The relation of the d-orthogonal polynomials to the Appell polynomials

AbstractWe are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials.A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite

Estimation of Hydrogen Production Using Wind Energy in Algeria

AbstractIn response to problems involved in the current crisis of petrol in Algeria, with the decrease in the price of the oil barrel, the rate of growth in domestic electricity demand and with an associated acceleration of global warming, as a result of significantly increased greenhouse gas (GHG) emissions, renewable energy seems today as a clean and strategic substitution for the next decades. However, the greatest obstacles which face electric energy comes from renewable energy systems are often referred to the intermittency of these sources as well as storage and transport problems, the need for their conversion into a versatile energy carrier in its use, storable, transportable and environmentally acceptable are required. Among all the candidates answering these criteria, hydrogen presents the best answer. In the present work, particular attention is paid to the production of hydrogen from wind energy. The new wind map of Algeria shows that the highest potential wind power was found in Adrar, Hassi-R’Mel and Tindouf regions. The data obtained from these locations have been analyzed using Weibull probability distribution function. The wind energy produced in these locations is exploited for hydrogen production through water electrolysis. The objective of this paper is to realize a technological platform allowing the evaluation of emergent technologies of hydrogen production from wind energy using four wind energy conversion systems of 600, 1250, 1500 and 2000kW rated capacity. The feasibility study shows that using wind energy in the selected sites is a promising solution. It is shown that the turbine “De Wind D7” is sufficient to supply the electricity and hydrogen with a least cost and a height capacity factor. The minimum cost of hydrogen production of 1.214 $/kgH2 is obtained in Adrar

On 2-orthogonal polynomials of Laguerre type

Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionals ω0 and ω1 (see Definition 1.1). Now, let {Qn}n≥0 be the sequence of polynomials defined by Qn:=(n+1)−1P′n+1,n≥0. When {Qn}n≥0 is, also, 2-orthogonal, {Pn}n≥0 is called classical (in the sense of having the Hahn property). In this case, both {Pn}n≥0 and {Qn}n≥0 satisfy a third-order recurrence relation (see below). Working on the recurrence coefficients, under certain assumptions and well-chosen parameters, a classical family of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionals ω0 and ω1 and obtain their weight functions which satisfy a second-order differential equation. From all these properties, we show that the resulting polynomials are an extention of the classical Laguerre's polynomials and establish a connection between the two kinds of polynomials

Some classical multiple orthogonal polynomials

Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system of several functions. We describe seven families of multiple orthogonal polynomials which have he same flavor as the very classical orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some open research problems and some applications

The Kontorovich-Lebedev transform as a map between dd-orthogonal polynomials

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $KL_{\alpha}$-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the $KL_{\alpha}$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $KL_{\alpha}$-transform is a $d$-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and $d$ is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.Comment: 27 page

The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

We show that multiple orthogonal polynomials for r measures $(\mu_1,...,\mu_r)$ satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices $\vec{n}\pm \vec{e}_j$, where $\vec{e}_j$ are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of measures $\mu_j$. We show how the Christoffel-Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials.Comment: 22 page

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

Aerodynamic investigation of the start-up process of H-type vertical axis wind turbines using CFD

In this study, a CFD start-up model has been built after conducting the sensitivity studies to evaluate the self-starting behaviour of the H-type vertical axis wind turbines (VAWTs). The self-starting behaviour of a well-investigated VAWT is used for the model validation, and then the details of aerodynamics of the start-up process have been examined. Finally, the effect of the moment of inertia and the blade number on the aerodynamic behaviour of the self-starting and power performance of the H-type VAWT are analysed. It has been found that in the critical region, where TSR<1, the contribution of the drag to the torque generation plays a significant role in the second and third quarters of the rotor revolution, where the azimuthal position varies between 100° and 253°. The results also show that increasing the turbine inertia did not show a noticeable effect on the start-up behaviour of the turbine and final rotational speed. However, an increase in the instantaneous turbine power during the start-up process after the optimum TSR is observed with decreasing the turbine inertia. The current findings also show that an increase in the blade number makes the turbine easier to start-up; however, this may reduce the turbine power coefficient