Acta Polytechnica doi:10.14311/AP.2016.56.0283 Acta Polytechnica 56(4):283–290, 2016 © Czech Technical University in Prague, 2016 available online at http://ojs.cvut.cz/ojs/index.php/ap ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS Jiří Hrivnák, Lenka Motlochová∗ Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, CZ-115 19 Prague, Czech Republic ∗ corresponding author: lenka.motlochova@fjfi.cvut.cz Abstract. The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of A1 and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of A2 and a specific version of two variable Jacobi polynomials. The connection with recently studied G2-polynomials is established. Formulas for connection between the four types of orbit functions of Bn or Cn and the (anti)symmetric multivariate cosine and sine functions are explicitly derived. Keywords: Weyl-orbit functions, Chebyshev polynomials, Jacobi polynomials, (anti)symmetric trigonometric functions. 1. Introduction Special functions associated with the root systems of simple Lie algebras, e.g. Weyl-orbit functions, play an important role in several domains of mathematics and theoretical physics, in particular in representa- tion theory, harmonic analysis, numerical integration and conformal field theory. The purpose of this pa- per is to link Weyl-orbit functions with various types of orthogonal polynomials, namely Chebyshev and Jacobi polynomials and their multivariate general- izations, and thus to motivate further development of the remarkable properties of these polynomials in connection with orbit functions. The collection of Weyl-orbit functions includes four different families of functions called C–, S–, Ss– and Sl–functions [6, 15, 16, 25]. They are induced from the sign homomorphisms of the Weyl groups of geometric symmetries related to the underlying Lie algebras. The symmetric C–functions and antisymmetric S– functions also appear in the representation theory of simple Lie algebras [32, 34]; the S–functions appear in the Weyl character formula and every character of irreducible representations of simple Lie algebra can be written as a linear combination of C–functions. Unlike C– and S–functions, Ss– and Sl–functions exist only in the case of simple Lie algebras with two different lengths of roots. A review of several pertinent properties of the Weyl- orbit functions is contained in [8, 15, 16, 25]. These functions possess symmetries with respect to the affine Weyl group – an infinite extension of the Weyl group by translations in dual root lattice. Therefore, we con- sider C–, S–, Ss– and Sl–functions only on specific subsets of the fundamental domain F of the affine Weyl group. Within each family, the functions are continuously orthogonal when integrated over F and form a Hilbert basis of squared integrable functions on F [25, 27]. They also satisfy discrete orthogonality relations which is of major importance for the pro- cessing of multidimensional digital data [8, 10, 27]. Using discrete Fourier-like transforms arising from discrete orthogonality, digital data are interpolated in any dimension and for any lattice symmetry afforded by the underlying simple Lie algebra. Several special cases of simple Lie algebras of rank two are studied in [28–30]. The properties of orbit functions also lead to nu- merical integration formulas for functions of several variables. They approximate a weighted integral of any function of several variables by a linear combi- nation of function values at points called nodes. In general, such formulas are required to be exact for all polynomial functions up to a certain degree [3]. Furthermore, the C–functions and S–functions of sim- ple Lie algebra A1 coincide, up to a constant, with the common cosine and sine functions respectively. They, are therefore, related to the extensively stud- ied Chebyshev polynomials and, consequently, to the integration formulas, quadratures, for the functions of one variable [5, 31]. In [24], it is shown that there are analogous formulas for numerical integration, for multivariate functions, that depend on the Weyl group of the simple Lie algebra An and the corresponding C– and S–functions. The resulting rules for functions of several variables are known as cubature formulas. The idea of [24] is extended to any simple Lie algebra in [9, 25, 26]. Optimal cubature formulas in the sense of the nodal points that are required are known only for S– and Ss–functions. 283 http://dx.doi.org/10.14311/AP.2016.56.0283 http://ojs.cvut.cz/ojs/index.php/ap Jiří Hrivnák, Lenka Motlochová Acta Polytechnica Besides the Chebyshev polynomials, the Weyl-orbit functions are related to other orthogonal polynomi- als. For example, orbit functions of A2 and C2 co- incide with two-variable analogues of Jacobi polyno- mials [21]. It can also be shown that the C–, S–, Ss– and Sl–functions arising in connection with sim- ple Lie algebras Bn and Cn become, up to a con- stant, (anti)symmetric multivariate cosine functions and (anti)symmetric multivariate sine functions [17]. Note that these generalizations lead to multivariate analogues of Chebyshev polynomials and are used to derive optimal cubature formulas. Therefore, this fact indicates that it might be possible to obtain such formulas for all families of orbit functions. As for Chebyshev polynomials, it is of interest to study the accuracy of approximations and interpolations using cubature formulas. This paper starts with a brief introduction to Weyl groups in Section 2. Then there is a review of the relations of the Weyl-orbit functions with other spe- cial functions which are associated with the Weyl groups. In Section 3.1, the connection of the C– and S–functions of one variable with Chebyshev polynomi- als is recalled. In Sections 3.2 and 3.4, we show that each family of Weyl-orbit functions corresponding to A2 and C2 can be viewed as a two-variable analogue of Jacobi polynomials [21]. In Section 3.4, we also provide the exact connection with generalizations of trigonometric functions [34]. 2. Weyl groups of simple Lie algebras In this section, we summarize the properties of the Weyl groups that are neeeded for definition of orbit functions. There are four series of simple Lie algebras An(n ≥ 1), Bn(n ≥ 3), Cn(n ≥ 2), Dn(n ≥ 4) and five exceptional simple Lie algebras E6, E7, E8, F4 and G2. Each of these algebras is connected with its corresponding Weyl group [1, 12, 13, 18, 34]. They are completely classified by Dynkin diagrams (see e.g. figure in [15]). A Dynkin diagram characterizes a set ∆ of simple roots α1, . . . ,αn generating an Euclidean space isomorphic to Rn with the scalar product de- noted by 〈·, ·〉. Each node of the Dynkin diagram represents one simple root αi. The number of links