Jacobi Polynomials Pn Research Articles

The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials

Exceptional Laguerre and Jacobi polynomials p n ( x ) p_n(x) are bispectral, in the sense that as functions of the continuous variable x x , they are eigenfunctions of a second order differential operator and as functions of the discrete variable n n , they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.

Jacobi Series for General Parameters via Hadamard Finite Part and Application to Nonhomogeneous Hypergeometric Equations

The representation of analytic functions as convergent series in Jacobi polynomials Pn(α,β) is reformulated using the Hadamard principal part of integrals for all α,β∈C∖{0,-1,-2,…}, α+β≠-2,-3,…. The coefficients of the series are given as usual integrals in the classical case (when Rα,Rβ>-1) or by their Hadamard principal part when they diverge. As an application it is shown that nonhomogeneous differential equations of hypergeometric type do generically have a unique solution which is analytic at both singular points in C.

Zeros of Jacobi polynomials P n ( α , β ) $ P_{n}^{(\alpha ,\beta)} $ , − 2 < α, β < − 1

The sequence of Jacobi polynomials $\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ is orthogonal on (− 1,1) with respect to the weight function (1 − x)α(1 + x)β provided α > − 1,β > − 1. When the parameters α and β lie in the narrow range − 2 < α, β < − 1, the sequence of Jacobi polynomials $\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ is quasi-orthogonal of order 2 with respect to the weight function (1 − x)α+ 1(1 + x)β+ 1 and each polynomial of degree n,n ≥ 2, in such a Jacobi sequence has n real zeros. We show that any sequence of Jacobi polynomials $\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ with − 2 < α, β < − 1, cannot be orthogonal with respect to any positive measure by proving that the zeros of Pn− 1(α,β) do not interlace with the zeros of Pn(α,β) for any $n \in \mathbb {N},$ n ≥ 2, and any α,β lying in the range − 2 < α, β < − 1. We also investigate interlacing properties satisfied by the zeros of equal degree Jacobi polynomials Pn(α,β),Pn(α,β+ 1) and Pn(α+ 1,β+ 1) where − 2 < α, β < − 1. Upper and lower bounds for the extreme zeros of quasi-orthogonal order 2 Jacobi polynomials Pn(α,β) with − 2 < α, β < − 1 are derived.

Approximation of Burger Equation by Orthogonal Collocation Method With Base Shifted Jacobi Polynomials P(α, β)n for Different Values of α and β

In the present study, orthogonal collocation method has been followed to discretize the non-linear Burger equation. Shifted Jacobi polynomials Pn(α, β)n (x) has been chosen as base functions. The problem is approximated for different values of αandβ. Numerical values obtained for different values of α and β have been compared to the exact ones. The stability analysis has been done by von Neumann approach. The technique is found to be unconditionally stable for any choice of base function.

On parameter derivatives of a family of polynomials in two variables

The purpose of the present paper is to give the parameter derivative representations of the form∂Pn,k(λ;x,y)∂λ=∑m=0n-1∑j=0mdn,j,mPm,j(λ;x,y)+∑j=0ken,j,kPn,j(λ;x,y)for a family of orthogonal polynomials of variables x and y, with λ being a parameter and 0⩽k⩽n;n,k=0,1,2,…. First, we shall present the representations of the parameter derivatives of the generalized Gegenbauer polynomials Cn(λ,μ)(x) with the help of the parameter derivatives of the classical Jacobi polynomials Pn(α,β)(x), i.e. ∂∂αPn(α,β)(x) and ∂∂βPn(α,β)(x). Then, by using these derivatives, we investigate the parameter derivatives for two-variable analogues of the generalized Gegenbauer polynomials. Furthermore, we discuss orthogonality properties of the parametric derivatives of these polynomials.

A note on parameter derivatives of the Jacobi polynomials on the triangle

The aim of the present paper is to obtain derivatives of the Jacobi polynomials with two variables on the triangle with respect to their parameters by using the parameter derivatives of the classical Jacobi polynomials Pn(α,β)(x). Furthermore, we give orthogonality properties of the parametric derivatives on the triangle.

The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials PN(α ,β ) ,α ,β > -1 are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore, we show that a large negative eigenvalue of order N2 appears for -1 1 . The eigenvalues for Legendre polynomials are directly related to the roots of the spherical Bessel and Hankel functions that are involved in solving Helmholtz equation inspherical coordinates.

