Appl. Math. Inf. Sci. 15, No. 3, 307-315 (2021) 307 Applied & Information Sciences An International Journal

http://dx.doi.org/10.18576/amis/150308

Discrete Bessel and Mathieu Functions

Kenan Uriostegui∗ and Kurt Bernardo Wolf Institute of Physical Sciences, National Autonomous University of M´exico, Morelos 62251, M´exico

Received: 2 Feb. 2021, Revised: 2 Apr. 2021, Accepted: 12 Apr. 2021 Published online: 1 May 2021

Abstract: The two-dimensional separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of translations and orthogonal transformations; we replace the latter by the discrete dihedral group of N discrete rotations and reflections. The in polar and elliptic coordinates is then used to define discrete Bessel and Mathieu functions, as approximants to the well-known continuous Bessel and Mathieu functions, as N-point Fourier transforms approximate the Fourier transform over the circle, with integrals replaced by finite sums. We find that these ‘discrete’ functions approximate the numerical values of their continuous counterparts very closely and preserve some key special function relations.

Keywords: Bessel functions, Mathieu functions, Fourier analysis, Group theory, Numerical approximation and analysis

16 1 Introduction 10− ) the corresponding continuous ones over a region, roughly 0 n + ρ < N. ≤ The role of the Euclidean group of translations, Several authors have introduced functions that reflections and rotations in the determination of the approximate the well-known continuous Bessel functions coordinate systems that separate the solutions of the Jn(ρ) for the purpose of reducing computation time, or to two-dimensional Helmholtz equation is well known from provide new classes of solutions to difference equations the work by Willard Miller Jr. [1]. This symmetry that will share some of their salient properties [4,5,6]. accounts for their separability in four coordinate systems: Our approach follows the well known approximation Cartesian, polar, parabolic and elliptic. Only the elliptic afforded by the N-point finite Fourier transform to the system is generic; when the two foci coalesce, this system integral Fourier transform over the circle. This is done for becomes the polar one with angular and radial polar and elliptic coordinates, and introduces both coordinates; when one focus departs to infinity the system ‘discrete’ Bessel and Mathieu functions. These functions, becomes parabolic; and when both foci do, it becomes we should emphasize, differ from those proposed in the Cartesian. works cited above, which are also distinct in definition The polar decomposition was used by Biagetti et al. and purpose among themselves. By construction, it will [2] to first introduce a discrete version of Bessel functions follow that under N ∞, these discrete functions become based on an expansion of plane waves into a finite number the continuous ones,→ although this limit requires further of polar components —that was not quite complete. This mathematical precision, as it may involve Gibbs-type was properly completed in Ref. [3], defining discrete N oscillation phenomena that we cannot address here. Bessel functions Bn (ρ), which approximate the usual continuous Bessel functions Jn(ρ) by replacing Fourier In Sect. 2, we present this discretization method and a series over a circle S 1 by the finite Fourier transform on resum´eof the results in Ref. [3] for Bessel functions, to N equidistant points on that circle, note that the discrete functions thus defined approximate the continuous ones remarkably well. In Sect. 3, we apply S 1 S 1 S 1 θm = 2πm/N, m 0,1,...,N 1 =: (N), (1) the strategy of replacing harmonic analysis on by (N) ∈{ − } to define discrete approximants to the Mathieu functions where m is counted modulo N. It was found that these of first and second kind in the elliptic coordinate system. discrete functions approximated very closely (of the order All relations are backed by numerical verification. In the

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concluding Sect. 4, we provide some further connections of a continuous function, or as the index for the list of and preliminary conclusions. components of an N-cyclic vector. In either case, the inner (1) (2) product of two discrete functions fn and fn is naturally

