Orthogonal polynomials and spectral-type differential equations

Abstract

We are interested in differential equations of the form ▷수식 삽입◁(원문을 참조하세요) having an orthogonal polynomial system of solutions. In 1929, Bochner showed that there are essentially (i.e.,up to a complex linear change of variable) four orthogonal polynomial systems that satisfy the differential equation (0.1) with $N=2.$ These orthogonal polynomial systems are Jacobi, Laguerre, Hermite, Bessel polynomials, which are called the classical orthogonal polynomials. Krall classifed all orthogonal polynomials satisfying fourth order differential equation of the form (0.1). In addition to recovering four classical orthogonal polynomials, he also found three new orthogonal polynomials which are now called the classical-type orthogonal polynomials. For N＞4, the complete classification of such differential equations remains open. Interests in such differential equations lie partly in the fact that they provide excellent examples to illustrate the general Titchmarsh-Weyl theory of singular boundary value problems. We first show that if a linear differential equation (0.1) has an orthogonal polynomial system of solutions, then the differential operator $L_N[ㆍ] $ must be symmetrizable, which is the key step to develop spectral analysis of such differential equations. Secondly, we show that for any orthogonal polynomials ${P_n(x)}_{n=0}^∞$ satisfying a differential equation (0.1) of order N (≥2) ${P_n(x)}_{n=0}^∞$ must be essentially Hermite polynomials if and only if the leading coefficient $ℓ_N(x)$ of $L_N[ㆍ]$ is a non-zero constant. Thirdly, let τ=σ+ν be a point mass(es) perturbation of a classical moment functional σ which is quasi-definite. We then find conditions for orthogonal polynomials ${Q_n(x)}_{n=0}^∞$ relative to τ to be Bochner-Krall orthogonal polynomials, that is, ${Q_n(x)}_{n=0}^∞$ are eigenfunctions of a finite order linear differential operator with polynomial coefficients of the type (0.1). For example, we show that σ cannot be the moment functional for Besse...