Acta Polytechnica doi:10.14311/AP.2018.58.0402 Acta Polytechnica 58(6):402–413, 2018 © Czech Technical University in Prague, 2018 available online at http://ojs.cvut.cz/ojs/index.php/ap MULTIDIMENSIONAL HYBRID BOUNDARY VALUE PROBLEM Marzena Szajewskaa,∗, Agnieszka Tereszkiewiczb a Institute of Mathematics, University of Bialystok, 1M Ciolkowskiego, PL-15-245 Bialystok, Poland b Bialystok University of Technology, Faculty of Civil and Environmental Engineering, 45 E Wiejska, PL-15-351 Bialystok, Poland ∗ corresponding author: m.szajewska@math.uwb.edu.pl Abstract. The purpose of this paper is to discuss three types of boundary conditions for few families of special functions orthogonal on the fundamental region. Boundary value problems are considered on a simplex F in the real Euclidean space Rn of dimension n > 2. Keywords: hybrid functions, Dirichlet boundary value problem, Neumann boundary value problem, mixed boundary value problem. 1. Introduction The boundary value problems, considered in the paper, is a generalization of [24] in which the authors pre- sented two-dimensional hybrids with mixed boundary value problems. Here we take a real Euclidean space Rn of dimension n on finite regions F ⊂ Rn that are polyhedral domains. The aim of this paper is to seek solutions of the Helmholtz equation with mixed bound- ary condition by analogy to two-dimensional cases. The solutions are presented as expansions into a series of special functions that satisfy required conditions at the (n− 1)-dimensional boundaries of F . The recent discovery of special functions [5, 10, 11, 16, 19, 23] makes realization of this idea easy and straightforward in any dimension. The new functions, called ’multi- dimensional hybrids’, satisfy the Dirichlet boundary condition on some parts of the boundary F and Neu- mann on the remaining ones. The methods used in the paper are the standard methods of separation of variables for differential equations (see for exam- ple [15, 18]) and the branching rule method for orbits of reflection groups (see for example [19, 24, 28]). The boundary value conditions play an important role in mathematics and physics. They are used, for exam- ple, in the theory of elasticity, electrostatics and fluid mechanics [4, 9, 26]. In §2, we present the well known Helmholtz equation and three types of boundary conditions. In § 3, we recall some facts about finite reflection groups. The next Section is devoted to special functions, projection matrices and branching rules. In § 5 we present 3D cases in details, namely B3,C3,C2×A1,G2×A1,A1× A1×A1. In the Appendix we list tables containing the values of functions on the boundaries of fundamental region. 2. Helmholtz equation and boundary conditions In this paper we consider the partial differential equation called the homogeneous Helmholtz equa- tion [15, 18, 25, and references therein]: ∆Ψ(x) = −w2Ψ(x), (1) where w-positive real constant, x = (y1, . . . ,yn) is given in Cartesian coordinates and ∆ = n∑ i=1 ∂2 ∂y2 i . Using a standard method of separation of variables for (1) (see for example [15]) and searching for the solutions in the form Ψ(x) = X1(y1) · · ·Xn(yn), we have the following differential equation X′′1 X2 · · ·Xn + X1X ′′ 2 · · ·Xn + · · · + X1X2 · · ·X′′n + w 2X1X2 · · ·Xn = 0. (2) By introducing −k21, . . . ,−k 2 n so-called separation con- stants, we get the solution of (2) in the form X11 (y1) = cos(k1 y1), ... X1n−1(yn−1) = cos(kn−1 yn−1), X1n(yn) = cos(kn yn), X21 (y1) = sin(k1 y1), ... X2n−1(yn−1) = sin(kn−1 yn−1), X2n(yn) = sin(kn yn), (3) where kn := √ w2 − ∑n−1 i=1 ki 2, ki 6= 0 for i = 1, . . . ,n. The way of choosing separation constants is not unique. In this paper ki, i = 1, . . . ,n−1 are selected according to a branching rule method [19, 24, 28], see next sections. Three types of boundary conditions. D: A Dirichlet boundary condition defines the value of the function itself Ψ(x) = f(x), for x ∈ ∂F, 402 http://dx.doi.org/10.14311/AP.2018.58.0402 http://ojs.cvut.cz/ojs/index.php/ap vol. 58 no. 6/2018 Multidimensional Hybrid Boundary Value Problem where f(x) is a given function defined on the bound- ary. N: A Neumann boundary condition defines the value of the normal derivative of the function ∂Ψ ∂n (x) = f(x), for x ∈ ∂F, where n denotes normal vector to the boundary ∂F . M: A mixed boundary condition defines the value of the function itself on one part of the boundary and the value of the normal derivative of the function on the other part of the boundary D: Ψ|∂F0 = f0, N: ∂Ψ ∂n ∣∣∣ ∂F1 = f1, where ∂F = ∂F0∪∂F1 and f0,f1 are given functions, defined on the appropriate boundary. 