N 1 2 Continuous and discrete Bessel functions (1) (2) − (1) (2) ( f , f )(N) := ∑ fn ∗ fn , (8) The Helmholtz equation for wavefields f (x,y) of (fixed) n=0 real wavenumber κ R is ∈ and it is clear that the N ∞ limit will lead back from the discrete to the continuum,→ with the approximationsand (∂ 2 + ∂ 2 + κ2) f (x,y)= 0, (2) x y limits familiar from Fourier theory. The Helmholtz equation (2) in polar coordinates, with ∂ ∂ ∂ and x y R2. In this section we follow z / z ( , ) multiplied by r2, the well≡ known case of polar∈ coordinates, 2 2 2 2 x = r cosθ, y = r sinθ, (3) (r ∂r + r∂r + ∂φ + κ ) f (r,φ)= 0, (9) r [0,∞), θ ( π,π]= S 1. ∈ ∈ − shows that solutions can be factored into a function of the A key assumption is a Hilbert space structure for the radius times a function of the angle as solutions f (x,y) by which one can write them as the f (r,φ) = R(r)Φ(φ), while (5) implies that solutions two-dimensional Fourier transform, Φ(φ) for the angular factor will determine a corresponding radial factor R(r). An orthonormal and 1 2 f (x,y)= dκx dκy expi(xκx +yκy) f (κx,κy). (4) complete set of eigenfunctions of ∂φ over the circle 2π ZZR2 S 1 1/2 e φ is the set of phases Φn(φ) := (2π)− exp(inφ), The Helmholtz equation (2) is then correspondingly with∈ integer n 0, 1,... , and inner products transformed to a conjugate space κ κ R2 where it (Φ ,Φ ) = δ .∈ When { ± the domain} of these functions is ( x, y) n n′ n,n′ 2 2 2 ∈ ◦ S 1 S 1 reads (κ κ κ ) f (κx,κy) = 0, which we can also restricted from φ to φm N as in (1), we retain − x − y ∈ ∈ ( ) refer to polar coordinates κx = κ cosφ, κy = κ sinφ, with the subset of N functions on the N points in S 1 , given e (N) the surface element dκx dκy = κ dκ dφ. The solutions to by the Fourier-transformed Helmholtz equation are thus 1 1 2πimn reduced by a Dirac δ-distributions in the radius [1], as Φ (N) φ : exp inφ exp (10) 1 n ( m) = ( m)= f (κx,κy) = √2πκ− δ(κ κ) f (φ), with f (φ) a √N √N  N  − ◦ ◦ function on the φ-circle S 1 of radius κ, that we write (N) e = Φn N (φm), again κ, understanding that ite is the fixed wavenumber. ± The Helmholtz solutions (4) thuse acquire the labeled by the cyclic subset n 0,1,...,N 1 , that are single-integral form also orthonormal under the common∈{ inner product− } (8) for S 1 discrete functions on (N), and complete:

1 N 1 f (x,y)= dφ expiκ(xcosφ + ysinφ) f (φ), N − S 1 (N) ( ) (N) (N) √2π Z ◦ (Φn ,Φ )(N) = δn,n , Φn (φm)∗ Φn (φm )= δm,m . n′ ′ ∑ ′ ′ (5) n=0 with the Hilbert space structure based on the inner product (11) of functions f (1)(φ) and f (2)(φ) on the circle, Returning to (5) with (x,y) in the polar coordinates (r,θ) ◦ ◦ of (4), and taking for f (φm) the basis functions (11) on S◦ 1 (1) (2) (1) (2) the discrete points of (N), we write the N solutions to ( f , f ) := dφ f (φ)∗ f (φ). (6) ◦ ◦ ◦ ZS 1 ◦ ◦ the discretized Helmholtz equation, labeled by cyclical n 0,1,...,N 1 , as ∈ It is here that we reduce the continuous circle Fourier { − } transform to the N-point discrete Fourier transform, from fn(r,θk) S 1 to S 1 , replacing integrals by summations and the (N) = 1 exp[iκr(cosθ cosφ + sinθ sinφ )]Φ (N)(φ ) S 1 S 1 √N ∑ k m k m n m continuous variable φ with φm (N), as m S 1 ∈ (N) ∈ ∈ 1 = N ∑ exp[iκr cos(θk φm)]exp(inφm) 2π N, m S 1 − dφ F (φ) F(φm), ↔ (7) ∈ (N) 1 ∑ ZS ◦ ↔ φm = 2πm/N, in θ π 2 m S 1 e ( k+ / ) ∈ (N) = N ∑ exp(iκr sinϕm)exp( inϕm), m S 1 − ∈ (N) for m 0,1,...,N 1 counted modulo N; the set of N (12) ∈{ − } 1 discrete angles φm are thus equidistant by 2π/N. The having replaced ϕm := θk φm + π in the summation over − 2 functions f (φm) fm can be interpreted as sample points the N discrete points on the circle. ≡

