Orthogonal Polynomials in Two Variables. A Further Analysis of the Polynomials Orthogonal over a Region Bounded by Two Lines and a Parabola

Sprinkhuizen-Kuyper, Ida G.

doi:10.1137/0507041

Cited by 26 publications

(29 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar results have been found for orthogonal polynomials in two dimensions [1,2]. For particular coefficients, these twodimensional orthogonal polynomials correspond to eigenfunctions of the radial part of the Laplace-Beltrami operator on certain rank 2 symmetric spaces.…”

supporting

confidence: 66%

See 1 more Smart Citation

Intertwining operators for solving differential equations, with applications to symmetric spaces

Anderson

Camporesi

1990

Commun.Math. Phys.

View full text Add to dashboard Cite

Abstract. The use of intertwining operators to solve both ordinary and partial differential equations is developed. Classes of intertwining operators are constructed which transform between Laplacians which are self-adjoint with respect to different non-trivial measures. In the two-dimensional case, the intertwining operator transforms a non-separable partial differential operator to a separable one. As an application, the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed.It has long been appreciated that one of the nice properties of the special functions is that there exist differential operators which transform between functions of the same type, changing the indices of the functions by integral amounts. This property is the source of formulae which provide a compact expression for certain special functions in terms of powers of a differential operator applied to elementary functions. It has also been known for some time that the use of fractional differential operators, or pseudo-differential operators, extends the set of transformations between functions, allowing the indices to be changed by non-integral amounts.Most of this common knowledge is for orthogonal polynomials and other special functions in one dimension. Similar results have been found for orthogonal polynomials in two dimensions [1,2]. For particular coefficients, these twodimensional orthogonal polynomials correspond to eigenfunctions of the radial part of the Laplace-Beltrami operator on certain rank 2 symmetric spaces. We present here a new approach to the construction and transformation among eigenfunctions of differential equations based on the construction of intertwining operators. This approach generalizes the classical operator transformations among special functions and is naturally applied in higher dimensions.The intertwining operator approach to eigenfunctions was motivated by the method used by Dowker [3] to find the heat kernel for a free particle propagating on a Lie group manifold. It was developed by one of the authors [4] to find the heat kernel of a free-particle propagating on an n-dimensional sphere. Intertwining operators have also been used to solve non-linear integrable systems [5], The intent here is to construct a few general classes of intertwining operators for one

show abstract

supporting

confidence: 66%

“…The root pairs UL I and (x. 2 ,a l and a 2 , and 2cn l and 2α 2 each form Weyl sets. The multiplicity of roots within a Weyl set must be the same, but different sets can have different multiplicities.…”

Section: Rank 2 O=>omentioning

confidence: 99%

Intertwining operators for solving differential equations, with applications to symmetric spaces

Anderson

Camporesi

1990

Commun.Math. Phys.

View full text Add to dashboard Cite

show abstract

“…In [8] Koornwinder defined and investigated an important class of bivariate orthogonal polynomials {P^'M(w(-a'ft)} that are orthogonal with respect to the weight function (u2 -4v)yW(u, v), where W(u, v) -u>(°> ft(x)w^a• ft(y) and vj(a>ft = (1 -jc)a(l + x)P is the Jacobi weight (see [9,20] for further analysis; the polynomials are denoted Pk'%'y in [8]). The cases y = ±\ correspond to our p^1'( with w being a Jacobi weight.…”

Section: Jrmentioning

confidence: 99%

On Bivariate Gaussian Cubature Formulae

Schmid¹,

Xu²

1994

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…(Of course, the BC 1 case is given by the classical Jacobi polynomials of §4.) They were introduced by the author in [10], and further elaborated in [16] and in (my main reference) [13].…”

Section: Lowering and Raising Jacobi Polynomialsmentioning

confidence: 99%