3. Finite reflection groups Our method is general and can be presented for any crystallographic finite reflection groups G of any rank and any dimension which are associated with sim- ple and semisimple Lie algebras/groups [1, 7, 10, 27]. There is a complete classification of finite reflection groups given by Dynkin diagrams [2, 3, 10]. These graphs provide the relative angles and relative length of the vectors of a set of simple roots of the root sys- tems. There are two kinds of root systems according to the number of roots with different lengths: sys- tems with one root length, and systems with two root lengths. A reflection r in a hyperplane orthogonal to the long/short root and passing through the origin of Rn be denoted by rl/rs respectively. Working with finite reflection groups, it is conve- nient to use four bases in Rn, namely natural e-, the simple root α-, co-root α̌- and weight ω-bases [2, 7, 10]. The co-root basis α̌ is defined by the formula α̌i = 2αi 〈αi|αi〉 . The ω-basis is dual to simple root basis. The relation- ship between considered bases is standard for group theory and is expressed by 〈α̌i|ωj〉 = δij. There are two types of fundamental region either simplex for simple Lie group G or prism for semisimple one. The simplex with n + 1 vertices has the following coordinates F = { 0, ω1 q1 , . . . , ωn qn } , where qi, i = 1, . . . ,n, called co-marks, can be found in [6, 10] for any simple Lie group G of any rank and any dimension. The fundamental region for prisms can be given in the following sense. Let G = G1 × G2, where G1,G2 are finite reflection groups. Let ω1, . . . ,ωk be a set of generating elements of G1 and ωk+1, . . . ,ωn of G2. Then the prism can be written as follows F = { 0, ωi qi , ωj qj , ωi qi + ωj qj } , where i = 1, . . . ,k and j = k + 1, . . . ,n and qi,qj are co-marks [6, 10]. Let ∂Fi be contained in the hyperplane generated by a set of orthogonal reflections r0,r1, . . . ,ri−1,ri+1, . . . ,rn, i = {0, . . . ,n}, where r0 is an affine reflection (it cor- responds to long reflection). If ri corresponds to the reflection orthogonal to the short/long root then we denote a part of the boundary by ∂Fs or ∂Fl respec- tively. In other words we can say that the boundary ∂F of the fundamental region F will be denoted by ∂Fl/∂Fs if its normal vector is perpendicular to the long/short root α respectively. 4. Special functions as a solution of Helmholtz equation There are four kinds of special functions of interest to us whose orthogonality on lattice fragment F is known for any simple Lie group [5, 6, 10, 16, 17, 19, 20, 23, and references therein]. The general formula for special functions (called orbit functions) [10, 11] corresponding to the finite reflection group G is given by ∑ w∈G σ(w)e2πi〈wλ|x〉, λ ∈ P +, x ∈ F (4) where the summation extends over the whole group G, P + denotes the set of dominant weights [10] and σ(w) = ±1 depends on the type of the orbit func- tion. The homomorphism σ : G → {±1} is a prod- uct of σ(rl),σ(rs) ∈ {±1}. There are four types of maps σ [16, 17]: σ(rl) = σ(rs) = 1 =⇒ C, σ(rl) = σ(rs) = −1 =⇒ S, σ(rl) = −1, σ(rs) = 1 =⇒ Sl, σ(rl) = 1, σ(rs) = −1 =⇒ Ss. (5) All four families of functions defined above are formed as finite sums of exponential terms. The first two families, namely C- and S-functions are generalized cosine and sine functions. They are symmetric and skew-symmetric with respect to the finite reflection group [6, 10, 16, 19–21, 23]. The other two, Ss- and Sl-functions [11, 12, 16, 17, 23] have analogous prop- erties as C- and S-functions. The main difference between them is their behaviour at the boundary of their domain of orthogonality in Rn. Every finite group G generated by reflections can be reduced to a subgroup A1×···×A1 using a branching 403 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica rule method described in [13, 14, 19, 22, 24, 28]. This method allows us to do the separation of variables for special functions (5) corresponding to group G. As a result, we have all the functions written as a product of sine and cosine functions. Remark 1. All four families of functions (5) pre- sented above are solutions of the Helmholtz equation (1) where w2 = 4π2〈λ|λ〉 with one of the three types of boundary conditions described in § 2. Projection matrix reduces any n-dimensional group G to a subgroup A1×. . .