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(N) Fig. 1: The ‘discrete’ Bessel functions Bn (ρ) on continuous intervals 0 ρ (2N 1) (gray lines), vs. the ‘continuous’ Bessel ≤ ≤ − functions Jn(ρ) (thin black lines), for orders n 0, 10, 30, 50 and point numbers N 21, 61, 101 . Heavy black lines replace both ∈ { } 16 ∈ { } where the ‘discrete’ and the ‘continuous’ Bessel functions differ by less than 10− .

Following Miller [1], the phase in front of (12), The distinction between even and odd cases of n, as done in(θ +π/2) n 2πink/N n (N) e k = i e = i √NΦn (θk), is extracted to in [3], is subtle but importantto obtain the correct result for write the functions as all n’s (cf.[2, Eq. (9)]). It results in the parity and cyclicity

n (N) (N) properties fn(r,θk)= i √NBn (κr)Φn (θk), (13) (N) where the radial factor Bn (ρ), ρ := κr, are the discrete (N) (N) n (N) n (N) Bn (ρ)= Bn N(ρ) = ( 1) B n(ρ) = ( 1) Bn ( ρ), Bessel functions. From (12) these functions are seen to be ± − − − − real and their parities, using coefficients cn,sn := 1,0 { } { } for n even or 0,1 for n odd, can be written as (N) Bn (0)= δn,0, (15) (N) { } Bn (ρ) (14)

1 which also hold for the continuous Bessel functions Jn(ρ) = ∑ exp(iρ sinϕm)[cn cos(nϕm) isn sin(nϕm)] N S 1 − of integer order [7]. m N ∈ A plane wave of wavenumber κ along the y-axis in a Helmholtz medium that allows only N equidistant 1 cosnϕm, n even, = exp(iρ sinϕm) directions of propagation on the circle can be obtained N ∑  isinnϕm, n odd. m S 1 − from (15) using the completeness relation (11) to expand ∈ N

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the middle term and write

N 1 (N) − (N) exp(iρ sinϕm)= B0 (ρ)+2 ∑ B2n (ρ)cos(2nϕm) n=1 N 1 − (N) +2i ∑ B2n+1(ρ)sin((2n+1)ϕm), n=0 (16) showing how the discrete Bessel functions can take the place of the continuous ones, cf. [8, Eq. KU120(13)]. In Ref. [3], we proved analytically and verified numerically that the following expressions for the discrete Bessel functions are exact analogues of those valid for continuous Bessel functions. Corresponding to [8, WA44] for odd N =: 2 j + 1, in Ref. [3], we proved the linear relations involving the even and odd-n discrete Bessel functions,

j B0(ρ)+ ∑ B2n(ρ)cos(2nϕm)= cos(ρ sinϕm), n=1 j 1 ∑ B2n+1(ρ)sin((2n+1)ϕm)= 2 sin(ρ sinϕm). n=0 (17) Fig. 2: Set of equally-spaced discrete points on (20) The quadratic formulas [9, 7.6.2, Eq. (6)] associated to S 1 § of the ‘angular’ coordinates ψm (N) for N = 21, and the name of Graf, were shown in Ref. [10] to derive from hyperbolas of the ‘radial’ coordinate{ } for ∈ ρ 0.5, 1, 1.5 . the rotation of spherical harmonics through Wigner-D ∈ { } functions, under contraction from the rotation to the Euclidean group. These relations, of group-theoretical origin, retain their validity under the discretization of the 3 Discrete Mathieu functions rotation subgroup, and lead to

