Rediscovering the contributions of Rodrigues on the representation of special functions

Following the pressure of Hermite in 1865 and Heine in 1868, the name of Rodrigues was definitively associated to his famous representation given in 1815. In this paper, we precise the contribution of Rodrigues, proving that his formula not only generates the Legendre polynomials, but also special Gegenbauer polynomials and even the corresponding second kind functions, which are never mentioned. The links with Hildebrandt polynomials and with the Bell polynomials are also briefly examined.     

Generalized Hermite polynomials

Hermite polynomials of several variables are defined by a generalization of the Rodrigues formula for ordinary Hermite polynomials. Several properties are derived, including the differential equation satisfied by the polynomials and their explicit expression. An application is given.     

Polynomial extensions of distributions and their applications in actuarial and financial modeling

The paper deals with orthogonal polynomials as a useful technique which can be attracted to actuarial and financial modeling. We use Pearson’s differential equation as a way for orthogonal polynomials construction and solution. The generalized Rodrigues formula is used for this goal. Deriving the weight function of the differential equation, we use it as a basic distribution density of variables like financial asset returns or insurance claim sizes. In this general setting, we derive explicit formulas for option prices as well as for insurance premiums. The numerical analysis shows that our new models provide a better fit than some previous actuarial and financial models.     

Generalized Hermite polynomials

Hermite polynomials of several variables are defined by a generalization of the Rodrigues formula for ordinary Hermite polynomials. Several properties are derived, including the differential equation satisfied by the polynomials and their explicit expression. An application is given.     

Connections between Romanovski and other polynomials

A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.     

Some Results on the Coefficients of Integrated Expansions of Ultraspherical Polynomials and their Integrals

The formula of expressing the coefficients of an expansion of ultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.     

Some Results on the Coefficients of Integrated Expansions of Ultraspherical Polynomials and their Integrals

The formula of expressing the coefficients of an expansion of ultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.     

Polynomials associated with a functional product of the Hermite type

We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second‐order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite‐type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues‐type formula and a four‐term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd.     

THE KRALL POLYNOMIALS: A NEW CLASS OF ORTHOGONAL POLYNOMIALS

Abstract

A new set of orthogonal polynomials is found that are solutions to a sixth order formally self adjoint differential equation. These polynomials are shown to generalize the Legendre and Legendre type polynomials. We also show that these polynomials satisfy many properties shared by the classical orthogonal polynomials of Jacobi, Laguerre and Hermite.     

A quadrature tau method for fractional differential equations with variable coefficients

In this article, we develop a direct solution technique for solving multi-order fractional differential equations (FDEs) with variable coefficients using a quadrature shifted Legendre tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the tau method for FDEs with variable coefficients is treated using the shifted Legendre–Gauss–Lobatto quadrature. Numerical results are given to confirm the reliability of the proposed method for some FDEs with variable coefficients.     

Hermite Matrix-Valued Functions Associated to Matrix Differential Equations

Some sequences of matrix polynomials have been introduced recently as solutions of certain second-order differential equations, which can be seen as appropriate generalizations, to the matrix setting, of classical orthogonal polynomials. In this paper, we consider families (in a complex parameter) of matrix-valued special functions of Hermite type, which arise as natural extensions of the aforementioned matrix polynomials of the same type. We show that such families are solutions of corresponding differential equations and enjoy several structural properties. In particular, they satisfy a Rodrigues formula expressed in terms of the Weyl fractional calculus. We also show that, unlike the scalar case, a second-order differential operator having such a family as a set of joint eigenfunctions need not be unique.     

The Dω—classical orthogonal polynomials

This is an expository paper; it aims to give an essentially self-contained overview of discrete classical polynomials from their characterizations by Hahn’s property and a Rodrigues’ formula which allows us to construct it. The integral representations of corresponding forms are given.     

Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices

ABSTRACT

By building a second-order adjoint difference equations on nonuniform lattices, the generalized Rodrigues type representation for the second kind solution of a second-order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a ‘generalized’ Rodrigues formula is also established.     

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.     

Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.      

Hobson's formula for Dunkl operators and its applications

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.     

The rodrigues type representations for a certain class of special functions

Summary An attempt is made here to present a systematic introduction to and several applications of a certain method of obtaining Rodrigues type representations for a fairly wide variety of sequences of special functions. The main results, contained in Theorems2 and3 below, are shown to apply not only to the Bessel polynomials, the classical orthogonal polynomials including, for instance, Hermite, Jacobi (and, of course, Gegenbauer, Legendre, and Tchebycheff), and Laguerre polynomials, and to their various generalizations studied in recent years, but also to such other special functions as the Bessel function and a certain class of generalized hypergeometric functions. Entrata in Redazione il 25 giugno 1977. This work was partially supported by the National Research Council of Canada under grants A-7353 and A-4027. For a preliminary report of this paper see Notices Amer. Math. Soc.,24 (1977), p. A-238, Abstract no. 77T-B43.     

Legendre Wavelets Direct Method for the Numerical Solution of Time-Fractional Order Telegraph Equations

In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann–Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

On some properties of q-Hahn multiple orthogonal polynomials

This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems.