13)

yields a system of two simultaneous differential equationsbound by a separation constant k, whose Pöschl–Teller formis most evident in variables μ � 1

2φ0 and ν � 1

2 ϑ0,

d2T �μ�dμ2

��4k2 � 1

4 sin2 μ� 1

4 cos2 μ

�T �μ� � 0;

d2S�ν�dν2

��4�E � 1� � 1–4 k2

4 sin2 ν� 1–4 k2

4 cos2 ν

�S�ν� � 0: (14)

Finally, as shown in Ref. [8] and determined by the boun-dary conditions, two quantum numbers n1; n2 ∈ Z�

0 are im-posed for the solutions on H�, yielding Gegenbauerpolynomials in cos ϑ 0 and Legendre polynomials in cos φ 0.These are

ϒIIn1 ;n2�ϑ 0;φ 0�≔Cn1 ;n2�sin ϑ 0�n1�1∕2�sin φ 0�1∕2

× Cn1�1n2 �cos ϑ 0�Pn1�cos φ 0�; (15)

with the normalization constant

Cn1 ;n2 ≔ 2n1n1!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2n1 � 1��n1 � n2 � 1�n2!

π�2n1 � n2 � 1�!

s; (16)

where the principal quantum number is n � n1 � n2 ∈ Z�0 ,

and with E � n�n� 2� as before. The orthonormality of thesesolutions is also over the half-sphere H� under the formallysame measure d2SII�ϑ 0;φ 0� � sin ϑ 0dϑ 0dφ 0, where the angleshave the range shown in Eq. (9).

On the disk in Cartesian coordinates r � �x; y�, thesolutions are

ΨIIn1 ;n2�x; y� � Cn1 ;n2�1 − x2�n1∕2Cn1�1

n2 �x�Pn1

�yffiffiffiffiffiffiffiffiffiffiffiffi

1 − x2p

�;

(17)

separated in the nonorthogonal coordinates x and y∕p�1 − x2�,

and normalized under the inner product on D in Eq. (5). Theseare shown in Fig. 2 (bottom).

3. EXPANSION BETWEEN I AND II SOLUTIONS

The two bases of solutions of the Zernike equation in thecoordinate systems I and II on the half-sphere, ϒI

n;m�ϑ;φ�

Fig. 2. Top: the basis of Zernike solutions ΨIn;m�r;ϕ� in Eq. (12),

normalized on the disk and classified by principal and angular momen-tum quantum numbers �n; m�. Since they are complex, we show ReΨI

n;m form ≥ 0 and ImΨIn;m form < 0. Bottom: the new real solutions

ΨIIn1 ;n2 �r;ϕ� in Eq. (17) of Zernike’s equation [Eq. (1)] in the coor-

dinate system II, classified by the quantum numbers �n1; n2� (as ifthey were two-dimensional quantum harmonic oscillator states—which they are not). (Figure by Cristina Salto–Alegre.)

1846 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article

in Eq. (11) and ϒIIn1 ;n2�ϑ 0;φ 0� in Eq. (16), with the same prin-

cipal quantum number n,

n1 � n2 � n � 2nr � jmj ∈ Z�0 ;

nr ; n1; n2 ∈ Z�0 ; m ∈ f−n; −n� 2;…; ng; (18)

whose projections on the disk are shown in Fig. 2, were ar-ranged into pyramids with rungs labeled by n, and containingn� 1 states each. They could be mistakenly seen as indepen-dent su(2) multiplets of spin j � 1

2 n because, as we said above,in system II they are not bases for this Lie algebra. Nevertheless,in each rung n, the two bases must relate through linear com-bination (the notation for the indices of the ϒI-function bases,here �n; m�, is different but equivalent to �nr ; m� used inRef. [9]):