2 j Elliptic coordinates on the plane generalize the previous B (ρ)B (ρ )= B (ρ + ρ ), (18) ∑ n n′ n ′ n′ ′ polar coordinates (4). They are defined in terms of n= 2 j − − Cartesian coordinates through keeping in mind the parity property (15) for the negative n- x = coshρ cosψ, y = sinhρ sinψ, (19) indices in the sum for odd N, addressing the vector rather ρ [0,∞), ψ ( π,π]= S 1. than spin representations of the rotation group. ∈ ∈ − In Fig. 1 we essentially repeat the figure in Ref. [3] where (ρ,ψ) are analogues of the previous polar where we compared the discrete and continuous Bessel coordinates r φ for which we retain the names as (N) ( , ) functions, Bn (ρ) and Jn(ρ), to support the claim that the ‘radial’ and ‘angular’ variables. For fixed ρ or for fixed approximation is indeed remarkable within an interval ψ, the locus of points (x,y) R2 that satisfy that is roughly 0 n + ρ < N. A similar set of figures is ∈ ≤ presented below for Mathiew functions. 2 2 2 2 2 2 2 2 (N) x /cosh ρ +y /sinh ρ = 1, x /cos ψ y /sin ψ = 1, Now, having N basis functions Bn (ρ), numbered by − (20) cyclic n modulo N, it is natural to inquire whether the draw families of confocal ellipses or hyperbolas argument ρ can or should be also discretized to the N respectively. At ρ = 0, ψ S 1 draws twice the line integer values ρk = k 0,1,...,N 1 . This was done in ∈ ∈{ − } between the two foci (x,y) = ( 1,0) for ψ = (0,π). The Ref. [2] while in [3] the plot in Fig. 1 marked these points major and minor semi-axes of± the ellipses are coshρ and (N) and used them to define a kernel Bn (ρk) for a ‘discrete sinhρ respectively, so their eccentricities are 1/coshρ, Bessel transform’ between two N-vectors of components that tend to circles when ρ ∞. On the other hand, for 1 → fn and fk. The fact is that while the angle ϕ is discretized fixed ψ S in each of the four quadrants, since ρ 0, naturally to N points on the circle, the radial coordinate ρ only one∈ of the four arms of the hyperbola is traversed.≥ is not subjecte to a similarly compelling set of points, but Thus, we expect four parity cases out of the two is valid and non-cyclic over the complex ρ-plane. The reflections across the x and y axes. Compare this with the same discretization process for the angular —but not the case of polar coordinates where r 0 but all reflection radial— coordinate will be applied to the Mathieu case axes are equivalent, so ( 1)n in (≥15) provides the two below. Bessel parity cases. −

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efficient numerical computation. For use below, we write them using as

n As 1 cos(sψ)cen(ψ,q) n = dψ , (24) Bs  π ZS 1  sin(sψ)sen(ψ,q) 

n 1 for n = 0, while A0 = (2π)− S 1 dψ cen(ψ,q), and Bn 0.6 The Mathieu functions (23R) are orthogonal under 0 ≡ the inner product (6) over the circle, (cem,cen) = πδm,n, ◦ (sem,sen) = πδm,n for n = 0 —zero otherwise, and ◦ 6 (cem,sen) = 0. We now◦ restrict the range of the angular coordinate ψ S 1 S 1 from to (N), shown for the elliptic coordinates in Fig. 2, in correspondence with the previous discrete phase functions in the Bessel case (11), and thus defining the ‘angular’ discrete Mathieu functions of the first type over 1 ψm S as ∈ (N)

(N) N 1 2n+p ce2n+p(ψm,q) − a2s+p cos((2s+p)ψm) (N) := ∑ , (25)  se (ψ ,q)   b2n+p sin((2s+p)ψ )  Fig. 3: Discrete vs. continuous ‘angular’ Mathieu functions for 2n+p m s=0 2s+p m (N) N = 41, q = 2. The values of the discrete functions cen (ψm,q) n n (N) with coefficients a , b . The finite N-point Fourier and sen (ψm,q) at ψm, 0 m N 1, are indicated by circles. s s ≤ ≤ − q q transform approximates them through the replacement (7) The continuous Mathieu functions cen(ψ, ) and sen(ψ, ) are n n marked by black lines in their full range 0 ψ < 2π. Their to the functions and coefficients As , Bs of the continuous 16 ≤ case in (24), as difference is less than 10− for all points ψm.

