New Separated Polynomial Solutions to the Zernike System on the Unit Disk and Interbasis Expansion

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

New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion

1,2,3 4, 1 GEORGE S. POGOSYAN, KURT BERNARDO WOLF, * AND ALEXANDER YAKHNO 1Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico 2Yerevan State University, Yerevan, Armenia 3Joint Institute for Nuclear Research, Dubna, Russia 4Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico *Corresponding author: [email protected]

Received 23 May 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 the unit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solution bases 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 be Clebsch–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 to explain the symmetry hidden in the equal spacing of n familiar from the oscillator model. In 1934, Frits Zernike published a paper that gave rise to phase- ’ contrast microscopy [1]. That paper presented a differential In Refs. [8,9] we have interpreted Zernike s Eq. (1) as defining a classical and a quantum system with a nonstandard equation of second degree to provide an orthogonal basis of “ ” − 1 Zˆ polynomial solutions on the unit disk to describe wavefront Hamiltonian 2 . This turns out to be interesting, because aberrations in circular pupils. This basis was also obtained in the classical system the trajectories turn out to be closed in Ref. [2] using the Schmidt orthogonalization process, as ellipses, and in the quantum system, this Hamiltonian partakes ’ in a cubic Higgs superintegrable algebra [10]. its authors noted that the reason to set up Zernike s differential “ ” equation had not been clearly justified. The two-dimensional The key to solve the system was to perform a vertical map D r x;y differential equation in r x;y that Zernike solved is from the disk in to a half-sphere in three-space ~r x;y;z, to be indicated as H ≔ fj~rj1;z ≥ 0g.On 2 2 ZbΨ r ≔ ∇ − r ∇ − 2r ∇ Ψ r −EΨ r ; H the issue of separability of solutions becomes clear: the · · (1) 1 orthogonal spherical coordinate system ϑ; φ, ϑ ∈ 0; 2 π, φ ∈ 2 on the unit disk D ≔ fjrj ≤ 1g and Ψr ∈ L D (once two −π; π on H, projects on the polar coordinates r;ϕ of D. parameters had been fixed by the condition of self-adjointness). But as shown in Fig. 1, the half-sphere can also be covered with The solutions found by Zernike are separable in polar coordi- other orthogonal and separated coordinate systems (i.e., those nates r;ϕ, with Jacobi polynomials of degrees nr in the radius whose boundary coincides with one fixed coordinate): where r ≔ jrj times trigonometric functions eimϕ in the angle ϕ. The the coordinate poles are along the x axis and the range of spheri- 0 0 solutions are thus classified by nr ;m, which add up to non- cal angles is ϑ ∈ 0; π and φ ∈ 0; π. Since the poles of the negative integers n 2nr jmj, providing the quantized spherical coordinates can lie in any direction of the x;y plane eigenvalues En nn 2 for the operator in Eq. (1). and rotated around them, we take the x-axis orientation as rep- The spectrum nr ;m or n; m of the Zernike system is ex- resenting the whole class of new solutions, which we identify by actly that of the two-dimensional quantum harmonic oscillator. the label II, to distinguish them from Zernike’s polar-separated This evident analogy with the quantum oscillator spectrum has solutions, which will be labeled I. H been misleading, however. Two-term raising and lowering op- The coordinate system II is orthogonal on but projects erators do not exist; only three-term recurrence relations have on nonorthogonal ones on D; the new separated solutions con- been found [3–7]. Beyond the rotational symmetry that ex- sist of Legendre and Gegenbauer polynomials [9]. Of course, plains the multiplets in fmg, no Lie algebra has been shown the spectrum fEng in Eq. (1) is the same as in the coordinate

1084-7529/17/101844-05 Journal © 2017 Optical Society of America Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A 1845 system I. Recall also that coordinates that separate a differential equation lead to extra commuting operators and constants of the motion. In this paper, we proceed to find the I–II interbasis expansions between the original Zernike and the newly found solution bases; its compact expression in terms of su(2) Clebsch–Gordan coefficients certainly indicates that some kind of deeper symmetry is at work. The solutions of the Zernike system [1] in the new coordi- I II I nate system, which we indicate by ϒ and ϒ on H, and Ψ and ΨII on D, are succinctly derived and written out in Section 2. In Section 3, we find the overlap between them, add some remarks in the concluding Section 4, and reserve for Appendix A some special-function developments.