×A1 [13, 19]. The branching rule allows one to divide any orbit of group G into a union of orbits of group A1. As an example see 3D cases described in § 5. Remark 2. The union of orbits which we get after reduction determine our choice of separating constants used in solution of Helmholtz equation (1). The behaviour of the functions C,S,Ss and Sl on the boundary ∂F can be summarize in the Tab. 1. D N ∂Fs ∂Fl ∂Fs ∂Fl Cλ(x) ∗ ∗ 0 0 Sλ(x) 0 0 ∗ ∗ Ssλ(x) 0 ∗ ∗ 0 Slλ(x) ∗ 0 0 ∗ Table 1. Behaviour of the functions C, S, Ss and Sl on the boundary ∂F for any finite refleciton group G where ∗ denotes any function non-equivalent to 0. For any group G considered in the paper C- functions fulfil the Dirichlet condition with value non- equivalent to 0 and the Neumann condition with 0 value on the whole boundary. The S-functions behave inversely. The Ss-functions fulfil the Dirichlet condi- tion with a value non-equivalent to 0 on the part of boundary denoted by ∂Fl and the Neumann condition with a value non-equivalent to 0 on the part of the boundary denoted by ∂Fs. The Sl functions behave inversely. In the case of C-functions we talk about Dirichlet boundary condition and S-functions - Neu- mann boundary condition. For Ss- and Sl-functions we talk about mixed boundary condition. In the next section we present 3D cases in details. 5. 3D finite reflection groups The 3 dimensional groups which we considered here are B3,C3,C2×A1,G2×A1,A1×A1×A1 [2, 7, 8, 10]. We use the following notation for coordinates: R3 3 λ = (a,b,c)ω = aω1 + bω2 + cω3, R3 3 x = (x1,x2,x3)α̌ = (y1,y2,y3)e, where indexes e, ω, and α̌ denote natural-, ω-, and α̌- basis, respectively. The action of the Laplace operator ∇ on the functions given in different bases can be found in [10]. In the next subsections we describe each case in details. For each case we present functions which are the solutions of Helmholtz equation (1). We give the exact forms of the projection matrices and branching rules which allow us to choose the separation constants used in (3). All functions described below fulfil one of the three types of boundary conditions described in §,2. 5.1. B3 and C3 groups The α-basis vectors in Cartesian coordinates are B3: C3: α1 := (1,−1, 0)e, α1 := 1 √ 2 (1,−1, 0)e, α2 := (0, 1,−1)e, α2 := 1 √ 2 (0, 1,−1)e, α3 := (0, 0, 1)e, α3 := 1 √ 2 (0, 0, 2)e. As one can easily notice the short root for B3 is α3 and for C3 are α1,α2. The fundamental regions F for B3 and C3 groups, written in ω-basis, have the vertices: FB3 = {0,ω1, 1 2ω2,ω3}, FC3 = {0,ω1,ω2,ω3}, and are shown in Fig. 1. Figure 1. The fundamental region F for B3 and C3 group. The reduction of B3 and C3 to a subgroup A1 × A1 ×A1 is given by the projection matrices PB3 =  1 1 01 1 1 0 2 1   , PC3 =  1 1 10 1 1 0 0 1   . Then the branching rule is the following: O(a,b,c) PB3−−→ O(2a+2b+c)O(2b+c)O(c) ∪O(2b+c)O(2a+2b+c)O(c) ∪O(2a+2b+c)O(c)O(2b+c) ∪O(c)O(2a+2b+c)O(2b+c) ∪O(2b+c)O(c)O(2a+2b+c) ∪O(c)O(2b+c)O(2a+2b+c), 404 vol. 58 no. 6/2018 Multidimensional Hybrid Boundary Value Problem O(a,b,c) PC3−−→ O(a+b+c)O(b+c)O(c) ∪O(b+c)O(a+b+c)O(c) ∪O(a+b+c)O(c)O(b+c) ∪O(b+c)O(c)O(a+b+c) ∪O(c)O(a+b+c)O(b+c) ∪O(c)O(b+c)O(a+b+c). According to Remarks 1 and 2 the separation con- stants for B3 and C3 group we can choose as −k21 = −π 2(2a + 2b + c)2, −k22 = −π 2(2b + c)2, −k23 = −π 2c2, (6) where w2 = 4π2(a2 + 2ab + 2b2 + ac + 2bc + 34c 2). The separation constants for C3 group are −k21 = −π 2(a + b + c)2, −k22 = −π 2(b + c)2, −k23 = −π 2c2, (7) where w2 = 4π2( 12a 2 + ab + b2 + ac + 2bc + 32c 2). The explicit forms of orbit functions for B3 and C3 group have form B3: Ca,b,c(x) = C2a+2b+c(x1)C2b+c(x2)Cc(x3) + C2b+c(x1)C2a+2b+c(x2)Cc(x3) + C2a+2b+c(x1)Cc(x2)C2b+c(x3) + Cc(x1)C2a+2b+c(x2)C2b+c(x3) + C2b+c(x1)Cc(x2)C2a+2b+c(x3) + Cc(x1)C2b+c(x2)C2a+2b+c(x3), Sla,b,c(x) = C2a+2b+c(x1)C2b+c(x2)Cc(x3) −C2b+c(x1)C2a+2b+c(x2)Cc(x3) −C2a+2b+c(x1)Cc(x2)C2b+c(x3) + Cc(x1)C2a+2b+c(x2)C2b+c(x3) + C2b+c(x1)Cc(x2)C2a+2b+c(x3) −Cc(x1)C2b+c(x2)C2a+2b+c(x3), Sa,b,c(x) = S2a+2b+c(x1)S2b+c(x2)Sc(x3) −S2b+c(x1)S2a+2b+c(x2)Sc(x3) −S2a+2b+c(x1)Sc(x2)S2b+c(x3) + Sc(x1)S2a+2b+c(x2)S2b+c(x3) + S2b+c(x1)Sc(x2)S2a+2b+c(x3) −Sc(x1)S2b+c(x2)S2a+2b+c(x3), Ssa,b,c(x) = S2a+2b+c(x1)S2b+c(x2)Sc(x3) + S2b+c(x1)S2a+2b+c(x2)Sc(x3) + S2a+2b+c(x1)Sc(x2)S2b+c(x3) + Sc(x1)S2a+2b+c(x2)S2b+c(x3) + S2b+c(x1)Sc(x2)S2a+2b+c(x3) + Sc(x1)S2b+c(x2)S2a+2b+c(x3); C3: Ca,b,c(x) = Ca+b+c(x1)Cb+c(x2)Cc(x3) + Cb+c(x1)Ca+b+c(x2)Cc(x3) + Ca+b+c(x1)Cc(x2)Cb+c(x3) + Cc(x1)Ca+b+c(x2)Cb+c(x3) + Cb+c(x1)Cc(x2)Ca+b+c(x3) + Cc(x1)Cb+c(x2)Ca+b+c(x3), Ssa,b,c(x) = Ca+b+c(x1)Cb+c(x2)Cc(x3) −Cb+c(x1)Ca+b+c(x2)Cc(x3) −Ca+b+c(x1)Cc(x2)Cb+c(x3) + Cc(x1)Ca+b+c(x2)Cb+c(x3) + Cb+c(x1)Cc(x2)Ca+b+c(x3) −Cc(x1)Cb+c(x2)Ca+b+c(x3), Sa,b,c(x) = Sa+b+c(x1)Sb+c(x2)Sc(x3) −Sb+c(x1)Sa+b+c(x2)Sc(x3) −Sa+b+c(x1)Sc(x2)Sb+c(x3) + Sc(x1)Sa+b+c(x2)Sb+c(x3) + Sb+c(x1)Sc(x2)Sa+b+c(x3) −Sc(x1)Sb+c(x2)Sa+b+c(x3) Sla,b,c(x) = Sa+b+c(x1)Sb+c(x2)Sc(x3) + Sb+c(x1)Sa+b+c(x2)Sc(x3) + Sa+b+c(x1)Sc(x2)Sb+c(x3) + Sc(x1)Sa+b+c(x2)Sb+c(x3) + Sb+c(x1)Sc(x2)Sa+b+c(x3) + Sc(x1)Sb+c(x2)Sa+b+c(x3). The functions on the right side of the above equations are special functions corresponding to group A1 Cµ(xi) = ∑ µ∈A1 e2πi〈µ|xi〉 = 2 cos(2πµxi), Sµ(xi) = ∑ µ∈A1 σ(µ)e2πi〈µ|xi〉 = 2i sin(2πµxi), where µ ∈ P +A1,xi ∈ FA1, i = 1, 2, 3. The coordinate xi respond to the i-th coordinate in A1 × A1 × A1. The functions Cµ(xi) and Sµ(xi) are the solutions of Helmholtz equation (1) in the form (3) in 1D case. For group B3 the functions C- and Sl- are real valued and S- and Ss are purely imaginary. In the case of C3, the functions C- and Ss- are real valued and S- and Sl are purely imaginary. The normal vectors shown on Fig. 2 are B3: C3: n1 = {− 1√2, 1√ 2 , 0}, n1 = {0, 0, 1}, n2 = {0,− 1√2, 1√ 2 }, n2 = {0, 1√2,− 1√ 2 }, n3 = {0, 0,−1}, n3 = { 1√2,− 1√ 2 , 0}, n4 = { 1√2, 1√ 2 , 0}, n4 = {1, 0, 0}. 405 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica Figure 2. Normal vectors for B3 and C3. In the case of B3 group normal vector n4 is perpen- dicular to the short simple root. The rest of them, namely n1,n2,n3 are perpendicular to the long simple roots. So the boundaries that correspond to normal vectors are ∂Fs for n1 and ∂Fl for the others. The values of the functions on the boundaries are summa- rized in the Appendix in Tab. 2. In the case of C3 it is a little bit different, the normal vectors n2,n3 are perpendicular to the short simple roots and n1,n4 - to the long simple roots. The values of the functions on the boundaries are given in the Appendix in Tab. 3. 5.2. C2 × A1 and G2 × A1 groups The α-basis vectors in Cartesian coordinates have the form C2 ×A1: G2 ×A1: α1 := 1 √ 2 (1,−1, 0)e, α1 := ( √ 2, 0, 0)e, α2 := 2 √ 2 (0, 2, 0)e, α2 := ( − 1 √ 2 , 1√ 6 , 0 ) e , α3 := 1 √ 2 (0, 0, 2)e, α3 := 1 √ 2 (0, 0, 2)e. The vertices of the fundamental regions F for C2 × A1,G2×A1 groups, shown in Fig. 3, written in ω-basis are FC2×A1 = {0,ω1,ω2,ω3,ω1 + ω3,ω2 + ω3}, FG2×A1 = { 0, 1 2 ω1,ω2,ω3, 1 2 ω1 + ω3,ω2 + ω3 } . Figure 3. The fundamental region F for C2 × A1 and G2 × A1 group. The groups C2 ×A1,G2 ×A1 can be reduced to a subgroup A1×A1×A1 using a branching rule method described in [19, 24]. The projection matrices and the branching rules are PC2×A1 =  1 1 00 1 0 0 0 1   , PG2×A1 =  1 1 03 1 0 0 0 1   , O(a,b)O(c) PC2 ×A1−−−−−→ O(a+b)O(b)O(c) ∪O(b)O(a+b)O(c), O(a,b)O(c) PG2 ×A1−−−−−→ O(a+b)O(3a+b)O(c) ∪O(2a+b)O(b)O(c) ∪O(a)O(3a+2b)O(c). The separation constants for C2 ×A1 are −k21 = −π 2(a+b)2, −k22 = −π 2b2, −k23 = −π 2c2, (8) where w2 = 4π2(a2 + ab + b2 + 12c 2) and for G2 ×A1 equal −k21 = −2π 2(2a + b)2, −l21 = −2π 2(a + b)2, −k22 = − 2 3π 2b2, −l22 = − 2 3π 2(3a + b)2, −k23 = −2π 2c2, −l23 = −2π 2c2, −m21 = −2π 2a2, −m22 = − 2 3π 2(3a + 2b)2, −m23 = −2π 2c2, (9) where w2 = 4π2(2a2 + 2ab + 23b 2 + 12c 2). The explicit forms of orbit functions are C2 ×A1: Ca,b,c(x) = Ca+b(x1)Cb(x2)Cc(x3) + Cb(x1)Ca+b(x2)Cc(x3), Ssa,b,c(x) = Ca+b(x1)Cb(x2)Cc(x3) −Cb(x1)Ca+b(x2)Cc(x3), Sa,b,c(x) = Sa+b(x1)Sb(x2)Sc(x3) −Sb(x1)Sa+b(x2)Sc(x3), Sla,b,c(x) = Sa+b(x1)Sb(x2)Sc(x3) + Sb(x1)Sa+b(x2)Sc(x3); G2 ×A1: Ca,b,c(x) = Ca(x1)C3a+2b(x2)Cc(x3) + Ca+b(x1)C3a+b(x2)Cc(x3) + C2a+b(x1)Cb(x2)Cc(x3), Sla,b,c(x) = Sa(x1)C3a+2b(x2)Sc(x3) −Sa+b(x1)C3a+b(x2)Sc(x3) + S2a+b(x1)Cb(x2)Sc(x3), Sa,b,c(x) = Sa(x1)S3a+2b(x2)Sc(x3) −Sa+b(x1)S3a+b(x2)Sc(x3) + S2a+b(x1)Sb(x2)Sc(x3), 406 vol. 58 no. 6/2018 Multidimensional Hybrid Boundary Value Problem Ssa,b,c(x) = Ca(x1)S3a+2b(x2)Cc(x3) −Ca+b(x1)S3a+b(x2)Cc(x3) −C2a+b(x1)Sb(x2)Cc(x3), where Cµ(xi),Sµ(xi) for i = 1, 2, 3 are the same as in the previous cases. For group C2 × A1 the functions C- and Ss- are real valued and S- and Sl are purely imaginary. In the case of G2 ×A1, the functions C- and Sl- are real valued and S- and Ss are purely imaginary. Figure 4. Normal vectors of F for C2 × A1, G2 × A1 groups. The normal vectors shown in Fig. 4 are C2 ×A1: G2 ×A1: n1 = {0, 0,−1}, n1 = {0, 0,−1}, n2 = {0,−1, 0}, n2 = { √ 3 2 ,− 1 2, 0}, n3 = {− 1√2, 1√ 2 , 0}, n3 = {−1, 0, 0}, n4 = {1, 0, 0}, n4 = {12, √ 3 2 , 0}, n5 = {0, 0, 1}, n5 = {0, 0, 1}. In the case of C2 ×A1 the group normal vector n3 is perpendicular to the short simple root. The rest of them, namely n1,n2,n4,n5 are perpendicular to the long simple roots. So the boundaries that correspond to normal vectors are ∂Fs for n3 and ∂Fl for the others. The values of the functions on the boundaries are summarized in Appendix in Tab. 4. In the case of G2 ×A1, the normal vector n2 corresponds to the short simple root so to the boundary ∂Fs and the rest of normal vectors to the long simple roots i.e. to the boundaries ∂Fl. The values of the functions on the boundaries are given in Appendix in Tab. 5. 5.3. A1 × A1 × A1 group Although the root system of A1 ×A1 ×A1 does not have two different lengths of roots, it is still an inter- esting case for us. The α-basis vectors in Cartesian coordinates have the form α1 := ( √ 2, 0, 0)e, α2 := (0, √ 2, 0)e, α3 := (0, 0, √ 2)e. According to (4) and (5) there are two families of special functions C and S. By the analogy to homo- morphism (5) we can define new families of functions. σ(r1) = σ(r2) = σ(r3) = 1 =⇒ CCC, σ(r1) = σ(r2) = σ(r3) = −1 =⇒ SSS, σ(r1) = σ(r2) = 1, σ(r3) = −1 =⇒ CCS, σ(r1) = σ(r2) = −1, σ(r3) = 1 =⇒ SSC, σ(r1) = σ(r3) = 1, σ(r2) = −1 =⇒ CSC, σ(r1) = −1, σ(r2) = σ(r3) = 1 =⇒ SCC, σ(r1) = 1, σ(r2) = σ(r3) = −1 =⇒ CSS, σ(r2) = −1, σ(r1) = σ(r3) = 1 =⇒ SCS, where CCC, SSS correspond to C and S-functions, re- spectively and the rest of them to Sl- and Ss-functions. All families of functions defined on the fundamental region FA1×A1×A1 = {0,ω1,ω2,ω3,ω1 + ω2, ω1 + ω3,ω2 + ω3,ω1 + ω2 + ω3}. fulfill mixed boundary condition (see Tab. 6). Figure 5. The fundamental region F with normal vectors of A1 × A1 × A1 group. The projection matrix is the identity matrix and then the choice of separation constants is trivial: −k21 = −π 2a2, −k22 = −π 2b2, −k23 = −π 2c2. (10) According to the branching rule O(a,b,c) PA1 ×A1 ×A1−−−−−−−−→ O(a)O(b)O(c) we have CCCa,b,c(x) := Ca(x1)Cb(x2)Cc(x3), SCSa,b,c(x) := Sa(x1)Cb(x2)Sc(x3), CSSa,b,c(x) := Ca(x1)Sb(x2)Sc(x3), SSCa,b,c(x) := Sa(x1)Sb(x2)Cc(x3), SSSa,b,c(x) := Sa(x1)Sb(x2)Sc(x3), CSCa,b,c(x) := Ca(x1)Sb(x2)Cc(x3), CCSa,b,c(x) := Ca(x1)Cb(x2)Sc(x3), SCCa,b,c(x) := Sa(x1)Cb(x2)Cc(x3), 407 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica where Cµ(xi),Sµ(xi) for i = 1, 2, 3 are the same as in the previous cases. The first four families of functions are real valued and the rest of them are pure imagi- nary. Normal vectors shown on Fig. 5 are n1 = {0, 0,−1}, n2 = {0,−1, 0}, n3 = {−1, 0, 0}, n4 = {0, 1, 0}, n5 = {1, 0, 0}, n6 = {0, 0, 1}. The values of the functions on the boundaries are shown in Appendix in Tab. 6. 6. Appendix In Tables 2–6 we collect the values of special functions on the boundaries of the fundamental region F for each of 3D finite reflection groups presented in the paper. References [1] Borel, A., and J. de Siebental, “Les sous-groupes fermés de rang maximum de groupes de Lie clos.” Comment. Math. Helv. 23, (1949): 200–221. [2] Bourbaki, N. Groupes et algèbres de Lie, Chapters IV, V, VI, Hermann, Paris, 1968. [3] Dynkin, E.B. “Semisimple subalgebras of semisimple Lie algebras.” AMS Trnanslations, Series 2, Vol. 6, (1957): 111–244. [4] Griffiths, D.J., and R. College, Introduction to electrodynamics, Prentice Hall, New Jersey, 1999. [5] Hakova, L., Hrivnak, J., and J. Patera, “Four families of Weyl group orbit functions of B3 and C3.” J. Math. Phys. 54, 083501 (2013). [6] Hrivnak, J., and J. Patera, “On discretization of tori of compact simple Lie groups.” J. Phys. A: Math. 