between two nodes corresponding to αi and αj respec- tively is equal to 〈αi,α∨j 〉〈αj,α ∨ i 〉, where α ∨ i ≡ 2αi 〈αi,αi〉 . Note that we use the standard normalization for the lengths of roots, namely 〈αi,αi〉 = 2 if αi is a long simple root. In addition to the basis of Rn consisting of the simple roots αi, it is convenient for our purposes to introduce the basis of fundamental weights ωj given by 〈ωj,α∨i 〉 = δij. This allows us to express the weight lattice P defined by P ≡{λ ∈ Rn | 〈λ,α∨i 〉 ∈ Z, i = 1, . . . ,n} as Z-linear combinations of ωj. The subset of domi- nant weights P + is standardly given as P + ≡ Z≥0ω1 + · · · + Z≥0ωn. We consider the usual partial ordering on P given by µ � λ if and only if λ − µ is a sum of simple roots with non-negative integer coefficients. To each simple root αi there is a corresponding reflection ri with respect to the hyperplane orthogonal to αi, ri(a) ≡ rαi(a) = a− 2〈a,αi〉 〈αi,αi〉 αi, for a ∈ Rn. The finite group W generated by such reflections ri, i = 1, . . . ,n is called the Weyl group. For the properties of the Weyl groups see e.g. [12, 14]. In the case of simple Lie algebras with two different lengths of the roots, we need to distinguish between short and long simple roots. Therefore, we denote by ∆s the set of simple roots containing only short simple roots and we denote by ∆l the set of long simple roots. We also define the following vectors: % ≡ n∑ i=1 ωi, % s ≡ ∑ αi∈∆s ωi, % l ≡ ∑ αi∈∆l ωi. (1) 3. Weyl-orbit functions Each type of Weyl-orbit function arises from the sign homomorphism of Weyl groups σ : W →{±1}. There exist only two different sign homomorphisms on W connected to simple Lie algebras with one length of the roots: identity, denoted by 1, and the determinant [8, 25]. They are given by their values on the generators ri of W as 1(ri) = 1 for all αi ∈ ∆, det(ri) = −1 for all αi ∈ ∆. In the case of simple Lie algebras with two different lengths of roots, i.e. Bn,Cn,F4 and G2, there are two additional sign homomorphisms denoted by σs and σl and given as σs(ri) = { −1 if αi ∈ ∆s, 1 if αi ∈ ∆l, σl(ri) = { 1 if αi ∈ ∆s, −1 if αi ∈ ∆l. Labelled by the parameter a ∈ Rn, the Weyl-orbit function of the variable b ∈ Rn corresponding to sign homomorphism σ is introduced via the formula ϕσa(b) = ∑ w∈W σ(w)e2πi〈w(a),b〉, a,b ∈ Rn. 284 vol. 56 no. 4/2016 Weyl Orbit Functions and (Anti)symmetric Trigonometric Functions Each sign homomorphism 1, det,σs,σl determines one family of complex valued Weyl-orbit functions, called C–, S– ,Ss– and Sl–functions respectively, and de- noted by C–functions: σ ≡ 1, ϕσ ≡ Φ, S–functions: σ ≡ det, ϕσ ≡ ϕ, Ss–functions: σ ≡ σs, ϕσ ≡ ϕs, Sl–functions: σ ≡ σl, ϕσ ≡ ϕl. Several remarkable properties of Weyl-orbit functions, such as continuous and discrete orthogonality are usu- ally achieved by restricting a to some subsets of the weight lattice P (see for example [8, 10, 15, 16, 27]). Note also that symmetric C–functions and antisym- metric S–functions appear in the theory of irreducible representations of simple Lie algebras [32, 34]. It is also convenient to use an alternative definition of the Weyl-orbit functions via sums over the Weyl group orbits [15, 16, 25]. These orbit sums Ca,Sa+%, Ssa+%s, Sla+%l differ from Φa, ϕa+%, ϕ s a+%s, ϕla+%l only by a constant Ca(b) = Φa(b) |StabW a| , Sa+%(b) = ϕa+%(b), Ssa+%s(b) = ϕsa+%s(b) |StabW (a + %s)| , Sla+%l(b) = ϕl a+%l(b) |StabW (a + %l)| with |StabW c| denoting the number of elements of W which leave c invariant. 