2. TWO COORDINATE SYSTEMS, TWO FUNCTION BASES The Zernike differential Eq. (1)inr x;y on the disk D can be “elevated” to a differential equation on the half-sphere H in Fig. 1. Top row: orthogonal coordinate systems that separate on the ~ H D Fig. 1 through first defining the coordinates ξ ξ1; ξ2; ξ3 by half-sphere . Bottom row: their vertical projection on the disk . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Left: spherical coordinates with their pole at the z axis; separated 2 2 ξ1 ≔ x;ξ2 ≔ y; ξ3 ≔ 1 − x − y ; (2) solutions will be marked by I. Right: spherical coordinates with their pole along the x axis, whose solutions are identified by II. The latter H D then relating the measures of and through maps on nonorthogonal coordinates on the disk that also separate dξ dξ dxdy d2r solutions of the Zernike equation. d2S ξ~ 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 (3) ξ3 1 − x − y 1 − jrj 2 2 V W r ≔ − r ∕81 − r , which has the form of a repulsive and the partial derivatives by ∂x ∂ξ − ξ1∕ξ3∂ξ 1 3 oscillator constrained to −1; 1, whose rather delicate and ∂y ∂ξ − ξ2∕ξ3∂ξ . 2 3 boundary conditions were addressed in Ref. [9]. H D ξ~ A. Map Between and Operators The coordinates can be now expressed in terms of the two mutually orthogonal systems of coordinates on the sphere [11], Due to the change in measure (3), the Zernike operator on D, b as shown in Fig. 1: Z in Eq. (1), must be subject to a similarity transformation by 2 ~ 2 the root of the factor between d Sξ and d r; thus we define System I∶ ξ1 sin ϑ cos φ; ξ2 sin ϑ sin φ; the Zernike operator on the half-sphere H and its solutions as π∕2 π ξ3 cos ϑ; ϑj0 ; φj−π; (8) Wb ≔ 1−21∕4Zb1 − jrj2−1∕4; 0 0 0 System II∶ ξ1 cos ϑ ; ξ2 sin ϑ cos φ ; ϒξ~ ≔ 1 − jrj21∕4Ψr: (4) 0 0 0 π 0 π ξ3 sin ϑ sin φ ; ϑ j0; φ j0: (9) In this way the inner products required for functions on the disk and on the sphere are related by In the following, we succinctly give the normalized solutions Z for the differential Eq. (6) in terms of the angles for H in 0 2 0 Ψ; Ψ D ≔ d rΨr Ψ r the coordinate systems I and II, and their projection as wave- D ZD fronts on the disk of the optical pupil. The spectrum of n; m 2 ~ ~ 0 ~ 0 quantum numbers that classify each eigenbasis, and d Sξϒξ ϒ ξ ≕ ϒ; ϒ H : (5) n ;n H 1 2, will indeed be formally identical with that of the two-dimensional quantum harmonic oscillator in polar and Perhaps rather surprisingly, the Zernike operator Wb in Eq. (4) ~ Cartesian coordinates, respectively. on ξ ∈ H has the structure of (−2 times) a Schrödinger Hamiltonian, B. Solutions in System I [Eq. (8)] ξ2 ξ2 ’ b ~ 1 2 ~ ~ Zernike s differential Eq. (1) is clearly invariant under rotations W ϒξ ΔLB 2 1 ϒξ−Eϒξ; (6) Wb 4ξ3 around the center of the disk, corresponding to rotations of in Eq. (6) around the ξ3 axis of the coordinate system (8) on the – which is a sum of the Laplace Beltrami operator sphere. Written out in those coordinates, it has the form of a Δ Lˆ 2 Lˆ 2 Lˆ 2 LB 1 2 3, where Schrödinger equation, Lˆ ≔ ξ ∂ − ξ ∂ ; Lˆ ≔ ξ ∂ − ξ ∂ ; Lˆ ≔ ξ ∂ − ξ ∂ I 2 I 1 3 ξ2 2 ξ3 2 1 ξ3 3 ξ1 3 2 ξ1 1 ξ1 1 ∂ ∂ϒ ϑ; φ 1 ∂ ϒ ϑ; φ sin ϑ (7) sin ϑ ∂ϑ ∂ϑ sin2θ ∂φ2 1 are the generators of a formal so(3) Lie algebra. The second E tan2ϑ 1 ϒIϑ; φ0; (10) summand in Eq. (6) represents a radial potential 4 1846 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article