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Phys., 6 (1965): 1534. 408 http://dx.doi.org/10.1007/s00041-014-9355-0 vol. 58 no. 6/2018 Multidimensional Hybrid Boundary Value Problem B 3 C S D N D N F 1 2( C 2b + c (x )C 2a + 2b + c (x )C c (z )+ C c (x )C 2b + c (z )C 2a + 2b + c (x ) 0 0 √ 2i (− k 1 S 2b + c (x )C 2a + 2b + c (x )S c (z )+ k 1 S c (x )C 2a + 2b + c (x )S 2b + c (z ) + C c (x )C 2b + c (x )C 2a + 2b + c (z )) − k 2 S c (x )C 2b + c (x )S 2a + 2b + c (z )+ k 2 S 2a + 2b + c (x )C 2b + c (x )S c (z ) − k 3 S 2a + 2b + c (x )C c (x )S 2b + c (z )+ k 3 S 2b + c (x )C c (x )S 2a + 2b + c (z )) F 2 2( C c (y )C 2b + c (y )C 2a + 2b + c (x )+ C c (x )C 2b + c (y )C 2a + 2b + c (y ) 0 0 √ 2i (k 1 S 2b + c (x )C 2a + 2b + c (y )S c (y )− k 1 S c (x )C 2a + 2b + c (y )S 2b + c (y ) + C c (y )C 2b + c (x )C 2a + 2b + c (y )) + k 2 S c (x )C 2b + c (y )S 2a + 2b + c (y )− k 2 S 2a + 2b + c (x )C 2b + c (y )S c (y ) + k 3 S 2a + 2b + c (x )S 2b + c (y )C c (y )− k 3 S 2b + c (x )S 2a + 2b + c (y )C c (y )) F 3 2( C c (y )C 2a + 2b + c (x )+ C 2b + c (y )C 2a + 2b + c (x )+ C c (x )C 2a + 2b + c (y ) 0 0 2i (− k 1 S 2b + c (x )S c (y )+ k 1 S c (x )S 2b + c (y ) C 2b + c (x )C 2a + 2b + c (y )+ C c (y )C 2b + c (x )+ C c (x )C 2b + c (y )) + k 2 S 2a + 2b + c (x )S c (y )− k 2 S c (x )S 2a + 2b + c (y ) − k 3 S 2a + 2b + c (x )S 2b + c (y )+ k 3 S 2b + c (x )S 2a + 2b + c (y )) F 4 C c (z )C 2b + c (1 − x )C 2a + 2b + c (x )+ C c (z )C 2b + c (x )C 2a + 2b + c (1 − x ) 0 0 √ 2 2 i( k 1 C 2a + 2b + c (x )S 2b + c (1 − x )S c (z )− k 2 C 2b + c (x )S 2a + 2b + c (1 − x )S c (z ) C c (x )C 2b + c (1 − x )C 2a + 2b + c (x )+ C c (x )C 2a + 2b + c (1 − x )C 2b + c (z ) + k 3 C c (x )S 2a + 2b + c (1 − x )S 2b + c (z )− k 3 C c (x )S 2b + c (1 − x )S 2a + 2b + c (z ) C c (1 − x )C 2b + c (z )C 2a + 2b + c (x )+ C c (1 − x )C 2b + c (x )C 2a + 2b + c (z ) − k 1 C 2a + 2b + c (x )S c (1 − x )S 2b + c (z )+ k 2 C 2b + c (x )S c (1 − x )S 2a + 2b + c (z ) + k 2 S 2a + 2b + c (x )C 2b + c (1 − x )S c (z )− k 1 S 2b + c (x )C 2a + 2b + c (1 − x )S c (z ) + k 1 S c (x )C 2a + 2b + c (1 − x )S 2b + c (z )− k 2 S c (x )C 2b + c (1 − x )S 2a + 2b + c (z ) − k 3 S 2a + 2b + c (x )C c (1 − x )S 2b + c (z )+ k 3 S 2b + c (x )C c (1 − x )S 2a + 2b + c (z )) B 3 S l D N F 1 0 √ 2i (k 2 C 2a + 2b + c (x )S 2b + c (x )C c (z )− k 1 C 2b + c (x )S 2a + 2b + c (x )C c (z ) + k 1 C c (x )S 2a + 2b + c (x )C 2b + c (z )− k 2 C c (x )S 2b + c (x )C 2a + 2b + c (z ) − k 3 C 2a + 2b + c (x )S c (x )C 2b + c (z )+ k 3 C 2b + c (x )S c (x )C 2a + 2b + c (z )) F 2 0 √ 2i (− k 2 C 2a + 2b + c (x )S 2b + c (y )C c (y )+ k 1 C 2b + c (x )S 2a + 2b + c (y )C c (y ) − k 1 C c (x )S 2a + 2b + c (y )C 2b + c (y )+ k 2 C c (x )S 2b + c (y )C 2a + 2b + c (y ) + k 3 C 2a + 2b + c (x )S c (y )C 2b + c (y )− k 3 C 2b + c (x )S c (y )C 2a + 2b + c (y )) F 3 2( C 2b + c (y )C 2a + 2b + c (x ) − C 2b + c (x )C 2a + 2b + c (x )+ C c (x )C 2a + 2b + c (y ) 0 − C c (x )C 2b + c (y ) − C c (y )C 2a + 2b + c (x )+ C c (y )C 2b + c (y )) F 4 0 √ 2 2 i( k 1 S 2a + 2b + c (x )C 2b + c (1 − x )C c (z )− k 2 S 2b + c (x )C 2a + 2b + c (1 − x )C c (z ) + k 3 S c (x )C 2a + 2b + c (1 − x )C 2b + c (z )− k 3 S c (x )C 2b + c (1 − x )C 2a + 2b + c (z ) − k 1 S 2a + 2b + c (x )C c (1 − x )C 2b + c (z )+ k 2 S 2b + c (x )C c (1 − x )C 2a + 2b + c (z ) + k 2 C 2a + 2b + c (x )S 2b + c (1 − x )C c (z )− k 1 C 2b + c (x )S 2a + 2b + c (1 − x )C c (z ) + k 1 C c (x )S 2a + 2b + c (1 − x )C 2b + c (z )− k 2 C c (x )S 2b + c (1 − x )C 2a + 2b + c (z ) − k 3 C 2a + 2b + c (x )S c (1 − x )C 2b + c (z )+ k 3 C 2b + c (x )S c (1 − x )C 2a + 2b + c (z )) B 3 S s D N F 1 2( S 2b + c (x )S 2a + 2b + c (x )S c (z )+ S c (x )S 2b + c (z )S 2a + 2b + c (x ) 0 + S c (x )S 2b + c (x )S 2a + 2b + c (z )) F 2 2( S c (z )S 2b + c (y )S 2a + 2b + c (x )+ S c (x )S 2b + c (y )S 2a + 2b + c (y ) 0 + S c (y )S 2b + c (x )S 2a + 2b + c (y )) F 3 0 2i (− k 3 S 2a + 2b + c (x )S 2b + c (y )− k 3 S 2b + c (x )S 2a + 2b + c (y ) − k 2 S c (x )S 2a + 2b + c (y )− k 1 S c (x )S 2b + c (y )− k 2 S 2a + 2b + c (x )S c (y )− k 1 S 2b + c (x )S c (y )) F 4 S c (z )S 2b + c (1 − x )S 2a + 2b + c (x )+ S c (z )S 2b + c (x )S 2a + 2b + c (1 − x ) 0 S c (x )S 2b + c (1 − x )S 2a + 2b + c (x )+ S c (x )S 2b + c (1 − x )S 2a + 2b + c (z ) S c (1 − x )S 2b + c (z )S 2a + 2b + c (x )+ S c (z )S 2b + c (1 − x )S 2a + 2b + c (z ) Table 2. The values of C-, S-, Sl- and Ss-functions on the boundaries of fundamental region F of B3. The separation constants ki, i = 1, 2, 3 are given by (6) in § 5.1. 