ϒIIn1 ;n2�ϑ 0;φ 0� �

Xnm�−n�2�

W n;mn1 ;n2ϒ

In;m�ϑ;φ�; (19)

wherePn

m�−n�2� indicates that m takes values separated by 2 asin Eq. (18). The relation between the primed and unprimedangles in Eqs. (8) and (9) is

cos ϑ 0 � sin ϑ cos φ; cos φ 0 � sin ϑ sin φffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − sin2ϑ cos2 φ

p : (20)

To find the linear combination coefficients W n;mn1 ;n2 in Eq. (19),

we compute first the relation (19) near to the boundary of thedisk and sphere, at ϑ � 1

2 π − ε for small ε, so that cos ϑ �− sin ε ≈ −ε and sin ϑ � cos ε ≈ 1 − 1

2 ε2. There, Eq. (20)

becomes

cos ϑ 0 ≈ cos φ; sin ϑ 0 ≈ sin φ;

cos φ 0 ≈ cos ε; sin φ 0 ≈ − sin ε∕sin φ: (21)

Hence, when ε → 0 is at the rim of the disk and sphere,after dividing Eq. (19) by

p�−ε� on both sides, this relationreads

Cn1 ;n2�sin φ�n1Cn1�1n2 �cos φ�Pn1�1�

�ffiffiffiffiffiffiffiffiffiffiffin� 1

π

r Xnm�−n�2�

W n;mn1 ;n2P

�jmj;0�nr �−1�eimφ; (22)

with nr � 12 �n − jmj�. Recalling that Pn1�1� � 1 and

P�jmj;0�nr �−1� � �−1�nr , we can now use the orthogonality of

the eimφ functions to express the interbasis coefficients as aFourier integral,

W n;mn1 ;n2 �

�−1�nrCn1 ;n2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ�n� 1�

p Zπ

−πdφ�sin φ�n1

× Cn1�1n2 �cos φ� exp�−imφ�: (23)

The integral Eq. (23) does not appear as such in the standardtables [12]; in Appendix A we derive the result and show that itcan be written in terms of a hypergeometric 3F 2 polynomialwhich is an su(2) Clebsch–Gordan coefficient with a specialstructure,

W n;mn1 ;n2 �

in1�−1��m�jmj�∕2n1!�n1 � n2�!�12 �n1 � n2 � m�

�!�12 �n1 − n2 − m�

�!

×

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2n1 � 1

n2!�2n1 � n2 � 1�!

s(24)

× 3F 2

�−n2; n1 � 1; − 1

2�n1 � n2 � m�

−n1 − n2; 12 �n1 − n2 − m� � 1

����1�� in1�−1��m�jmj�∕2Cn1 ;0

12n;−

12m;

12n;

12m; (25)

where we have used the notation of Varshalovich et al. inRef. [13] that couples the su(2) states �j1; m1� and �j2; m2�to �j;m�, as Cj;m

j1 ;m1;j2 ;m2≡ Cj1 ;j2 ;j

m1 ;m2 ;m ≡ hj1; m1; j2; m2j�j1; j2�j; mi.One can then use the orthonormality properties of the

Clebsch–Gordan coefficients to write the transformationinverse to Eq. (19) as

ϒIn;m�ϑ;φ� �

Xnn1�0

eW n1 ;n2n;m ϒII

n1 ;n2�ϑ 0;φ 0�; (26)

eW n1 ;n2n;m � �−i�n1�−1��m�jmj�∕2Cn1 ;0

12n;−

12m;

12n;

12m

(27)

with n1 � n2 � n. The relation between the unprimed andprimed angles of the coordinate systems I and II is the inverseof Eq. (20), namely,

cos ϑ� sin ϑ 0 sin φ 0; cos φ� cos ϑ 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − sin2ϑ 0 sin2φ 0p : (28)

4. CONCLUDING REMARKS

The new polynomial solutions of the Zernike differentialEq. (1) can be of further use in the treatment of generallyoff-axis wavefront aberrations in circular pupils. While theoriginal basis of Zernike polynomials ΨI

n;m�r;ϕ� serves natu-rally for axis-centered aberrations, the new basis ΨII

n1 ;n2�r;ϕ�in Eq. (17) includes, for n1 � 0, plane wave trains with n2nodes along the x axis of the pupil, which are proportionalto Un2�x�, the Chebyshev polynomials of the second kind.

