Generating Functions of Chebyshev Polynomials in Three Variables

Abstract

In this paper, we obtain generating functions of three-variable Chebyshev polynomials (of the first as well as of the second type) associated with the root system of the A 3 Lie algebra. Bibliography: 21 titles.To Petr Petrovich Kulish on the occasion of his 70th birthday 1. The aim of the present paper is to obtain generating functions of Chebyshev polynomials in three variables. Chebyshev polynomials in several variables associated with the root systems of simple Lie algebras were intensively studied in the last decades [1][2][3][4][5][6]. Such polynomials are applied in various areas of mathematics, as well as in physics. Examples of such applications with references can be found in the papers [7][8][9][10][11][12][13][14][15][16].Multivariate Chebyshev polynomials are natural generalizations of the classical ones. The classical Chebyshev polynomials T n (x) of the first kind are defined by the following formula (see, for example, [17]):( 1 ) they satisfy the following three-term recurrence relation:The classical Chebyshev polynomials U n (x) of the second kind are defined by the formulawhich is different from (1), but they satisfy the same recurrence relation (2). The initial conditions for the above polynomials are T 0 (x) = 1, T 1 (x) = x, U 0 (x) = 1, and U 1 (x) = 2x;( 3 ) together with the recurrence relation (2), they determine the polynomials uniquely without reference to (1) and (2). It is known that the function cos nφ can be treated as the invariant mean of the exponential function over the Weyl group of the A 1 algebra root system:Transition to a function that is the invariant mean over the Weyl group W of another root system leads us to generalization of the classical Chebyshev polynomials to polynomials in several variables [1][2][3][4]. Let us shortly recall the way in which such W -invariant functions can be constructed. Let R(L) be a reduced root system of any simple Lie algebra L. Such a system is a set of vectors in the d-dimensional Euclidean space E d supplied with scalar product (., .). The system R is uniquely determined by the base of simple roots α i , i = 1, . . . , d, and by a finite group W (R) which is generated by base reflections. This group is called the Weyl group.

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

“…The present work finishes the series of the articles ( [1], [2], [3]) which were initiated by P. Kulish. In these works we propose the method of constructing of generating functions for Chebyshev polynomials in several variables (of the first and second kinds), associated with simple Lie algebras. The method was tested on examples of polynomials related to Lie algebras A 2 , C 2 , A 3 and G 2 (in the last case only for the polynomials of the first kind).…”

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 .…”