1 2 with a potential V W ϑ− 8 tan ϑ. Clearly this will separate into a differential equation in φ with a separating constant m2, where m ∈ Z ≔ f0; 1; 2; …g and solutions ∼eimφ. This separation constant then enters into a differential equation in ϑ that also has the form of a one-dimensional Schrödinger equation with an effective potential of the I 2 1 2 1 2 Pöschl–Teller type V eff ϑm − 4csc ϑ − 4 sec ϑ, whose 1 solutions with the proper boundary conditions at ϑ 2 π are Jacobi polynomials. On the half-sphere, the solutions to Eq. (6) are thus rffiffiffiffiffiffiffiffiffiffiffi n 1 ϒI ϑ; φ ≔ sin ϑ jmj cos ϑ 1∕2 n;m π jmj;0 imφ × P1 cos 2ϑe ; (11) 2n−jmj where n ∈ Z0 ≔ f0; 1; 2; …g is the principal quantum number corresponding to E n nn 2 in Eq. (1). The index of the Jacobi polynomial is the radial quantum number that counts 1 the number of radial nodes, nr ≔ 2 n − jmj ∈ Z0 . Thus, in each level n, the range of angular momenta are m ∈ f−n; −n 2; …;ng. These solutions are orthonormal over 2 I the half-sphere H under the measure d S 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, the original solutions of Zernike, orthonormal under the inner product in Eq. (5), are rffiffiffiffiffiffiffiffiffiffiffi n 1 m ;0 I nr jmj j j 2 imϕ Ψn;mr;ϕ ≔ −1 r Pn 1 − 2r e ; (12) π r which are shown in Fig. 2 (top).

I C. Solutions in System II [Eq. (9)] Fig. 2. Top: the basis of Zernike solutions Ψn;mr;ϕ in Eq. (12), normalized on the disk and classified by principal and angular momen- The Zernike differential equation in the form of Eq. (6), after n; m 0 0 tum quantum numbers . Since they are complex, we show Re replacement of the second coordinate system ϑ ; φ in Eq. (9) I I Ψn;m for m ≥ 0 and Im Ψn;m for m<0. Bottom: the new real solutions on the half-sphere, acting on functions separated as ΨII r;ϕ ’ n1;n2 in Eq. (17) of Zernike s equation [Eq. (1)] in the coor- n ;n II 1 dinate system II, classified by the quantum numbers 1 2 (as if ϒ ϑ0; φ0pffiffiffiffiffiffiffiffiffiffiffi Sϑ0T φ0; (13) — sin ϑ0 they were two-dimensional quantum harmonic oscillator states which they are not). (Figure by Cristina Salto–Alegre.) yields a system of two simultaneous differential equations bound by a separation constant k, whose Pöschl–Teller form μ 1 φ0 ν 1 ϑ0 is most evident in variables 2 and 2 , where the principal quantum number is n n1 n2 ∈ Z0 , d2T μ 1 1 2 and with E nn 2 as before. The orthonormality of these 2 4k 2 2 T μ0; dμ 4 sin μ 4 cos μ solutions is also over the half-sphere H under the formally 2 II 0 0 0 0 0 d2S ν 1–4 k2 1–4 k2 same measure d S ϑ ; φ sin ϑ dϑ dφ , where the angles 4 E 1 S ν 0: dν2 4 sin2 ν 4 cos2 ν (14) have the range shown in Eq. (9). On the disk in Cartesian coordinates r x;y, the Finally, as shown in Ref. [8] and determined by the boun- solutions are dary conditions, two quantum numbers n1;n2 ∈ Z0 are im- n 1 y H II 2 n1∕2 1 ffiffiffiffiffiffiffiffiffiffiffiffi posed for the solutions on , yielding Gegenbauer Ψn ;n x;yC n ;n 1 − x C n2 xPn p ; 1 2 1 2 1 1 − x2 polynomials in cos ϑ0 and Legendre polynomials in cos φ0. These are (17) p 2 ϒII ϑ0; φ0 ≔ C sin ϑ0 n11∕2 sin φ0 1∕2 separated in the nonorthogonal coordinates x and y∕ 1 − x , n1;n2 n1;n2 and normalized under the inner product on D in Eq. (5). These n11 C n cos ϑ0 P cos φ0 ; × 2 n1 (15) are shown in Fig. 2 (bottom). with the normalization constant sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3. EXPANSION BETWEEN I AND II SOLUTIONS 2n1 1n1 n2 1n2! C ≔ 2n1 n ! ; n1;n2 1 (16) The two bases of solutions of the Zernike equation in the π2n1 n2 1! I coordinate systems I and II on the half-sphere, ϒn;mϑ; φ Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A 1847

