Integral representations of quasi hypergeometric functions, K. Aomoto generating functions associ... more Integral representations of quasi hypergeometric functions, K. Aomoto generating functions associated with dihedral groups, C.F. Dunkl some relations for partitions into four squares, M.D. Hirschhorn and J.A. Sellers on a nonlinear recurrence related to Nevai polynomials, D. Kaminski the Brahmagupta matrix and its applications to tiling, R. Rangarajan and E.R. Suryanarayan solitons and coulomb plasmas, similarity reductions and special functions, V.P. Spiridonov orthogonal polynomials and their asymptotic behaviour, R. Wong a product formula for Jacobi polynomials, Y. Xu. (Part contents)
The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables.... more The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form qm=t−n with 2≤n≤N. For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the...
In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of... more In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.
Symmetry, Integrability and Geometry: Methods and Applications, 2021
There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting ... more There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170...
In this paper, we will develop a family of non-conforming “Crouzeix–Raviart” type finite elements... more In this paper, we will develop a family of non-conforming “Crouzeix–Raviart” type finite elements in three dimensions. They consist of local polynomials of maximal degree [Formula: see text] on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally, we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.
ABSTRACT Orthogonal polynomials and special functions of classical type often have geometrical in... more ABSTRACT Orthogonal polynomials and special functions of classical type often have geometrical interpretations for a restricted set of parameter values. The differential-difference operators constructed by the author [Trans. Am. Math. Soc. 311, No. 1, 167–183 (1989; Zbl 0652.33004)] provide a geometric setting in terms of reflection groups for a all parameter values. This paper begins with motivating examples involving Gegenbauer and Jacobi polynomials and a concise description of differential-difference operators. There is a detailed presentation of some novel generating functions and polynomials associated with the symmetry groups of regular polygons with an odd number of sides.
Integral representations of quasi hypergeometric functions, K. Aomoto generating functions associ... more Integral representations of quasi hypergeometric functions, K. Aomoto generating functions associated with dihedral groups, C.F. Dunkl some relations for partitions into four squares, M.D. Hirschhorn and J.A. Sellers on a nonlinear recurrence related to Nevai polynomials, D. Kaminski the Brahmagupta matrix and its applications to tiling, R. Rangarajan and E.R. Suryanarayan solitons and coulomb plasmas, similarity reductions and special functions, V.P. Spiridonov orthogonal polynomials and their asymptotic behaviour, R. Wong a product formula for Jacobi polynomials, Y. Xu. (Part contents)
The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables.... more The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form qm=t−n with 2≤n≤N. For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the...
In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of... more In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.
Symmetry, Integrability and Geometry: Methods and Applications, 2021
There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting ... more There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170...
In this paper, we will develop a family of non-conforming “Crouzeix–Raviart” type finite elements... more In this paper, we will develop a family of non-conforming “Crouzeix–Raviart” type finite elements in three dimensions. They consist of local polynomials of maximal degree [Formula: see text] on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally, we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.
ABSTRACT Orthogonal polynomials and special functions of classical type often have geometrical in... more ABSTRACT Orthogonal polynomials and special functions of classical type often have geometrical interpretations for a restricted set of parameter values. The differential-difference operators constructed by the author [Trans. Am. Math. Soc. 311, No. 1, 167–183 (1989; Zbl 0652.33004)] provide a geometric setting in terms of reflection groups for a all parameter values. This paper begins with motivating examples involving Gegenbauer and Jacobi polynomials and a concise description of differential-difference operators. There is a detailed presentation of some novel generating functions and polynomials associated with the symmetry groups of regular polygons with an odd number of sides.
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