3.1. Case A1 The symmetric C–functions and antisymmetric S– functions of A1 are, up to a constant, the common cosine and sine functions [15, 16], Ca(b) = 2 cos (2πa1b1), Sa(b) = 2i sin (2πa1b1), where a = a1ω1, b = b1α∨1 . It is well known that such functions appear in the definition of the exten- sively studied Chebyshev polynomials [5, 31]. Several types of Chebyshev polynomials are widely used in mathematical analysis, in particular, as efficient tools for numerical integration and approximations. The Chebyshev polynomials of the first, second, third and fourth kind are denoted by Tm(x), Um(x), Vm(x) and Wm(x), respectively. If x = cos(θ), then for any m ∈ Z≥0 it holds that Tm(x) ≡ cos(mθ), Um(x) ≡ sin ((m + 1)θ) sin(θ) , Vm(x) ≡ cos (( m + 12 ) θ ) cos (1 2θ ) , Wm(x) ≡ sin (( m + 12 ) θ ) sin (1 2θ ) . Therefore, for specific choices of parameter a1 and 2πb1 = θ, we can view the Weyl-orbit functions of A1 as these Chebyshev polynomials. Recall also that the Chebyshev polynomials are actually, up to a constant cα,β, special cases of Ja- cobi polynomials P (α,β)m (x), m ∈ Z≥0. The Jacobi polynomials are given as orthogonal polynomials with respect to the weight function (1 −x)α(1 + x)β, −1 < x < 1, where the parameters α,β are subjects to the condi- tion α,β > −1 [4, 33]. In particular, it holds that Tm(x) = c−12 ,−12 P (−12 ,− 1 2 ) m (x), Um(x) = c 1 2 , 1 2 P ( 12 , 1 2 ) m (x), Vm(x) = c−12 , 12 P (−12 , 1 2 ) m (x), Wm(x) = c 1 2 ,− 1 2 P ( 12 ,− 1 2 ) m (x). For both Chebyshev polynomials and Jacobi polyno- mials, there exist various multivariate generalizations, see for example [3, 21, 22]. In Sections 3.2, 3.3 and 3.4, we identify some of the two-variable analogous orthogonal polynomials with specific Weyl-orbit func- tions. 3.2. Case A2 Since, for A2 the two simple roots are of the same length, there are only two corresponding families of Weyl-orbit functions, C– and S–functions. For a = a1ω1 +a2ω2 and b = b1α∨1 +b2α∨2 the explicit formulas of C–functions and S–functions are given by Ca(b) = 1 |StabW a| ( e2πi(a1b1+a2b2) + e2πi(−a1b1+(a1+a2)b2) + e2πi((a1+a2)b1−a2b2) + e2πi(a2b1−(a1+a2)b2) + e2πi((−a1−a2)b1+a1b2) + e2πi(−a2b1−a1b2) ) , Sa(b) = e2πi(a1b1+a2b2) −e2πi(−a1b1+(a1+a2)b2) −e2πi((a1+a2)b1−a2b2) + e2πi(a2b1−(a1+a2)b2) + e2πi((−a1−a2)b1+a1b2) −e2πi(−a2b1−a1b2) with the values |StabW a| given in Table 1 of [9]. They are related to the generalized cosine and sine functions TC k and TSk studied in [23] and defined as TC k(t) = 1 3 ( e iπ 3 (k2−k3)(t2−t3) cos k1πt1 + e iπ 3 (k2−k3)(t3−t1) cos k1πt2 + e iπ 3 (k2−k3)(t1−t2) cos k1πt3 ) , TSk(t) = 1 3 ( e iπ 3 (k2−k3)(t2−t3) sin k1πt1 + e iπ 3 (k2−k3)(t3−t1) sin k1πt2 + e iπ 3 (k2−k3)(t1−t2) sin k1πt3 ) , 285 Jiří Hrivnák, Lenka Motlochová Acta Polytechnica Figure 1. The region of orthogonality bounded by the three-cusped deltoid. where t = (t1, t2, t3) ∈ R3 with t1 + t2 + t3 = 0 and k = (k1,k2,k3) ∈ Z3 with k1 + k2 + k3 = 0. The explicit correspondence Ca = 6 |StabW a| TC k, Sa = 6TSk is obtained by the following change of variables and parameters, k1 = a1, k2 = a2, k3 = −a1 −a2, t1 = 2b1 − b2, t2 = −b1 + 2b2, t3 = −b1 − b2. It is possible to express Ca and Sa+%/S% with a ∈ P + as polynomials in Cω1 and Cω2 [1]. Taking into account that Cω1 = Cω2 , one can pass to real variables by making a natural change of variables, these x = Cω1 + Cω2 2 = cos 2πb1 + cos 2πb2 + cos 2π(b1 − b2), y = Cω1 −Cω2 2i = sin 2πb1 − sin 2πb2 − sin 2π(b1 − b2). Since C–functions and S–functions are continuously orthogonal, their polynomial versions inherit the orthogonality property. One can verify that the corresponding polynomials are special cases of two- dimensional analogues of Jacobi polynomials orthogo- nal with respect to the weight function wα(x,y) = ( −(x2 + y2 + 9)2 + 8(x3 − 3xy2) + 108 )α on the region bounded by the three-cusped deltoid called Steiner’s hypocycloid [21] with the boundary given by −(x2 + y2 + 9)2 + 8(x3 − 3xy2) + 108 = 0, see Fig. 1. More precisely, the polynomials Ca and Sa+%/S% correspond to the choices α = −12 and α = 1 2 respectively. 3.3. Case G2 Since, for G2 its two simple roots are of different lengths, all four families of C–, S–, Ss and Sl– functions are obtained [15, 16, 25]. The symmetric and antisymmetric orbit functions are given by the follow- ing formulas for a = a1ω1 +a2ω2 and b = b1α∨1 +b2α∨2 , Ca(b) = 2 |StabW a| ( cos 2π(a1b1 + a2b2) + cos 2π(−a1b1 + (3a1 + a2)b2) + cos 2π((a1 + a2)b1 −a2b2) + cos 2π((2a1 + a2)b1 − (3a1 + a2)b2) + cos 2π((−a1 −a2)b1 + (3a1 + 2a2)b2) + cos 2π((−2a1 −a2)b1 + (3a1 + 2a2)b2) ) , Sa(b) = 2 ( cos 2π(a1b1 + a2b2) − cos 2π(−a1b1 + (3a1 + a2)b2) − cos 2π((a1 + a2)b1 −a2b2) + cos 2π((2a1 + a2)b1 − (3a1 + a2)b2) + cos 2π((−a1 −a2)b1 + (3a1 + 2a2)b2) − cos 2π((−2a1 −a2)b1 + (3a1 + 2a2)b2) ) . The hybrid cases can be expressed as Ssa(b) = 2i |StabW a| ( sin 2π(a1b1 + a2b2) + sin 2π(−a1b1 + (3a1 + a2)b2) − sin 2π((a1 + a2)b1 −a2b2) − sin 2π((2a1 + a2)b1 − (3a1 + a2)b2) − sin 2π((−a1 −a2)b1 + (3a1 + 2a2)b2) − sin 2π((−2a1 −a2)b1 + (3a1 + 2a2)b2) ) , Sla(b) = 2 ( sin 2π(a1b1 + a2b2) − sin 2π(−a1b1 + (3a1 + a2)b2) + sin 2π((a1 + a2)b1 −a2b2) − sin 2π((2a1 + a2)b1 − (3a1 + a2)b2) − sin 2π((−a1 −a2)b1 + (3a1 + 2a2)b2) + sin 2π((−2a1 −a2)b1 + (3a1 + 2a2)b2) ) . These functions have been studied in [22], under the notation CC k, SSk, SC k and CSk with CC k(t) = 1 3 ( cos π(k1 −k3)(t1 − t3) 3 cos πk2t2 + cos π(k1 −k3)(t2 − t1) 3 cos πk2t3 + cos π(k1 −k3)(t3 − t2) 3 cos πk2t1 ) , SSk(t) = 1 3 ( sin π(k1 −k3)(t1 − t3) 3 sin πk2t2 + sin π(k1 −k3)(t2 − t1) 3 sin πk2t3 + sin π(k1 −k3)(t3 − t2) 3 sin πk2t1 ) , 286 vol. 56 no. 4/2016 Weyl Orbit Functions and (Anti)symmetric Trigonometric Functions SC k(t) = 1 3 ( sin π(k1 −k3)(t1 − t3) 3 cos πk2t2 + sin π(k1 −k3)(t2 − t1) 3 cos πk2t3 + sin π(k1 −k3)(t3 − t2) 3 cos πk2t1 ) , CSk(t) = 1 3 ( cos π(k1 −k3)(t1 − t3) 3 sin πk2t2 + cos π(k1 −k3)(t2 − t1) 3 sin πk2t3 + cos π(k1 −k3)(t3 − t2) 3 sin πk2t1 ) , where the variable t = (t1, t2, t3) ∈ R3H = {t ∈ R 3 | t1 + t2 + t3 = 0} and parameter k = (k1,k2,k3) ∈ Z3 ∩ R3H. Indeed, performing the following change of variables and pa- rameters, t1 = −b1 + 3b2, t2 = 2b1 − 3b2, t3 = −b1, k1 = a1 + a2, k2 = a1, k3 = −2a1 −a2, we obtain the following relations. Ca = 12 |StabW a| CC k, Sa = −12SSk, Ssa = 12i |StabW a| SC k, Sla = 12i |StabW a| CSk. In [22], the functions Ca,Sa+%/S%,Ssa+%s/Ss%s and Sl a+%l/S l %l are expressed as two-variable polynomials in variables x = 1 6 Cω2 = 1 3 ( cos 2πb2 + cos 2π(b1 − b2) + cos 2π(−b1 + 2b2) ) , y = 1 6 Cω1 = 1 3 ( cos 2πb1 + cos 2π(−b1 + 3b2) + cos 2π(2b1 − 3b2) ) and it is shown that they are orthogonal within each family with respect to a weighted integral on the region (see Fig. 2) containing points (x,y) satisfying (1 + 2y − 3x2)(24x3 −y2 − 12xy − 6x− 4y − 1) ≥ 0 with the weight function wα,β(x,y) equal to (1 + 2y − 3x2)α(24x3 −y2 − 12xy − 6x− 4y − 1)β with parameters α = β = − 1 2 for C–functions, α = β = 1 2 for S–functions, α = 1 2 , β = − 1 2 for Ss–functions, α = − 1 2 , β = 1 2 for Sl–functions. Figure 2. The region of orthogonality for the case G2. 