Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a new method for evaluating integrals that include orthogonal polynomials. The method is illustrated by obtaining the following integral result that involves the Bessel function and associated Laguerre polynomial: integral(infinity)(0) x(v)e(-x/2) J(v)(mu x)L-n(2v)(x)dx = 2(v)Ga...

A simple method for approximation of all-pole recursive digital filters, directly in digital domain, is described. Transfer function of these filters, referred to as Ultraspherical filters, is controlled by order of the Ultraspherical polynomial, ν. Parameter ν, restricted to be a non-negative real number (ν ≥ 0), controls ripple peaks in the passband of the magnitude response and enables a tra...

In this paper, we prove that if f ( x ) = ∑ k 0 n a is polynomial with real zeros only, then the sequence { } satisfies following inequalities + 1 2 − c / ≤ , where . This inequality equivalent to higher order Turán inequality. It holds for coefficients of Riemann ξ -function, ultraspherical, Laguerre and Hermite polynomials, partition function. Moreover, as corollary, function p increasing ≥ 5...

In this paper D-optimal designs for the weighted polynomial regresŽ 2 . n sion model of degree p with efficiency function 1 x are presented. Interest in these designs stems from the fact that they are equivalent to locally D-optimal designs for inverse quadratic polynomial models. For the unrestricted design space and p n, the D-optimal designs put equal masses on p 1 points which coincide with...

1. Macdonald polynomials Macdonald polynomials P λ (x; q, t) are orthogonal symmetric polynomials which are the natural multivariable generalisation of the continuous q-ultraspherical polyno-mials C n (x; β|q) [2] which, in their turn, constitute an important class of hyper-geometric orthogonal polynomials in one variable. Polynomials C n (x; β|q) can be obtained from the general Askey-Wilson p...

Abstract In this work, a technique for finding approximate solutions ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is hybrid between Galerkin and collocation methods. Also, can be extended to fractional integro-differential (FIDEs) optimal control problems (FOCPs). spatial approximations with their derivatives are based on shifted ultraspherical pol...