The purpose of this work is to study the classification of the second order partial differential equations having an orthogonal polynomials as solutions in d variables
In 1967, Krall and Sheffer posed and partially solved the following problem: Classify, up to a complex linear change of variables, all orthogonal polynomial systems that arise as eigenfunctions of the differential equation in two variables.
Krall and Sheffer first found necessary and sufficient conditions for weak orthogonal polynomials to satisfy an admissible differential equation in two variables
Later, Littlejohn investigate the same problem and made an important observation that the recurrence relations found by Krall and Sheffer can be restated in a much simpler closed form.
First, we find d variable version of a characterization of orthogonal polynomials satisfying the differential equation via the so-called structure relation, which was first proved for classical orthogonal polynomials in one variable by AI-Salam and Chihara. Also, In the particular case when the second order differential equations belongs to the basic class, we characterize the differential equations which have a product of d classical orthogonal polynomials in one variable as solutions and find conditions under which derivatives of orthogonal polynomial solutions to the differential equation are also orthogonal polynomials satisfying the same type of differential equations.
Littlejohn noted that all differential equations found by Krall and Sheffer are elliptic in the region of orthogonality in case the orthogonalizing weight is known. However,the type of differential operators can not be determined properly from the specific forms of the equations given Krall and Sheffer since the type of a differential operator is not preserved under a complex change of variables. In fact, Krall and Sheffer found at least one differential equation, which is hyperbolic everywhere and has orthogonal polynomials as solutions.
In Chapter 4, we prove that if the differential equation (1.1) in two variables has orthogonal polynomials as solutions, then the second order differential equation can not be parabolic in any nonempty open subset of the plane and must be symmetrizable. We also establish Rodrigues type formula for orthogonal polynomial solutions to the differential equation in two variables.
Krall and Sheffer classified the second order differential equations in two variables, up to a complex linear change of variables , which have orthogonal polynomials as solutions, but complex linear change of variables does not preserve the positive-definiteness of orthogonality and the type of the second order differential equation in two variables. In Chapter 5, we classify completely, up to a real change of variables, the second order differential equations in two variables which have centrally symmetric orthogonal polynomials as solutions together with explicit representations of orthogonal polynomial solutions.
Finally, we discuss the relations between the cubature formulae on the three regions $B^d$, $S^d$, and $T^d$.
1967년에 Krall과 Sheffer는 $\mathbb{R}^2$ 공간상에서 직교다항식을 해로 갖는 2차 편미분 방정식을 분류하는 문제를 제기하고 부분적인 해답을 얻었다. 그 후에, Littlejohn이 Krall과Sheffer의 결과를 훨씬 더 간단한 형태로 표현했다.
본 연구는 먼저 3 장에 서 Krall과 Sheffer 제기한 문제를 확장하여 $\mathbb{R}^d$ 공간 상에서 직교다항식이 2차 편미분 방정식의 해가되기 위한 필요 충분 조건, 즉 AI-Salam과 Chihara의 판별법에 대한 다변수 버션과 몇가지 필요조건들을 제시한다.
4장에서는 $\mathbb{R}^1$ 공간 상에서의 직교다항식의 곱으로 표시되는 $\mathbb{R}^d$ 공간 상에서의 직교다항식을 해로 갖는 편미분 방정식을 분류하고 2차의 직교 다항식을 만족하는 직교 다항식의 미분으로 구성된 다항식이 같은 형태의 2차 편미분방정식을 만족하기 위한 충분조건, 즉 Hahn-Sonine 정리에 대한 다변수 버젼을 구한다. 또한 $\mathbb{R}^1$ 공간 상에서의 Rodrigues 공식 $\mathbb{R}^2$ 공간상에서의 Rodrigues 공식으로 확장한다.
그리고 5장에서는 $\mathbb{R}^2$ 공간 상에서의 대칭 직교다항식을 해로 가지는 2차 편미분 방정식을 분류하고 각각의 미분방정식에 대한 직교다항식의 구체척인 표현을 제시한다. 마지막으로 5장에서는 단위구(Unit Sphere), 단위공(Unit Ball), 단체(單體:Simplex) 상에서의 입체구적공식들 사이의 관계에 대하여 기술한다.