We find that the Zernike system is also very relevant forstudies of “nonstandard” symmetries described by Higgs alge-bras. While rotations in the basis of spherical harmonics isdetermined through the Wigner-D functions [13] of the rota-tion angles on the sphere, here the boundary conditions of thedisk and sphere allow for only a 1

2 π rotation of the z axis toorientations in the x-y plane, and the basis functions do notrelate through Wigner D-functions, but Clebsch–Gordan co-efficients of a special type. Since the classical and quantumZernike systems analyzed in Refs. [8,9] have several newand exceptional properties, we surmise that certain applicationsmust also be of interest.

APPENDIX A: THE INTERBASIS INTEGRAL ANDCLEBSCH–GORDAN COEFFICIENTS

The integral in Eq. (23) does not seem to be in the literature,although similar integrals appear in a paper of Kildyushov [14]to calculate his three coefficients. Thus, let us solve ab initio,with λ � n1 and ν � n2, integrals of the kind

Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A 1847

I λ;mν ≔Z

π

−πdφ sinλ φCλ�1

ν �cos φ�e−imφ; (A1)

where λ; ν ∈ f0; 1; 2;…g.We write the trigonometric function and the Gegenbauer

polynomial in their Fourier series expansions,

sinλ φ� eiλφ

�2i�λ �1− e−2iφ�λ � 1

�2i�λXλk�0

�−1�kλ!k!�λ− k�!e

i�λ−2k�φ;

(A2)

Cλ�1ν �cos φ� �

Xνl�0

�λ� l�!l !�ν − l�!

�λ� ν − l�!�λ!�2 e−i�ν−2l�φ: (A3)

Substituting these expansions in Eq. (A1), using the orthog-onality of the eiκφ functions and thereby eliminating one of thetwo sums, we find a 3F 2 hypergeometric series for unitargument,

I λ;mν � 2π

�2i�λ�λ� ν�!

ν!

�−1��λ−ν−m�∕2�12 �λ − ν − m�

�!�12 �λ� ν� m�

�!

× 3F 2

−ν; λ� 1; − 1

2 �λ� ν� m�−λ − ν; 1

2 �λ − ν − m� � 11

!: (A4)

Multiplying this by the coefficients Cn1 ;n2 in Eq. (23), onefinds the first expression in Eq. (24).

In order to relate the previous result with the su(2) Clebsch–Gordan coefficients in Eq. (25), we use the formula in [13][Eq. (21), Section 8.2] for the particular case at hand, and arelation between 3F 2-hypergeometric functions,

3F 2

�a;b; cd ; e

����1��Γ�d �Γ�d − a −b�Γ�d − a�Γ�d −b�3F 2

�a;b; e − c

a�b−d �1; e

����1�;(A5)

to write these particular symmetric coefficients as

C γ;0α;−β;α;β �

�2α�!γ!�α� β�!�γ − α − β�!

×

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2γ � 1

�2α − γ�!�2α� γ � 1�!

s

× 3F 2

�−2α� γ; γ � 1; −α − β

−2α; γ − α − β� 1

����1�: (A6)

Finally, upon replacement of α � 12 n � 1

2 �n1 � n2�,β � 1

2m, and γ � n1, the expression (24) reduces to

Eq. (25) times the phase and sign.The interbasis expansion coefficients binding the two bases in

Eq. (19) and Fig. 2 can be seen as �n� 1� × �n� 1�matrices W�n� � kW n;m

n1 ;n2k with composite rows �n1; n2� andcolumns �n; m� for each rung n1 � n2 � n ∈ Z�

0 , on �n� 1�-dimensional column vectors of functions as ϒII�ϑ 0;φ 0� �W�n�ϒI�ϑ;φ�. The elements W n;m

n1 ;n2 in Eq. (25) are the productof phases

in1 ; �−1�12�m�jmj� �

� �−1�m; m > 0;1; m ≤ 0

(A7)

times the special Clebsch–Gordan coefficients Cn1 ;012n;−

12m;

12n;

12m. For

even n and odd n1, Cn1 ;012n;0;

12n;0

� 0.