II ϒ ϑ0; φ0 in1 −1 mjmj∕2n ! n n ! in Eq. (11) and n1;n2 in Eq. (16), with the same prin- n;m 1 1 2 W n ;n cipal quantum number n, 1 2 1 1 2 n1 n2 m ! 2 n1 − n2 − m ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 n2 n 2nr jmj ∈ Z0 ; 2n 1 1 nr;n1;n2 ∈ Z0 ;m∈ f−n; −n 2; …;ng; (18) × (24) n2!2n1 n2 1! whose projections on the disk are shown in Fig. 2, were ar- 1 −n2;n1 1; − n1 n2 m ranged into pyramids with rungs labeled by n, and containing F 2 1 n 1 × 3 2 1 states each. They could be mistakenly seen as indepen- −n1 − n2; 2 n1 − n2 − m1 1 dent su(2) multiplets of spin j 2 n because, as we said above, n ;0 in1 −1 mjmj∕2C 1 ; 1n;−1m;1n;1m (25) in system II they are not bases for this Lie algebra. Nevertheless, 2 2 2 2 n in each rung , the two bases must relate through linear com- where we have used the notation of Varshalovich et al. in ϒI bination (the notation for the indices of the -function bases, Ref. [13] that couples the su(2) states j1;m1 and j2;m2 n; m n ;m j;m j1;j2;j here , is different but equivalent to r used in j;m C ≡ C m ;m ;m ≡ j ;m ; j ;m j ;j j; m to ,as j1;m1;j2;m2 1 2 h 1 1 2 2j 1 2 i. Ref. [9]): One can then use the orthonormality properties of the Xn Clebsch–Gordan coefficients to write the transformation ϒII ϑ0; φ0 W n;m ϒI ϑ; φ ; n1;n2 n1;n2 n;m (19) inverse to Eq. (19)as m−n2 Xn I e n1;n2 II 0 0 P ϒn;mϑ; φ W n;m ϒn ;n ϑ ; φ ; (26) n 1 2 m n 0 where m−n2 indicates that takes values separated by 2 as 1 in Eq. (18). The relation between the primed and unprimed n ;n n ;0 We 1 2 −i n1 −1 mjmj∕2C 1 n;m 1n;−1m;1n;1m (27) angles in Eqs. (8) and (9)is 2 2 2 2 sin ϑ sin φ with n1 n2 n. The relation between the unprimed and cos ϑ0 sin ϑ cos φ; cos φ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (20) primed angles of the coordinate systems I and II is the inverse 1 − sin2ϑ cos2 φ of Eq. (20), namely, n;m cos ϑ0 To find the linear combination coefficients W n ;n in Eq. (19), 1 2 cos ϑ sin ϑ0 sin φ0; cos φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: (28) we compute first the relation (19) near to the boundary of the 1 − sin2ϑ0 sin2 φ0 1 disk and sphere, at ϑ 2 π − ε for small ε, so that cos ϑ 1 2 − sin ε ≈ −ε and sin ϑ cos ε ≈ 1 − 2 ε . There, Eq. (20) becomes 4. CONCLUDING REMARKS The new polynomial solutions of the Zernike differential 0 0 cos ϑ ≈ cos φ; sin ϑ ≈ sin φ; Eq. (1) can be of further use in the treatment of generally cos φ0 ≈ cos ε; sin φ0 ≈ − sin ε∕sin φ: off-axis wavefront aberrations in circular pupils. While the (21) I original basis of Zernike polynomials Ψn;mr;ϕ serves natu- II ε → 0 rally for axis-centered aberrations, the new basis Ψn ;n r;ϕ Hence, when isp at the rim of the disk and sphere, 1 2 after dividing Eq. (19)by −ε on both sides, this relation in Eq. (17) includes, for n1 0, plane wave trains with n2 reads nodes along the x axis of the pupil, which are proportional to U n x, the Chebyshev polynomials of the second kind. n 1 2 C sin φ n1 C 1 cos φ P 1 n1;n2 n2 n1 We find that the Zernike system is also very relevant for rffiffiffiffiffiffiffiffiffiffiffi “ ” Xn studies of nonstandard symmetries described by Higgs alge- n 1 n;m jmj;0 imφ W n ;n Pn −1e ; (22) bras. While rotations in the basis of spherical harmonics is π 1 2 r D m−n2 determined through the Wigner- functions [13] of the rotation angles on the sphere, here the boundary conditions of the 1 1 with nr n − jmj. Recalling that Pn 11 and disk and sphere allow for only a 2 π rotation of the z axis to m ;0 2 1 Pj j −1 −1 nr x-y nr , we can now use the orthogonality of orientations in the plane, and the basis functions do not the eimφ functions to express the interbasis coefficients as a relate through Wigner D-functions, but Clebsch–Gordan co- Fourier integral, efficients of a special type. Since the classical and quantum n Z Zernike systems analyzed in Refs. [8,9] have several new −1 r C n ;n π W n;m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 dφ sin φ n1 and exceptional properties, we surmise that certain applications n1;n2 2 πn 1 −π must also be of interest.

