Orthogonal polynomials satisfying partial differential equations belonging to the basic class
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초록
We classify all partial differential equations with polynomial coefficients in x and y of the form A(x)uxx + 2B(x, y)uxy + C(y)uyy + D(x)ux + E(y)uy = λnu, which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial differential equations in d-variables.
Abstract
We classify all partial differential equations with polynomial coefficients in x and y of the form A(x)uxx + 2B(x, y)uxy + C(y)uyy + D(x)ux + E(y)uy = λnu, which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial differential equations in d-variables.
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