On differential equations for Sobolev-type Laguerre polynomials

We obtain all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials. This generalizes the results found in 1990 by the first and second author in the case of the generalized Laguerre polynomials defined by T.H. Koornwinder in 1984.Comment: 45 page

Linear perturbations of differential of difference operators with polynomials as eigenfunctions

AbstractThis paper deals with one-parameter linear perturbations of a family of polynomials {Pn(x)}n=0∞ with deg[Pn(x)] = n of the form Pnμ(x) = Pn(x) + μQn(x), where μ is a real parameter and {Qn(x)}n=0∞ are polynomials with deg[Qn(x)] ⩽ n. Let the polynomials {Pn(x)}n=0∞ be eigenfunctions of a linear differential or difference operator L+μA with eigenvalues {λn}n=0∞. The purpose of this p to derive necessary and sufficient conditions for the polynomials {Qn(x)}n=0∞ such that the polynomials {Pnμ(x)}n=0∞ are eigenfunctions of a linear difference or differential operator (possibly of infinite order) of the form L + μA with eigenvalues {λn + μαn}n=0∞

Difference Equations for Generalized Meixner Polynomials

AbstractIn this paper is introduced a system of polynomials orthogonal with respect to the classical discrete weight function for Meixner polynomials with an extra point mass added at x=0. A difference operator of infinite order is constructed for which these new polynomials are eigenfunctions and a second-order difference equation is given with polynomial coefficients, n-dependent and of at most degree 2, which these polynomials satisfy

Closed-form dynamic response of damped mass–spring cascades

AbstractClosed-form responses due to step forces of finite and semi-infinite damped mass–spring cascades with variable boundary mass are derived. By convolution arbitrary responses may be obtained. Eigenfrequencies, eigenmodes and critical damping of nonuniform and uniform finite cascades are found, showing a decreasing critical damping with increasing eigenfrequency. Further the frequency-response functions of the cascades are given for three cases