n N 1 (N) n as 1 − cos(sψm)cen (ψm,q) 1 As n := ∑ (N) n , bs  N m=0  sin(sψm)sen (ψm,q)  ≃ 2Bs  The Helmholtz (2), written in the (26) elliptic coordinates (20), is clearly separable, n n n 0 for s = 0, while a0 = A0, b0 = 0, and also bs = 0. The6 last relation in (26) is an approximate equality, [(∂ 2 + κ2 cosh2ρ) + (∂ 2 κ2 cos2ψ)] f (ρ,ψ)= 0, (21) ρ ψ − the validity of which is contingent upon the numerical computation and comparison between the lower- and so that solutions can be written in the product form upper-case coefficients within a range of their indices in, f (ρ,ψ) P(ρ)Ψ(ψ). Dividing by f , one obtains two say, 0 n,s N 1, which is reflected in turn by the coupled∼ equations in ρ and ψ, the latter is an eigenvalue discrete≤ and continuous≤ − Mathieu functions themselves. In equation in the angular coordinate, Fig. 3 we compare a sample of continuous angular 2 1 2 Mathieu functions of the first kind with their discrete (∂ψ 2qcos2ψ)Ψ(ψ,q)= νΨ(ψ,q), q := 4 κ , − (22) approximations from Eq. (25). In favor of the thus defined known as the Mathieu differential equation. The angular discrete Mathieu functions, we note that they satisfy coordinate ψ is periodic and a well-known solution orthogonality relations under the discrete inner product method consists in expanding solutions of (22) in the (8), namely

Fourier basis exp(inψ) over all integer n. This defines (N) (N) 1 (N) (N) 1 ∼ (ce ,ce )(N) = Nδ , (sen ,se )(N) = Nδ , the Mathieu functions of the first kind ce ψ q and n n 2 n,n′ n =0 2 n,n′ n( , ) ′ ′6 sen(ψ,q) with integer n [11], characterized by a parity (N) (N) (cen ,se )(N) = 0. (27) index p 0,1 for even and odd cases [8, 8.61], and n′ distinct for∈{ even} and odd indices. In a two-line§ expression By construction, the parities of the discrete Mathieu (N) (N) all cases can be written as functions are also even cen ( ψm,q)= cen (ψm,q), or (N) (N) − odd se ψ q se ψ q . ∞ 2n+p n ( m, )= n ( m, ) ce (ψ,q) A2s p cos((2s+p)ψ) − − 2n+p = + . (23) At this point it is illuminating to inquire into the  se (ψ,q)  ∑  2n+p  manner in which the discrete functions approximate the 2n+p s=0 B2s p sin((2s+p)ψ) + (N) continuous ones. Consider for example how ce0 (ψm,q), The parities are even cen( ψ,q) = cen(ψ,q), odd whose definition (25) allows us to compute it for − 1 sen( ψ,q) = sen(ψ,q), and se0(ψ,q) 0. The continuous ψm S , matches ce0(ψ,q) in the whole ψ − n −n ≡ ∈ coefficients As , Bs are found introducing this expansion range. In Fig. 4, we do so for small N, noting that where into (22) to find recursion relations [8, 8.62] that lead to the continued ψm lines of the former take their values, §