3.4. Cases Bn and Cn It is shown in this section that the C–, S–, Ss– and Sl–functions arising from Bn and Cn are re- lated to the symmetric and antisymmetric multivari- ate generalizations of trigonometric functions [17]. The symmetric cosine functions cos+λ (x) and the an- tisymmetric cosine functions cos−λ (x) of the variable x = (x1, . . . ,xn) ∈ Rn are labelled by the parameter λ = (λ1, . . . ,λn) ∈ Rn and are given by the following explicit formulas, cos+λ (x) ≡ ∑ σ∈Sn n∏ k=1 cos(πλσ(k)xk), cos−λ (x) ≡ ∑ σ∈Sn sgn(σ) n∏ k=1 cos(πλσ(k)xk), where Sn denotes the symmetric group consisting of all permutations of numbers 1, . . . ,n, and sgn(σ) is the signature of σ. The symmetric sine functions sin+λ (x) and the antisymmetric sine functions sin − λ (x) are defined similarly, sin+λ (x) ≡ ∑ σ∈Sn n∏ k=1 sin(πλσ(k)xk), sin−λ (x) ≡ ∑ σ∈Sn sgn(σ) n∏ k=1 sin(πλσ(k)xk). Firstly, consider the Lie algebra Bn and an orthonor- mal basis {e1, . . . ,en} of Rn such that αi = ei −ei+1 for i = 1, . . . ,n− 1 and αn = en. If we determine any a ∈ Rn by its coordinates with respect to the basis {e1, . . . ,en}, a = (a1, . . . ,an) = a1e1 + · · · + anen, then it holds for the generators ri, i = 1, . . . ,n−1 and rn of the Weyl group W(Bn) of Bn that ri(a1, . . . ,ai,ai+1, . . . ,an) = (a1, . . . ,ai+1,ai, . . . ,an), rn(a1, . . . ,an−1,an) = (a1, . . . ,an−1,−an). 287 Jiří Hrivnák, Lenka Motlochová Acta Polytechnica Therefore, W(Bn) consists of all permutations of the coordinates ai with possible sign alternations of some of them, and we actually have that W (Bn) is isomor- phic to (Z/2Z)n o Sn [12]. This implies that Φa(b) = ∑ w∈W(Bn) e2πi〈w(a),b〉 = ∑ σ∈Sn n∏ k=1 ∑ lk=±1 e2πi(lkaσ(k)bk) = ∑ σ∈Sn n∏ k=1 ( e2πiaσ(k)bk + e−2πiaσ(k)bk ) = 2n ∑ σ∈Sn n∏ k=1 cos(2πaσ(k)bk) = 2n cos+a (2b). Since det is a homomorphism on W(Bn), we also obtain ϕa(b) = ∑ w∈W(Bn) det(w)e2πi〈w(a),b〉 = ∑ σ∈Sn det(σ) n∏ k=1 ∑ lk=±1 lke 2πi(lkaσ(k)bk) = ∑ σ∈Sn det(σ) n∏ k=1 ( e2πiaσ(k)bk −e−2πiaσ(k)bk ) = (2i)n ∑ σ∈Sn det(σ) n∏ k=1 sin(2πaσ(k)bk) = (2i)n sin−a (2b). Similar connections are valid for Ss–functions and Sl–functions, ϕsa(b) = (2i) n sin+a (2b), ϕ l a(b) = 2 n cos−a (2b). Since Lie algebras Bn and Cn are dual to each other, we can deduce that the symmetric and antisymmetric generalizations are also connected to the Weyl-orbit functions of Cn. In order to obtain explicit relations, one can proceed by analogy with case Bn and intro- duce an orthogonal basis {f1, . . . ,fn} such that for i = 1, . . . ,n− 1 〈fi,fi〉 = 1 2 , αi = fi −fi+1 and αn = 2fn. We denote by ãi the coordinates of any point a ∈ Rn with respect to the basis {f1, . . . ,fn}, i.e. a = (ã1, . . . , ãn) = ã1f1 + · · · + ãnfn. The generators ri, i = 1, . . . ,n− 1 and rn of the Weyl group W(Cn) corresponding to Cn are also given by ri(ã1, . . . , ãi, ãi+1, . . . , ãn) = (ã1, . . . , ãi+1, ãi, . . . , ãn), rn(ã1, . . . , ãn−1, ãn) = (ã1, . . . , ãn−1,−ãn). Thus, proceeding