1. Introduction
The purpose of this article is to develop generating functions of fourteen types of bivariate generalized Chebyshev polynomials [
1,
2,
3]. There exist two families of polynomials corresponding to the Lie algebra
, four to the algebra
and eight to the algebra
. Explicit formulas for the polynomials are deduced from their generating functions.
The four kinds of the of classical univariate Chebyshev polynomials [
4] constitute a fundamental part of polynomial numeric methods. Inherent relation of these polynomials to the standard trigonometric functions forms the cornerstone of their theoretical and practical applications. As cosine images of a finite part of the equidistant lattice [
4], Chebyshev nodes play a special role. Orbit functions related to the crystallographic root systems of Weyl groups [
5,
6] serve as multidimensional generalizations of the trigonometric functions and induce specific multivariate versions of Chebyshev polynomials [
2,
3]. Symmetric and antisymmetric orbit functions occur as a standard tool in Lie theory and the form of the corresponding two kinds of polynomials, which appears already in [
1], specializes for the algebra
to univariate Chebyshev polynomials of the first and second kind.
For any root system with two root-lengths, the concept of a sign homomorphism produces two additional classes of hybrid character polynomials [
3]. All four polynomial classes constitute special cases of the Heckman–Opdam polynomials [
7]. Discrete orthogonality relations of all four sorts of orbit functions over distinct finite fragments of Weyl group invariant lattices are developed in [
8,
9,
10,
11]. Specific generalized cosine images of the finite fragments of multidimensional lattices form the sets of generalized Chebyshev nodes [
3,
12,
13,
14,
15,
16]. Moreover, analysis of intrinsic discrete orthogonality relations of orbit functions leads to a special type of admissible shifts of the weight lattices [
17]. The admissible shift of the weight lattice doubles the number of polynomial families of
and specializes for the algebra
to the Chebyshev polynomials of the third and fourth kinds [
4]. Generating functions of the multivariate Chebyshev polynomials are closely linked to the character generators of simple Lie groups.
During the last century, the approach of generating functions was developed to resolve many diverse problems in Lie theory and in the theory of finite groups. A wide range of applications of generating functions in Lie theory can be traced to the definition of the generating function for the characters of the representation [
18]. Subsequently, the theory was developed further for the specific types of the Lie groups [
19,
20,
21,
22]. The generating functions have a unique capability to provide answers to questions that are inaccessible to any other methods. Typically, a generating function of a simple Lie group
G is a rational function of several formal variables built to solve a series of analogous problems like decomposition of the product of two irreducible finite dimensional representations of
G into the direct sum of them, or reduction of any finite irreducible representations of
G to the direct sum of representations of a particular subgroup
. A number of other problems in group representation theory are listed in the reference [
23].
Developed into the power series, the coefficients of the series provide answers to infinite number of computational problems involving the same Lie group [
18,
23,
24,
25]. A practical difficulty often is the complexity of the generating functions for the higher ranks of
G. So far, the generating functions practically for all problems are explicitly derived by hand computation. The derivation becomes particularly complicated when one wants to have the generating function in a positive form that also provides the integrity basis for each problem [
26] or in polynomial form involving fundamental character functions. Direct calculation of generating functions as rational polynomial functions in fundamental characters is utilized in the present paper. With efficient tools for symbolic computation available in recent years, many more generating functions could be calculated.
Constructed generating functions and explicit formulas in the present article serve as theoretical and practical means for handling the corresponding bivariate generalized Chebyshev polynomials. Explicit evaluating formulas for the polynomials, derived from the explicit form of the generating functions, permit straightforward calculation and computer implementation, considerably more efficient than current recursive algorithms [
14]. Cubature formulas for numerical integration, polynomial interpolation and approximation methods corresponding to several current cases of polynomials are recently intensively studied [
3,
12,
13,
14,
15,
16]. Since viability of these polynomial methods is a direct consequence of the discrete orthogonality relations of the underlying orbit functions, extensibility of these techniques to all fourteen cases of the studied bivariate polynomials is guaranteed.
The paper is organized as follows. In
Section 2, the fourteen cases of the bivariate Chebyshev polynomials are explicitly constructed and the lowest reference polynomials listed for each case. In
Section 3, generating functions are explicitly evaluated from their generic forms and tabulated for each case. In
Section 4, the
K-polynomials are introduced, their form for each algebra calculated and their utilization as components in evaluating formulas presented. Concluding remarks and follow-up questions are contained in
Section 5.