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(N) S 1 Fig. 4: Comparison between the ‘discrete’ Mathieu functions ce0 (ψm,q) whose arguments are continued to ψm (gray line) vs. ∈ S 1 the ‘continuous’ Mathieu function cen(ψ,q) (black line), for point numbers N 5,11,21 and q = 2. The discrete points ψm (N) lie at a subset of the intersections marked with circles. ∈ { } ∈

they intersect the properly continuous line of the latter. The coefficients in front of the summation will be now Although the two lines intersect also at other points, the determined through considering specific values for the two lines remain notably distinct. The figure shows that ‘angular’ coordinate ψ ψm, comparing them with approximation is not valid over presumably small ranges expressions of the continuous↔ Mathieu functions of the around these intersections, but only at the prescribed second kind obtained from integrals that are tabulated in ψm = 2πm/N points. We intend to elaborate on such and Ref. [8, 6.92]. There, the exponential factors appear with similar limits elsewhere. only a single§ summand in the exponent, either sine or 1 Proceeding now as we did in (12), but using the cosine. This occurs in (29) only for ψm = 0 or 2 π, (N) S 1 discrete Mathieu functions of the first kind cen (ψm,q) although the latter is not in the set (N) if ψ0 = 0, since N (N) and sen (ψm,q) in place of the plain phase functions was assumed to be odd. Φ (N)(φ ), we again have Helmholtz solutions f (ρ,ψ ) Let us first consider the case of even parity p = 0 and n m n k 1 on the plane characterized by parities and sub-indices n, the angle ψ = 2 π in (29), where the previous remark whose radial factor will be the discrete ‘radial’ Mathieu applies. Based on the close approximation between the functions of the second kind, to be indicated discrete and continuous Mathieu functions, we may (N) (N) correspondingly as Cen (ρ,q) and Sen (ρ,q), simply replace the latter for the former, so that the two lines in that expression read c f2n+p(ρ,ψk) s (N) (N) f (ρ,ψk) Ce (ρ,q) c N 1 ce (ψm,q)  2n+p+1  2n K2n − 2n (N) = s ∑ (N) (N) K N 1  Se2n+1(ρ,q)   2n+1  m=0  se2n+1(ψm,q)  1 − cen (ψm,q) = N ∑ (N) exp[iκ(xcosψm + ysinψm)  sen (ψm,q)  exp(2i√q sinhρ sinψm). m=1 × (N) (N) (30) c (q)Ce (ρ,q)ce (ψ ,q) =: n n n k , When this summation formula is compared with the s q Se(N) ρ q se(N) ψ q  n( ) n ( , ) n ( k, )  integral expressions tabulated in [8, 6.92], namely (28) §

where cn(q) and sn(q) are constants. Using the elliptic 2n Ce2n(ρ,q) ce2n(0,q)/2π A0 coordinates with a discretized angular part, = 2n+1 Se2n+1(ρ,q) ise (0,q)/2π B √q    − 2′ n+1 1  x(ρ,ψk) = coshρ cosψk and y(ρ,ψk) = sinhρ sinψk in ce2n(ψ,q) (20), the phase exponent is then iκ = 2i√q times dψ exp(2i√q sinhρ sinψ), × ZS 1  se2n+1(ψ,q)  xcosψm + ysinψm (31) = coshρ cosψk cosψm + sinhρ sinψk sinψm. we conclude that the constants in the summation (30), after identifying 2π N, A2n = a2n and B2n+1 2b2n+1, are As was done before in (12) and (13), we extract the new ↔ 0 0 1 ≃ 1 discrete ‘radial’ functions using the orthogonality (27) of ise 0 q the previous discrete ‘angular’ Mathieu functions, as c ce2n(0,q) s 2′ n+1( , ) K2n = , K2n 1 = − . (32) a2nN + 2b2n+1N√q (N) 0 1 Ce2n+p(ρ,q) Se(N) ρ q  2n+p+1( , )  where sen′ (0,q) := dsen(ψ,q)/dψ ψ=0. In Fig. 5 we (N) (N) | 1/N c (q)ce (ψ ,q) N 1 ce (ψ ,q) compare a sample of the discrete and continuous ‘radial’ 2n+p 2n+p k − 2n+p m = (N) ∑ (N) Mathieu functions, noting that the two lines are quite  1/Ns2n+p+1(q)se (ψk,q)   se (ψm,q)  2n+p+1 m=0 2n+p+1 coincident in the range ρ [0,π), but the commercial ∈ exp[2i√q(coshρ cosψk cosψm + sinhρ sinψk sinψm)]. Mathematica algorithm oscillate wildly beyond π. Again, × (29) we can only justify this statement numerically.