n11 C n cos φ exp−imφ: (23) × 2 APPENDIX A: THE INTERBASIS INTEGRAL AND – The integral Eq. (23) does not appear as such in the standard CLEBSCH GORDAN COEFFICIENTS tables [12]; in Appendix A we derive the result and show that it The integral in Eq. (23) does not seem to be in the literature, can be written in terms of a hypergeometric 3F 2 polynomial although similar integrals appear in a paper of Kildyushov [14] which is an su(2) Clebsch–Gordan coefficient with a special to calculate his three coefficients. Thus, let us solve ab initio, structure, with λ n1 and ν n2, integrals of the kind 1848 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article

Z π n1;0 λ;m λ λ1 −imφ times the special Clebsch–Gordan coefficients C 1 1 1 1 .For I ν ≔ dφ sin φC ν cos φe ; (A1) 2n;−2m;2n;2m −π n n C n1;0 0 even and odd 1, 1n;0;1n;0 . where λ; ν ∈ f0; 1; 2; …g. 2 2 The Zernike polynomials come in complex conjugate pairs, We write the trigonometric function and the Gegenbauer I I II ϒn;m ϒn;−m, while the ϒn ;n ’s are real. The linear combina- polynomial in their Fourier series expansions, 1 2 tions afforded by the W matrices above indeed yield real λ eiλφ 1 X −1kλ! functions because sinλ φ 1 −e−2iφλ eiλ−2kφ; 2i λ 2i λ k!λ − k! k0 n ;0 n ;0 C 1 −1 n2 C 1 : 1n;−1m;1n;1m 1n;1m;1n;−1m (A8) (A2) 2 2 2 2 2 2 2 2 Xν λ l ! λ ν − l ! λ1 −iν−2lφ C ν cos φ e : (A3) l!ν − l! λ!2 l0 Funding. Dirección General Asuntos del Personal Académico, Universidad Nacional Autónoma de México Substituting these expansions in Eq. (A1), using the orthog- “ ” onality of the eiκφ functions and thereby eliminating one of the (DGAGA, UNAM) ( Óptica Matemática IN101115); Universidad de Guadalajara-Consejo Nacional de Ciencia y two sums, we find a 3F 2 hypergeometric series for unit argument, Tecnología (CONACyT) (PRO-SNI-2017). 2π λ ν ! −1 λ−ν−m∕2 λ;m Acknowledgment. I ν We thank Prof. Natig M. Atakishiyev 2iλ ν! 1 1 2 λ − ν − m ! 2 λ ν m ! for his interest in the matter of interbasis expansions, and ! acknowledge the technical help from Guillermo Krötzsch 1 −ν; λ 1; − 2 λ ν m (ICF-UNAM) and Cristina Salto-Alegre with the figures. × 3F 2 1 : (A4) −λ − ν; 1 λ − ν − m1 G. S. P. and A. Y. thank the support of project PRO-SNI- 2 2017 (Universidad de Guadalajara). K. B. W. acknowledges C Multiplying this by the coefficients n1;n2 in Eq. (23), one the support of UNAM-DGAPA Project Óptica Matemática finds the first expression in Eq. (24). PAPIIT-IN101115. 