409 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica C3 C S D N D N F1 2(Cb+c(x)Cc(y)+Ca+b+c(x)Cc(y) 0 0 −2i(k1Sb+c(x)Sc(y)+k2Sa+b+c(x)Sc(y) +Ca+b+c(x)Cb+c(y)+Cc(x)Cb+c(y) −k3Sa+b+c(x)Sb+c(y)+k1Sc(x)Sb+c(y) +Cc(x)Ca+b+c(y)+Cb+c(x)Ca+b+c(y)) −k2Sc(x)Sa+b+c(y)+k3Sb+c(x)Sa+b+c(y) F2 2(Ca+b+c(x)Cb+c(z)Cc(z) 0 0 i √ 2(k1Cc(z)Sb+c(x)Ca+b+c(z)−k1Sc(x)Sb+c(z)Ca+b+c(z) +Cb+c(x)Ca+b+c(z)Cc(z) −k2Sc(z)Cb+c(z)Sa+b+c(x)+k2Sc(x)Cb+c(z)Sa+b+c(z) +Cc(x)Ca+b+c(z)Cb+c(z)) −k3Cc(z)Sb+c(x)Sa+b+c(z)+k3Cc(z)Sb+c(z)Sa+b+c(x)) F3 2(Cc(z)Cb+c(y)Ca+b+c(y) 0 0 −i √ 2(k1Sc(z)Sb+c(y)Ca+b+c(y)−k1Sc(y)Sb+c(z)Ca+b+c(y) +Cc(y)Cb+c(z)Ca+b+c(y) −k2Sc(z)Cb+c(y)Sa+b+c(y) + k2Sc(y)Cb+c(y)Sa+b+c(z) +Cc(y)Cb+c(y)Ca+b+c(z)) −k3Cc(y)Sb+c(y)Sa+b+c(z)+k3Cc(y)Sb+c(z)Sa + b + c(y)) F4 Ca+b+c( 1√2 )Cb+c(y)Cc(z) 0 0 i(k1Sc(z)Ca+b+c( 1√2 )Sb+c(y)−k1Sc(y)Ca+b+c( 1√ 2 )Sb+c(z) +Cb+c( 1√2 )Ca+b+c(y)Cc(z) −k2Cb+c( 1√ 2 )Sc(z)Sa+b+c(y)+k2Cb+c( 1√2 )Sc(y)Sa+b+c(z) +Cc( 1√2 )Ca+b+c(y)Cb+c(z) −k3Cc( 1√ 2 )Sb+c(y)Sa+b+c(z)+k3Cc( 1√2 )Sb+c(z)Sa+b+c(y)) C3 Sl D N F1 0 −2i(k1Sb+c(x)Cc(y)+k2Sa+b+c(x)Sc(y) +k3Sa+b+c(x)Sb+c(y)+k1Sc(x)Sb+c(y) +k2Sc(x)Sa+b+c(y)+k3Sb+c(x)Sa+b+c(y)) F2 2(Sa+b+c(x)Sb+c(z)Sc(z)+Sb+c(x)Sa+b+c(z)Sc(z) 0+Sc(x)Sa+b+c(z)Sb+c(z)) F3 2(Sc(z)Sb+c(y)Sa+b+c(y) + Sc(y)Sb+c(z)Sa+b+c(y) 0+Sc(y)Sb+c(y)Sa+b+c(z)) F4 0 i(k1Sc(z)Ca+b+c( 1√2 )Sb+c(y)+k1Sc(y)Ca+b+c( 1√ 2 )Sb+c(z) +k2Cb+c( 1√2 )Sc(z)Sa+b+c(y)+k2Cb+c( 1√ 2 )Sc(y)Sa+b+c(z) +k3Cc( 1√2 )Sb+c(y)Sa+b+c(z)+k3Cc( 1√ 2 )Sb+c(z)Sa+b+c(y)) C3 Ss D N F1 −2(Cb+c(x)Ca+b+c(y)−Cc(y)Ca+b+c(x) 0+Cb+c(y)Ca+b+c(x)+Cc(x)Ca+b+c(y) +Cc(y)Cb+c(x)−Cc(x)Cb+c(y)) F2 0 i √ 2(k1Cc(z)Cb+c(x)Sa+b+c(z)−k1Cc(x)Cb+c(z)Sa+b+c(z) −k2Cc(z)Sb+c(z)Ca+b+c(x)+k2Cc(x)Sb+c(z)Ca+b+c(z) +k3Sc(z)Cb+c(z)Ca+b+c(x)−k3Sc(z)Cb+c(x)Ca+b+c(z)) F3 0 −i √ 2(k1Cc(z)Cb+c(y)Sa+b+c(y)−k1Cc(y)Cb+c(z)Sa+b+c(y) −k2Cc(z)Sb+c(y)Ca+b+c(y)+k2Cc(y)Sb+c(y)Ca+b+c(z) +k3Sc(y)Cb+c(z)Ca+b+c(y)−k3Sc(y)Cb+c(y)Ca+b+c(z)) F4 Cc(z)Ca+b+c( 1√2 )Cb+c(y)−Cc(y)Ca+b+c( 1√ 2 )Cb+c(z) 0−Cb+c( 1√2 )Cc(z)Ca+b+c(y)+Cc( 1√ 2 )Cb+c(z)Ca+b+c(y) +Cb+c( 1√2 )Cc(y)Ca+b+c(z)−Cc( 1√ 2 )Cb+c(y)Ca+b+c(z) Table 3. The values of C-, S-, Sl- and Ss-functions on the boundaries of fundamental region F of C3. The separation constants ki, i = 1, 2, 3 are given by (7) in § 5.1. 410 vol. 58 no. 6/2018 Multidimensional Hybrid Boundary Value Problem C2×A1 C S D N D N F1 2(Ca+b(x)Cb(y)+Cb(x)Ca+b(y)) 0 0 −2ik3(Sa+b(x)Sb(y)−Sb(x)Sa+b(y)) F2 2(Ca+b(x)Cc(z)+Cb(x)Cc(z)) 0 0 −2ik2Sa+b(x)Sc(z)+ik1Sb(x)Sc(z) F3 2Ca+b(y)Cb(y)Cc(z) 0 0 −i √ 2(k1Ca+b(y)Sb(y)Sc(z)−k2Cb(y)Sa+b(y)Sc(z)) F4 Ca+b( √ 2 2 )Cb(y)Cc(z)+Cb( √ 2 2 )Ca+b(y)Cc(z) 0 0 ik1Ca+b( √ 2 2 )Sb(y)Sc(z)−ik2Cb( √ 2 2 )Sa+b(y)Sc(z) F5 Ca+b(x)Cb(y)Cc( √ 2 2 )+Cb(x)Ca+b(y)Cc( √ 2 2 ) 0 0 ik3(Sa+b(x)Sb(y)Cc( √ 2 2 )−Sb(x)Sa+b(y)Cc( √ 2 2 )) C2×A1 Sl D N F1 0 −2ik3(Sa+b(x)Sb(y)+Sb(x)Sa+b(y)) F2 0 −2i(k2Sa+b(x)Sc(z)+k1Sb(x)Sc(z)) F3 2Sa+b(y)Sb(y)Sc(z) 0 F4 0 i(k1Ca+b( √ 2 2 )Sb(y)Sc(z)+k2Cb( √ 2 2 )Sa+b(y)Sc(z)) F5 0 ik3(Sa+b(x)Sb(y)Cc( √ 2 2 )+Sb(x)Sa+b(y)Cc( √ 2 2 )) C2×A1 Ss D N F1 2(Ca+b(x)Cb(y)−Cb(x)Ca+b(y)) 0 F2 2(Ca+b(x)Cc(z)−Cb(x)Cc(z)) 0 F3 0 −i √ 2(k1Sa+b(y)Cb(y)Cc(z)−k2Sb(y)Ca+b(y)Cc(z)) F4 Ca+b( √ 2 2 )Cb(y)Cc(z)−Cb( √ 2 2 )Ca+b(y)Cc(z) 0 F5 Ca+b(x)Cb(y)Cc( √ 2 2 )−Cb(x)Ca+b(y)Cc( √ 2 2 ) 0 Table 4. The values of C-, S-, Sl- and Ss-functions on the boundaries of fundamental region F of C2 × A1. The separation constants ki, i = 1, 2, 3 are given by (8) in § 5.2. 