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Fig. 5: Discrete vs. continuous ‘radial’ Mathieu functions in the interval 0 ρ < 3.3, for N 5,11,21 and here for q = 2. The (N) (N) ≤ ∈ { } ‘discrete’ functions Cen (ρ,q) and Sen (ρ,q) with the (continuous) argument ρ (gray line), are compared with the ‘continuous’ 16 functions Cen(ρ,q) and Sen(ρ,q) (thin black line). As before, where both coincide within 10− they are replaced by a thick black line. The radial Mathieu functions, when computed with the commercial Mathematica algorithm, oscillate wildly after an upper value that decreases with increasing values of q.

Next we consider the case of odd parity p = 1 at the exponent appear as 1/√2, we can write value ψ = ψ0 = 0. The upper line in (30) reads N 1 (N) (N) s − (N) Ce2n+1(ρ,q) (33) Se2n+2(ρ,q)= K2n+2 ∑ se2n+2(ψm,q) (36) N 1 m=0 c − (N) = K2n+1 ∑ ce2n+1(ψm,q)exp(2i√qcoshρ cosψm), exp[i 2q(coshρ cosψm + sinhρ sinψm)]. × m=0 p that we compare with the integral for the continuous For the corresponding continuous case, we could not find Mathieu functions of the second kind in [8, 6.92], a corresponding integral in [8, 6.92], so we cannot give a § § s namely closed expression for the coefficient K2n+2 in (37). The lack of a similar plane-wave integral expression for the Ce2n+1(ρ,q) continuous Mathieu functions Se2n+2(ρ,q) has been noted also in Ref. [12] without explanation. However, we ice ( 1 π,q) 2′ n+1 2 have checked numerically that the simile of the discrete to = 2n+1 (34) 2πA1 √q continuous functions approximation provided by

dψ ce2n+1(ψ,q)exp(2i√qcoshρ cosψ), ZS 1 (N) × Se (ρ,q) ise2n 2(iρ,q)= Se2n 2(ρ,q), (37) 2n+2 ≃− + + where ce2′ n+1(ψ,q) is the derivative of the Mathieu function. Again exploiting the correspondences (7), which is an equality for continuous functions, cf.[8, Eqs. 2π N and A2n+1 2a2n+1, we conclude the constant in 8.611.4, 8.631.4]. For 0 < ρ < 2 the difference in (37) ↔ 1 ≃ 1 14 (37) to be less than 10− . We should note that generally the discrete ‘radial’ and ‘angular’ Mathieu functions for pure ice ( 1 π,q) c 2′ n+1 2 imaginary arguments are not related to similar equalities K2n+1 = 2n+1 . (35) 2a1 N√q of their continuous integral expressions, because the summation definitions in (25) involve hyperbolic The remaining case to be considered is that of odd parity functions. In particular, say, (N) and even index, namely for Se2n+2(ρ,q). This presents a problem because the summation (29) is identically zero for (N) (N) 1 Se2n+p+1(ρ,q) = ise2n+p+1(iρ,q) (38) both ψk = 0 and 2 π due to the parities of the terms in the 6 − 1 N 1 sum. It is different from zero for 0 < ψk < 2 π however, − 2n+p+1 1 = ∑ b2m+p+1 sinh[(2m+p+1)ρ]. so if we choose ψk = 4 π, where both summands in the m=0

c 2021 NSP Natural Sciences Publishing Cor. 314 K. Uriostegui, K. B. Wolf: Discrete Bessel and Mathieu functions