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 a REFERENCES relation between 3F 2-hypergeometric functions, “ 1. F. Zernike, Beugungstheorie des Schneidenverfahrens und a;b;c Γ d Γ d − a − b a;b;e − c seiner verbesserten Form, der Phasenkontrastmethode,” Physica – 3F 2 1 3F 2 1 ; 1, 689 704 (1934). d;e Γd − aΓd − b a b − d 1;e 2. A. B. Bhatia and E. Wolf, “On the circle polynomials of Zernike and ” (A5) related orthogonal sets, Math. Proc. Cambridge Philos. Soc. 50, 40–48 (1954). to write these particular symmetric coefficients as 3. T. H. Koornwinder, “Two-variable analogues of the classical orthogonal polynomials,” in Theory and Application of Special Functions, – γ;0 2α!γ! R. A. Askey, ed. (Academic, 1975), pp. 435 495. C 4. E. C. Kintner, “On the mathematical properties of the Zernike polyno- α;−β;α;β α β!γ − α − β! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mials,” Opt. Acta 23, 679–680 (1976). “ ” 2γ 1 5. A. Wünsche, Generalized Zernike or disc polynomials, J. Comput. × Appl. Math. 174, 135–163 (2005). 2α − γ!2α γ 1! 6. B. H. Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 38, 2487–2489 −2α γ; γ 1; −α − β (2013). × 3F 2 1 : (A6) “ −2α; γ − α − β 1 7. M. E. H. Ismail and R. Zhang, Classes of bivariate orthogonal polynomials,” arXiv:1502.07256 (2016). “ 1 1 8. G. S. Pogosyan, K. B. Wolf, and A. Yakhno, Superintegrable classical Finally, upon replacement of α 2 n 2 n1 n2, Zernike system,” J. Math. Phys. 58, 072901 (2017). 1 β 2 m, and γ n1, the expression (24) reduces to 9. G. S. Pogosyan, C. Salto-Alegre, K. B. Wolf, and A. Yakhno, Eq. (25) times the phase and sign. “Quantum superintegrable Zernike system,” J. Math. Phys. 58, The interbasis expansion coefficients binding the two bases in 072101 (2017). n 1 n 1 10. P. W. Higgs, “Dynamical symmetries in a spherical geometry,” J. Eq. (19)andFig.2 can be seen as × Phys. A 12, 309–323 (1979). W W n;m n ;n matrices n k n1;n2 k with composite rows 1 2 and 11. G. S. Pogosyan, A. N. Sissakian, and P. Winternitz, “Separation of columns n; m for each rung n1 n2 n ∈ Z0 ,onn 1- variables and Lie algebra contractions: applications to special func- dimensional column vectors of functions as ϒIIϑ0; φ0 tions,” Phys. Part. Nucl. 33, S123–S144 (2002). I n;m 12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and W n ϒ ϑ; φ. The elements W n ;n in Eq. (25) are the product 1 2 Products, 7th ed. (Elsevier, 2007). of phases 13. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonski, Quantum Theory of Angular Momentum (World Scientific, 1988). m 14. M. S. Kildyushov, “Hyperspherical functions of ‘three’ type in the n 1 m m −1 ;m>0; i 1 ; −1 2 j j ” – 1;m≤ 0 (A7) n-body problem, J. Nucl. Phys. (Yad. Fiz.) 15, 187 208 (1972) (in Russian).