This paper explores the theory of spherical harmonics for measures invariant under finite reflection groups. The author introduces a set of first-order differential-difference operators, which are analogous to the first-order partial derivatives in the ordinary theory of spherical harmonics. These operators are homogeneous of degree \(-1\) and commute with each other. The paper focuses on the case of \(\mathbb{R}^2\) and dihedral groups, where analogues of the Cauchy-Riemann equations apply to Gegenbauer and Jacobi polynomial expansions. The analysis involves the underlying structure based on finite Coxeter groups, which are finite groups acting on Euclidean space through reflections. The weight functions for orthogonality are products of powers of linear functions restricted to the unit sphere, and they must be invariant under the group action. The key result is that a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second-order differential-difference operator. The paper constructs a commutative set of first-order differential-difference operators associated with the second-order operator mentioned. It also determines the adjoints of these operators, which are fundamental selfadjoint operators with eigenvalues related to the character table of the reflection groups. The paper studies the Cauchy-Riemann equations for dihedral groups and provides explicit formulas for Gegenbauer and Jacobi polynomials. The main topics covered include the definition and commutativity proof of the operators, the determination of their adjoints, and the study of the Cauchy-Riemann equations for dihedral groups. The paper includes detailed proofs and examples, particularly for the hyperoctahedral group \(W_N\) and the dihedral groups \(D_n\), \(D_{2m+1}\), \(D_2\), and \(D_{2m}\).This paper explores the theory of spherical harmonics for measures invariant under finite reflection groups. The author introduces a set of first-order differential-difference operators, which are analogous to the first-order partial derivatives in the ordinary theory of spherical harmonics. These operators are homogeneous of degree \(-1\) and commute with each other. The paper focuses on the case of \(\mathbb{R}^2\) and dihedral groups, where analogues of the Cauchy-Riemann equations apply to Gegenbauer and Jacobi polynomial expansions. The analysis involves the underlying structure based on finite Coxeter groups, which are finite groups acting on Euclidean space through reflections. The weight functions for orthogonality are products of powers of linear functions restricted to the unit sphere, and they must be invariant under the group action. The key result is that a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second-order differential-difference operator. The paper constructs a commutative set of first-order differential-difference operators associated with the second-order operator mentioned. It also determines the adjoints of these operators, which are fundamental selfadjoint operators with eigenvalues related to the character table of the reflection groups. The paper studies the Cauchy-Riemann equations for dihedral groups and provides explicit formulas for Gegenbauer and Jacobi polynomials. The main topics covered include the definition and commutativity proof of the operators, the determination of their adjoints, and the study of the Cauchy-Riemann equations for dihedral groups. The paper includes detailed proofs and examples, particularly for the hyperoctahedral group \(W_N\) and the dihedral groups \(D_n\), \(D_{2m+1}\), \(D_2\), and \(D_{2m}\).