Laguerre Polynomials Generalized to a Certain Discrete Sobolev Inner Product Space Original Research Article

We are concerned with the set of polynomials {SM, Nn} which are orthogonal with respect to the discrete Sobolev inner product 〈 f, g 〉 = ∫∞0w(x) f(x) g(x) dx + Mf(0) g(0) + Nf′(0) g′(0), where w is a weight function, M ≥ 0, N ≥ 0. We show that these polynomials can be described as a linear combination of standard polynomials which are orthogonal with respect to the weight functions w(x), x2w(x), and x4w(x). The location of the zeros of SM, Nn is given in relation to the position of the zeros of the standard polynomials.