The Zernike polynomials come in complex conjugate pairs,ϒI

n;m � ϒI�n;−m, while the ϒII

n1 ;n2 ’s are real. The linear combina-tions afforded by the W matrices above indeed yield realfunctions because

Cn1 ;012n;−

12m;

12n;

12m

� �−1�n2Cn1 ;012n;

12m;

12n;−

12m: (A8)

Funding. Dirección General Asuntos del PersonalAcadémico, Universidad Nacional Autónoma de México(DGAGA, UNAM) (“Óptica Matemática” IN101115);Universidad de Guadalajara-Consejo Nacional de Ciencia yTecnología (CONACyT) (PRO-SNI-2017).

Acknowledgment. We thank Prof. Natig M. Atakishiyevfor his interest in the matter of interbasis expansions, andacknowledge the technical help from Guillermo Krötzsch(ICF-UNAM) and Cristina Salto-Alegre with the figures.G. S. P. and A. Y. thank the support of project PRO-SNI-2017 (Universidad de Guadalajara). K. B. W. acknowledgesthe support of UNAM-DGAPA Project Óptica MatemáticaPAPIIT-IN101115.

REFERENCES

1. F. Zernike, “Beugungstheorie des Schneidenverfahrens undseiner verbesserten Form, der Phasenkontrastmethode,” Physica1, 689–704 (1934).

2. A. B. Bhatia and E. Wolf, “On the circle polynomials of Zernike andrelated orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50,40–48 (1954).

3. T. H. Koornwinder, “Two-variable analogues of the classical orthogo-nal polynomials,” in Theory and Application of Special Functions,R. A. Askey, ed. (Academic, 1975), pp. 435–495.

4. E. C. Kintner, “On the mathematical properties of the Zernike polyno-mials,” Opt. Acta 23, 679–680 (1976).

5. A. Wünsche, “Generalized Zernike or disc polynomials,” J. Comput.Appl. Math. 174, 135–163 (2005).

6. B. H. Shakibaei and R. Paramesran, “Recursive formula to com-pute Zernike radial polynomials,” Opt. Lett. 38, 2487–2489(2013).

7. M. E. H. Ismail and R. Zhang, “Classes of bivariate orthogonal poly-nomials,” arXiv:1502.07256 (2016).

8. G. S. Pogosyan, K. B. Wolf, and A. Yakhno, “Superintegrable classicalZernike system,” J. Math. Phys. 58, 072901 (2017).

9. G. S. Pogosyan, C. Salto-Alegre, K. B. Wolf, and A. Yakhno,“Quantum superintegrable Zernike system,” J. Math. Phys. 58,072101 (2017).

10. P. W. Higgs, “Dynamical symmetries in a spherical geometry,” J.Phys. A 12, 309–323 (1979).

11. G. S. Pogosyan, A. N. Sissakian, and P. Winternitz, “Separation ofvariables and Lie algebra contractions: applications to special func-tions,” Phys. Part. Nucl. 33, S123–S144 (2002).

12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series andProducts, 7th ed. (Elsevier, 2007).

13. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonski, QuantumTheory of Angular Momentum (World Scientific, 1988).

14. M. S. Kildyushov, “Hyperspherical functions of ‘three’ type in then-body problem,” J. Nucl. Phys. (Yad. Fiz.) 15, 187–208 (1972)(in Russian).

1848 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article