as before, we derive the following. Φa(b) = 2n cos+a (b), ϕa(b) = (2i) n sin−a (b), ϕsa(b) = 2 n cos−a (b), ϕ l a(b) = (2i) n sin+a (b). Figure 3. The region of orthogonality bounded by two lines and parabola. Note that the Ss–functions are related to cos−a and the Sl–functions are related to sin+a in the case of Cn, whereas the Ss–functions correspond to sin+a and the Sl–functions correspond to cos−a if we consider the simple Lie algebra Bn. This follows from the fact that the short (long) roots of Cn are dual to the long (short) roots of Bn. Setting n = 2, the construction of the polynomials PI,+(k1,k2) ≡ cos + (k1,k2) , PI,−(k1,k2) ≡ cos−(k1+1,k2) cos−(1,0) , PIII,+(k1,k2) ≡ cos+(k1+ 12 ,k2+ 12 ) cos+( 12 , 12 ) , PIII,−(k1,k2) ≡ cos−(k1+ 32 ,k2+ 12 ) cos−( 32 , 12 ) labelled by k1 ≥ k2 ≥ 0 and in the variables X1 ≡ cos+(1,0)(x1,x2) = cos(x1) + cos(x2), X2 ≡ cos+(1,1)(x1,x2) = 2 cos(x1) cos(x2) yields special cases of two-variable polynomials built in [19–21]. These polynomials are constructed by orthogonalization of monomials 1,u,v,u2,uv,v2, . . . of generic variables u,v with respect to the weight function (1 −u + v)α(1 + u + v)β(u2 − 4v)γ in the domain bounded by the curves 1 −u + v = 0, 1 + u + v = 0 and u2 − 4v = 0, see Fig. 3. The pa- rameters α,β,γ are required to satisfy the conditions α,β,γ > −1, α + γ + 32 > 0 and β + γ + 3 2 > 0. The resulting polynomials with the highest term um−kvk are denoted by pα,β,γm,k (u,v), where m ≥ k ≥ 0. The polynomial variables X1 and X2 are related to the variables u and v of [19–21] by X1 = u, X2 = 2v and it can easily be shown that • PI,+(k1,k2) coincides, up to a constant, with p α,β,γ k1,k2 (u,v) for α = β = γ = −12 , • PIII,+(k1,k2) coincides, up to a constant, with p α,β,γ k1,k2 (u,v) for α = γ = −12 and β = 1 2 , 288 vol. 56 no. 4/2016 Weyl Orbit Functions and (Anti)symmetric Trigonometric Functions • PI,−(k1,k2) coincides, up to a constant, with p α,β,γ k1,k2 (u,v) for α = β = −12 and γ = 1 2 , • PIII,−(k1,k2) coincides, up to a constant, with p α,β,γ k1,k2 (u,v) for α = −12 and β = γ = 1 2 . 4. Concluding remarks (1.) Symmetric and antisymmetric cosine functions can be used to construct multivariate orthogonal polynomials analogous to the Chebyshev polyno- mials of the first and third kind. The method of construction is based on decomposition of the prod- ucts of these functions and is fully described in [7]. To build polynomials analogous to the Chebyshev polynomials of the second and fourth kind, it seems that the symmetric and antisymmetric generaliza- tions of sine functions have to be analysed. This hypothesis is supported by the decomposition of the products of two-dimensional sine functions which can be found in [11]. (2.) Another approach to generalization of the multi- variate polynomials related to the Weyl-orbit func- tions stems from the shifted orthogonality of the orbit functions developed in [2]. This generalization encompasses shifts of the points of the sets over which the functions are discretely orthogonal, and also shifts of the labeling weights. As a special case it contains for A1 all four kinds of Chebyshev poly- nomials. The existence of analogous polynomials obtained through this approach and their relations to already known generalizations deserves further study. (3.) 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