We study linear transformations $$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$$ of the form $$T[x^n]=P_n(x)$$ where $$\{P_n(x)\}$$ is a real orthogonal polynomial system. With $$T=\sum \tfrac{Q_k(x)}{k!}D^k$$ , we seek to understand behavior transformation T by studying roots $$Q_k(x)$$ . prove four main things. First, show that only case are constant and an system when $$P_n(x)$$ shifted set ...

This paper considers the computational efficiency of using generalised function parameterisations for multi-parametric quadratic programming (mp-QP) solutions to MPC. Earlier work demonstrated the potential of Laguerre parameterisations for improving computational efficiency in that the parametric solutions either required fewer regions and/or gave larger volumes. This paper considers the poten...

Abstract. We use a multidimensional extension of Bailey’s transform to derive two very general q-generating functions, which are q-analogues of a paper by Exton [7]. These expressions are then specialised to give more practical formulae, which are q-analogues of generating relations for Karlssons generalised Kampe de Fériet function. A number of examples are given including q-Laguerre polynomia...

Let Cλ n(x), n = 0, 1, . . . , λ > −1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (−1, 1) with respect to the weight function (1−x2)λ−1/2. Denote by xnk(λ), k = 1, . . . , n, the zeros of Cλ n(x) enumerated in decreasing order. In this short note we prove that, for any n ∈ IN , the product (λ+1)xn1(λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some ...

Orthogonality of the Jacobi and of Laguerre polynomials, P (α,β) n and L (α) n , is established for α, β ∈ C \ Z−, α + β 6= −2,−3, . . . using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann-Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when Rα,Rβ > −1).

and Applied Analysis 3 f : [0, τ] ×N ×N → N is smooth enough and there are constants a, c ∈ N, with c ̸ = − 1, such that sup (t,z,u)∈q τ 󵄨󵄨󵄨󵄨f (t, z, u) − az − cu 󵄨󵄨󵄨󵄨 < ∞, (12) where q τ = [0, τ] ×N ×N. (2) The interior controllability of the semilinear Laguerre equation

"A simple result deserves a simple proof" I Objects lying in four different boxes are rearranged in such a way that the number of objects where the bottom is a rearrangement of the top. The weight W (T) of the multiset permutation T is defined to be (-I) ~-where U, is the number of columns of the form I: 111122333444 , x # y. For example, if n-= 441223133411 then U, = 7 and w (n-) = (-~) ~ =-1....

Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian...

In this paper we study the controllability of the controlled Laguerre equation and the controlled Jacobi equation. For each case, we found conditions which guarantee when such systems are approximately controllable on the interval [0, t 1 ]. Moreover, we show that these systems can never be exactly controllable.

a new ARX-Laguerre representation is recently built to model the dynamics of complex processes [1, 2]. The ARX-Laguerre models have proven their ability to accurately suit the behavior of systems. In this work, the model is exploited to diagnose the system by detecting its defaults. In this paper we build a proportional observer based on the ARX-Laguerre model. Therefore, the designed observer ...