Orthogonality and asymptotics of Pseudo-Jacobi polynomials for non-classical parameters

The family of general Jacobi polynomials Pn(α,β) where α,β∈C can be characterised by complex (non-Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials Pn(α,β) where α,β∈R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another special subclass of Jacobi polynomials Pn(α,β) with α,β∈C, β=α¯ which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {Pnα,α¯}n=0∞ is the only other subclass in the general Jacobi family (beside the classical Jacobi polynomials) that has n real zeros for every n=0,1,2,… for certain values of α∈C. For some parameter ranges Pseudo-Jacobi polynomials are fully orthogonal, for others there is only complex (non-Hermitian) orthogonality. We summarise the orthogonality and quasi-orthogonality properties and study the zeros of Pseudo-Jacobi polynomials, providing asymptotics, bounds and results on the monotonicity and convexity of the zeros.

POLYNOMIALS EXPANSIONS FOR SOLUTION OF WAVE BESSEL EQUATION IN QUANTUM CALCULUS

In this paper, using the q2-Laplace transform early introduced by Abdi,1we study q-wave Bessel polynomials related with the q-Bessel operator Δq,α. We show in particular that they are linked to the q-little Jacobi polynomials pn(x;α, β|q2).

Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials Pn(α, β), with α,β > - 1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1, 1). This property is no longer val...

Some generating functions of modified Jacobi polynomials

The present note deals with the derivation of novel estensions of some bilateral and mixed trilateral generating functions of modified Jacobi polynomials Pn ^a ’ (x) from the stand point of one parameter group of continuous transformations. Some particular cases of interest as wel as the applicatîons of our results are laid down.

Some multilinear generating functions

The multiple hypergeometric generating function (1.3) below, due to H. M. Srivastava and J. P. Singhal [Acad. Roy. Belg. Bull. Cl. Sci. (5)58 (1972), 1238–1247], applies readily to deduce multilinear generating functions for thespecial Jacobi polynomialsP n (α-n,β) (x), P n (α,β-n) (x) orP n (α-n,β-n) (x), the Laguerre polynomialsL n (α) (x), thebiorthogonal polynomialsZ n α (x; k) of J.D.E. Konhauser [Pacific J. Math. 21 (1967), 303–314], and so on, and indeed also for any suitable products of these polynomials. The present paper is motivated by the need for a multiple hypergeometric generating function, analogous to (1.3), which could apply to yield multilinear generating functions for theunrestricted Jacobi polynomialsP n (α,β) (x). Several interesting generalizations of the multiple hypergeometric generating function (1.3), and of its analogue (5.6) thus obtained, are given; many of these generalizations are shown to apply also to derive multilinear generating functions for the classical Hermite polynomialsH n(x) and for their various known generalizations considered, among others, by F. Brafman [Canad. J. Math. 9 (1957), 180–187] and by H. W. Gould and A. T. Hopper [Duke Math. J. 29 (1962), 51–63]. The multilinear generating functions (1.19), (1.22), (1.23), (1.25), (1.30), (3.3), (4.1), (4.2), (4.8), (5.5), (5.6), (6.3), (6.4) and (6.6) below are believed to be new.

Polynomial Approximation and Growth of Generalized Axisymmetrig Potentials

Generalized axisymmetric potentials Fα (GASP) are regular solutions to the generalized axisymmetric potential equation(1.1)in some neighborhood Ω of the origin where they are subject to the initial data(1.2)along the singular line y = 0. In Ω, these potentials may be uniquely expanded in terms of the complete set of normalized ultraspherical polynomials(1.3)defined from the symmetric Jacobi polynomials Pn(α, α)(ξ) of degree n with parameter α as Fourier series(1.4)

Linearization of the Product of Jacobi Polynomials. III

In a series of papers [1; 2; 3; 4] the operation of linearizing the product of two Jacobi polynomials Pn(α, β)(x), α, β &gt; –1, has been investigated and the existence of a natural Banach algebra associated with the linearization coefficients has been proven. This was proven for α + β + 1 ≧ 0 in [3] and for a slightly larger region in [4]. It was shown in [4] that such a Banach algebra does not exist for . The method used in [1; 3; 4] was to prove the non-negativity of the expansion coefficients from which the existence of the Banach algebra easily follows. However, as shown in [4], the coefficients for a subset of can be negative infinitely often and so a different method must be used for these values of α and β.