4 Concluding remarks [3] K. Uriostegui and K.B. Wolf, Discrete Bessel functions and transform, (submitted) arXiv:2005.06076 [math-ph] (2020). The expansion of Helmholtz plane waves in series of [4] R.H. Boyer, Discrete Bessel functions, Journal of radial Bessel and angular trigonometric functions has its Mathematical Analysis and Applications, 2, 509–524, discrete analogue in Eq. (16), which tells us that the (1961). wavefield due to a finite number N of plane waves at [5] M. Bohner and T. Cuchta, The Bessel difference equation, equidistant direction angles can be expanded in discrete Proceeding of the American Mathematical Society, 145, Bessel radial functions and corresponding trigonometric 1567–1580, (2017). angular functions. A similar statement will hold when the [6] A. Slav´ık, Discrete Bessel functions and partial differential equations, Journal of Difference Equations wavefield is expanded in discrete Mathieu functions with and Applications, 24, 425–437, (2017). the phases determined by the points on an as [7] G.N. Watson, Theory of Bessel Functions, Cambridge depicted in Fig. 2. Conceivably such fields can be University Press, London UK: 14–37, (1922). produced in resonant two-dimensional micro-cavities fed [8] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series by a number of activation channels. and Products, A. Jeffrey and D. Zwillinger, Academic Press, We recognize that the full treatment and exploration California USA: 659–958, (2007). of properties for the discrete Bessel and Mathieu function [9] A. Erd´elyi et al., Higher Transcendental Functions (Based presented here is not exhaustive, but that it should be on notes by H. Bateman), Vol. 2, McGraw-Hill, New York sufficient to indicate that the approximation method USA: 1–78, (1953). consisting in the replacement of a continuous closed [10] P. Winternitz, K.B. Wolf, G.S. Pogosyan, and A.N. subgroup of the symmetry group of a partial differential Sissakian, Graf’s addition theorem obtained from SO(3) equation by a finite discrete group is definitely of interest. contraction, Theoretical and Mathematical Physics, 129, In the present case of two dimensions, the orthogonal 1501–1503, (2001). group was reduced to the dihedral group. In three [11] N.W. McLachlan, Theory and Application of Mathieu dimensions, the symmetry Euclidean symmetry group Functions, Oxford University Press, London UK: 10–28, could reduce its three-dimensional rotation subgroup by (1947). any of its polyhedral subgroups, whose functions may [12] L. Chaos-Cador and E. Ley-Koo, Mathieu functions, serve to describe wavefields with a corresponding subset evaluation, and generating functions, Revista Mexicana de of wave propagation directions. Here we have presented a F´ısica, 48, 67–75, (2002). set of exact relations, others whose approximation closeness was estimated through numerical computation, and others that have been only suggested by that Kenan Uriostegui approach, and for which we expect to present further received the MSc and results from ongoing work. PhD degree at the Instituto en Ciencias F´ısicas of the Universidad Nacional Acknowledgments Aut´onoma de M´exico. His main interests include unitary We thank the support of the Universidad Nacional transformations on discrete Aut´onoma de M´exico through the PAPIIT-DGAPA systems, approximation and project AG100120 Optica´ Matematica´ . K.U. expresses numerical analysis, algebraic gratitude to CONACyT-M´exico. structures in optomechanical systems and advances in quantum .

Competing interests

The authors declare that they have no competing interests.

References

[1] W. Miller Jr., The Helmholtz equation, in Symmetry and Separation of Variables, Encyclopedia of Mathematics, Vol. 4, Cambridge University Press, London UK: 1–47, (1984). [2] G. Biagetti, P. Crippa, L. Falaschetti, and C. Turchetti, Discrete Bessel functions for representing the Class of Finite Duration Decaying Sequences, IEEE 24th European Signal Processing Conference, 2126–2130, (2016).

c 2021 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 15, No. 3, 307-315 (2021) / www.naturalspublishing.com/Journals.asp 315

Kurt Bernardo Wolf is presently Investigador Titular (equivalent to full professor) at the Universidad Nacional Aut´onoma de M´exico, which he entered in 1971. He was the first director of Centro Internacional de Ciencias A.C. (1986–1994), organizing 14 national and international schools, workshops and conferences. He is author of: Integral Transforms in Science and Engineering (Plenum, New York, 1979) and Geometric Optics on Phase Space (Springer-Verlag, Heidelberg, 2004), and has been editor of 12 proceedings volumes. His work includes some 170 research articles in refereed journals, 58 chapters and in extenso contributions to proceedings, two books on scientific typography in Spanish and essays for the general public.

c 2021 NSP Natural Sciences Publishing Cor.