The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Abstract

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2 3Dedicated to the memory of our dear friend Peter KulishThe generating function of the second kind bivariate Chebyshev polynomials associated with the simple Lie algebra G 2 is constructed by the method proposed in [1] and [2].

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

“…There are two fundamental means of calculation of the polynomials. The recurrence relations construction [13,15] is summarized for bivariate cases in this paper whereas the generating function method is developed recently in [16,17]. If the problem should be regarded as discretization of known polynomials of two continuous variables, then very few such polynomials can be discretized by the method developed in this paper.…”

Generating Functions for Orthogonal Polynomials of A2, C2 and G2

“…The current fourteen generating functions of bivariate polynomials include for completeness the five bivariate cases from [31][32][33]. The present calculation procedure as well as the procedure in [31][32][33] is based on case-by-case analysis of the given Lie algebra. Expressing the general form of character generators [19] in rational polynomial form potentially yields the polynomial generating functions υ (1,0) for any case.…”

“…and by recursive formulas, the Formula (44) indeed represents efficient and straightforward means of evaluation of any given polynomial U(j,k) λ . • Two distinct renormalizations of polynomials U (j,k)λ , inherited from normalizations of the underlying orbit functions, are mainly used throughout the literature[2,3,[31][32][33]. Between the normalized orbit functions(13), summed over the entire Weyl group W, and orbit functions ϕ(j)λ , added over the group orbit O(λ) only, holds the following relation:ϕ (j) λ = h λ ϕ (j) λ ,where h λ = |Stab W λ| denotes the order of the stabilizer of λ ∈ R 2 in the group W. Thus, the two polynomials U are intertwined asU (j,k) λ = h λ+ (j,k) h (j,k) U (j,k)λ .…”