411 Marzena Szajewska, Agnieszka Tereszkiewicz Acta Polytechnica G 2 × A 1 C S D N D N F 1 2( C a (x )C 3a + 2b (y )+ C a + b (x )C 3a + b (y ) 0 0 − i2 k 3 (S a (x )S 3a + 2b (y )− S a + b (x )S 3a + b (y )+ S 2a + b (x )S b (y )) + C 2a + b (x )C b (y )) F 2 (C a (x )C 3a + 2b (√ 3x )+ C a + b (x )C 3a + b (√ 3x ) 0 0 2i (− m 2 S a (x )C 3a + 2b (√ 3x )+ l 2 S a + b (x )C 3a + b (√ 3x ) + C 2a + b (x )C b (√ 3x )) C c (z ) − k 2 S 2a + b (x )C b (√ 3x )) S c (z ) F 3 2( C 3a + 2b (y )+ C 3a + b (y )+ C b (y )) C c (z ) 0 0 − 2i (m 1 S 3a + 2b (y )− l 1 S 3a + b (y )+ k 1 S b (y )) S c (z ) F 4 C a (x )C 3a + 2b (− √ 3 3 x + √ 6 3 )C c (z ) 0 0 i( m 1 2 C a (x )S 3a + 2b (− √ 3 3 x + √ 6 3 )S c (z )− l 1 2 C a + b (x )S 3a + b (− √ 3 3 x + √ 6 3 )S c (z ) + C a + b (x )C 3a + b (− √ 3 3 x + √ 6 3 )C c (z ) + k 1 2 C 2a + b (x )S b (− √ 3 3 x + √ 6 3 )S c (z )+ m 2 √ 3 2 S a (x )C 3a + 2b (− √ 3 3 x + √ 6 3 )S c (z ) + C 2a + b (x )C b (− √ 3 3 x + √ 6 3 )C c (z ) − l 2 √ 3 2 S a + b (x )C 3a + b (− √ 3 3 x + √ 6 3 )S c (z )+ k 2 √ 3 2 S 2a + b (x )C b (− √ 3 3 x + √ 6 3 )S c (z )) F 5 (C a (x )C 3a + 2b (y )+ C a + b (x )C 3a + b (y )+ C 2a + b (x )C b (y )) C c (√ 2 2 ) 0 0 ik 3 (S a (x )S 3a + 2b (y )− S a + b (x )S 3a + b (y )+ S 2a + b (x )S b (y )) C c (√ 2 2 ) G 2 × A 1 S l D N F 1 0 2i m 3 (− S a (x )C 3a + 2b (y )+ S a + b (x )C 3a + b (y )− S 2a + b (x )C b (y )) F 2 (S a (x )C 3a + 2b (√ 3x )+ S a + b (x )C 3a + b (√ 3x )− S 2a + b (x )C b (√ 3x )) S c (z ) 0 F 3 0 2i (m 1 C 3a + 2b (y )− l 1 C 3a + b (y )+ k 1 C b (y )) S c (z ) F 4 0 i( − k 1 2 C b (√ 6 3 − √ 3x 3 )S c (z )C a + b (x )+ l 1 2 S c (z )C a + b (x )C 3a + b (√ 6 3 − √ 3x 3 ) + m 1 2 C a (x )S c (z )C 3a + 2b (√ 6 3 − √ 3x 3 ) − 1 2 √ 3k 2 S b (√ 6 3 − √ 3x 3 )S c (z )S 2a + b (x ) + 1 2 √ 3l 2 S c (z )S a + b (x )S 3a + b (√ 6 3 − √ 3x 3 )+ 1 2 √ 3m 2 S a (x )S c (z )S 3a + 2b (√ 6 3 − √ 3x 3 )) F 5 0 ik 3 (S a (x )S 3a + 2b (y )− S a + b (x )S 3a + b (y )+ S 2a + b (x )S b (y )) C c (√ 2 2 ) G 2 × A 1 S s D N F 1 2( C a (x )S 3a + 2b (y )− C a + b (x )S 3a + b (y )− C 2a + b (x )C b (y )) 0 F 2 0 2i C c (z )( m 1 C a (x )C 3a + 2b (√ 3x )− l 2 C a + b (x )C 3a + b (√ 3x ) − k 2 C 2a + b (x )C b (√ 3x )) − k 2 C 2a + b (x )C b (√ 3x )) F 3 2( S 3a + 2b (y )C c (z )− S 3a + b (y )C c (z )− S b (y )C c (z )) 0 F 4 (C a (x )S 3a + 2b (− √ 3 3 x + √ 6 3 )− C a + b (x )S 3a + b (− √ 3 3 x + √ 6 3 )− C 2a + b (x )S b (− √ 3 3 x + √ 6 3 )) C c (z ) 0 F 5 C a (x )S 3a + 2b (y )C c (√ 2 2 )− C a + b (x )S 3a + b (y )C c (√ 2 2 )− C 2a + b (x )S b (y )C c (√ 2 2 ) 0 Table 5. The values of C-, S-, Sl- and Ss-functions on the boundaries of fundamental region F of G2 × A1. The separation constants ki, li, mi, i = 1, 2, 3 are given by (9) in § 5.2. 412 vol. 58 no. 6/2018 Multidimensional Hybrid Boundary Value Problem A1×A1×A1 CCC SSS D N D N F1 Ca(x)Cb(y)Cc(0) 0 0 − √ 2πik3Sa(x)Sb(y)Cc(0) F2 Ca(x)Cb(0)Cc(z) 0 0 − √ 2πik2Sa(x)Cb(0)Sc(z) F3 Ca(0)Cb(y)Cc(z) 0 0 − √ 2πik1Ca(0)Sb(y)Sc(z) F4 Ca(x)Cb( 1√2 )Cc(z) 0 0 √ 2πik2Sa(x)Cb( 1√2 )Sc(z) F5 Ca( 1√2 )Cb(y)Cc(z) 0 0 √ 2πik1Ca( 1√2 )Sb(y)Sc(z) F6 Ca(x)Cb(y)Cc( 1√2 ) 0 0 √ 2πik3Sa(x)Sb(y)Cc( 1√2 ) A1×A1×A1 CCS SSC D N D N F1 0 − √ 2πik3Ca(x)Cb(y)Cc(0) Sa(x)Sb(y)Cc(0) 0 F2 Ca(x)Cb(0)Sc(z) 0 0 − √ 2πik2Sa(x)Cb(0)Cc(z) F3 Ca(0)Cb(y)Sc(z) 0 0 − √ 2πik1Ca(0)Sb(y)Cc(z) F4 Ca(x)Cb( 1√2 )Sc(z) 0 0 √ 2πik2Sa(x)Cb( 1√2 )Cc(z) F5 Ca( 1√2 )Cb(y)Sc(z) 0 0 √ 2πik1Ca( 1√2 )Sb(y)Cc(z) F6 0 √ 2πik3Ca(x)Cb(y)Cc( 1√2 ) Sa(x)Sb(y)Cc( 1√ 2 ) 0 A1×A1×A1 CSC SCS D N D N F1 Ca(x)Sb(y)Cc(0) 0 0 − √ 2πik3Sa(x)Cb(y)Cc(0) F2 0 − √ 2πik2Ca(x)Cb(0)Cc(z) Sa(x)Cb(0)Sc(z) 0 F3 Ca(0)Sb(y)Cc(z) 0 0 − √ 2πik1Ca(0)Cb(y)Sc(z) F4 0 √ 2πik2Ca(x)Cb( 1√2 )Cc(z) Sa(x)Cb( 1√ 2 )Sc(z) 0 F5 Ca( 1√2 )Sb(y)Cc(z) 0 0 √ 2πik1Ca( 1√2 )Cb(y)Sc(z) F6 Ca(x)Sb(y)Cc( 1√2 ) 0 0 √ 2πik3Sa(x)Cb(y)Cc( 1√2 ) A1×A1×A1 SCC CSS D N D N F1 Sa(x)Cb(y)Cc(0) 0 0 − √ 2πik3Ca(x)Sb(y)Cc(0) F2 Sa(x)Cb(0)Cc(z) 0 0 − √ 2πik2Ca(x)Cb(0)Sc(z) F3 0 − √ 2πik1Ca(0)Cb(y)Cc(z) Ca(0)Sb(y)Sc(z) 0 F4 Sa(x)Cb( 1√2 )Cc(z) 0 0 √ 2πik2Ca(x)Cb( 1√2 )Sc(z) F5 0 √ 2πik1Ca( 1√2 )Cb(y)Cc(z) Ca( 1√ 2 )Sb(y)Sc(z) 0 F6 Sa(x)Cb(y)Cc( 1√2 ) 0 0 √ 2πik3Ca(x)Sb(y)Cc( 1√2 ) Table 6. The values of six families of functions on the boundaries of fundamental region F of A1 × A1 × A1. The separation constants ki, i = 1, 2, 3 are given by (10) in § 5.3. 413 Acta Polytechnica 58(6):402–413, 2018 1 Introduction 2 Helmholtz equation and boundary conditions 3 Finite reflection groups 4 Special functions as a solution of Helmholtz equation 5 3D finite reflection groups 5.1 B3 and C3 groups 5.2 C2A1 and G2A1 groups 5.3 A1 A1 A1 group 6 Appendix References