On the \(D_\omega\)-classical orthogonal polynomials. (English) Zbl 1541.33011
Summary: We investigate the \(D_\omega\)-classical orthogonal polynomials, where \(D_\omega\) is the weighted difference operator. So, we address the problem of finding the sequence of orthogonal polynomials such that their \(D_\omega\)-derivatives is also orthogonal polynomials. To solve this problem we adopt a different approach to those employed in this topic. We first begin by determining the coefficients involved in their recurrence relations, and then providing an exhaustive list of all solutions. When \(\omega = 0\), we rediscover the classical orthogonal polynomials of Hermite, Laguerre, Bessel and Jacobi. For \(\omega =1\), we encounter the families of discrete classical orthogonal polynomials as particular cases.
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 33E30 | Other functions coming from differential, difference and integral equations |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 39A12 | Discrete version of topics in analysis |
| 39A70 | Difference operators |
Keywords:
classical orthogonal polynomials; discrete orthogonal polynomials; recurrence relations; difference operators; difference equationsReferences:
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