We look for differential equations of the form ∞ ∑ i=0 ci(x)y (x) = 0, where the coefficients {ci(x)} ∞ i=0 are continuous functions on the real line and where {ci(x)} ∞ i=1 are independent of n, for the generalized Jacobi polynomials { P n (x) } ∞ n=0 and for generalized Laguerre polynomials { L n (x) } ∞ n=0 which are orthogonal with respect to an inner product of Sobolev type. We use a metho...

We look for differential equations of the form ∞ ∑ i=0 ci(x)y (x) = λny(x), where the coefficients {ci(x)} ∞ i=0 do not depend on n, for the generalized Jacobi polynomials { P n (x) } ∞ n=0 found by T.H. Koornwinder in 1984 and for generalized Laguerre polynomials { L n (x) } ∞ n=0 which are orthogonal with respect to an inner product of Sobolev type. We introduce a method which makes use of co...

We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Pólya– Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial a0 + a1z + · · · + anzn has only real and non-positive zeros, then so does the polynomial a 2 0 +(a 1 − a0a2)z ...

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, fea...

Laguerre function has many advantages such as good approximation capability for different systems, low computational complexity and the facility of on-line parameter identification. Therefore, it is widely adopted for complex industrial process control. In this work, Laguerre function based adaptive model predictive control algorithm (AMPC) was implemented to control continuous stirred tank rea...

The main goal of this article is to explore a new type polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix through operational techniques. generating function and representations for family polynomials will be established. In addition, these specific are interpreted in terms quasi-monomiality. extended versions introduced, their characteristics explored using integral transform. ...

We investigate the average characteristic polynomial E [∏N i=1(z−xi) ] where the xi’s are real random variables drawn from a Biorthogonal Ensemble, i.e. a determinantal point process associated with a bounded finite-rank projection operator. For a subclass of Biorthogonal Ensembles, which contains Orthogonal Polynomial Ensembles and (mixed-type) Multiple Orthogonal Polynomial Ensembles, we prov...

We characterize the non-classical 4-dimensional elation Laguerre planes as precisely those 4-dimensional Laguerre planes of Kleinewillinghöfer type I.D.1. Furthermore, in the class of 2or 4-dimensional Laguerre planes or finite Laguerre planes of odd order, the non-miquelian elation Laguerre planes are precisely the Laguerre planes of Kleinewillinghöfer type D. MSC 2000: 51H15, 51B15, 51E25