Monotonicity of zeros of Laguerre polynomials

Denote by x_nk(α), k=1,…,n, the zeros of the Laguerre polynomial . We establish monotonicity with respect to the parameter α of certain functions involving x_nk(α). As a consequence we obtain sharp upper bounds for the largest zero of .     

On differential equations for Sobolev-type Laguerre polynomials

The Sobolev-type Laguerre polynomials are orthogonal with respect to the inner product

where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form

where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.

     

On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial is irreducible over the rationals for all n?1 and has Galois group A_n if n+1 is an odd square, and S_n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L_n^(α)(x) can be embedded in a double cover.     

Moment representations of the exceptional X1‐Laguerre orthogonal polynomials

Exceptional orthogonal Laguerre polynomials can be viewed as an extension of the classical Laguerre polynomials per excluding polynomials of certain order(s) from being eigenfunctions for the corresponding exceptional differential operator. We are interested in the (so‐called) Type I X₁‐Laguerre polynomial sequence , and , where the constant polynomial is omitted. We derive two representations for the polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We introduce adjusted moments and find a second, more elegant formula. We deduce a recursion formula for the moments and the adjusted ones. The adjusted moments are also expressed via a generating function. We observe a certain detachedness of the first two moments from the others.     

Extension of Laguerre polynomials with negative arguments

We consider the irreducibility of polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{\left(\alpha \right)}\left(x\right)$ vanishes if and only if $n\ge \left|\alpha \right|=?\alpha$. Therefore we assume that $\alpha =?n?s?1$ where $s$ is a non-negative integer. Let $g\left(x\right)={(?1)}^n{L}_n^{(?n?s?1)}\left(x\right)=\sum _{j=0}^n{a}_j\frac{x^j}{j!}$ and more general polynomial, let $G\left(x\right)=\sum _{j=0}^n{a}_j{b}_j\frac{x^j}{j!}$ where $b_j$ with $0\le j\le n$ are integers such that $\left|b_0\right|=\left|b_n\right|=1$. Schur was the first to prove the irreducibility of $g\left(x\right)$ for $s=0$. It has been proved that $g\left(x\right)$ is irreducible for $0\le s\le 60$. In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either $G\left(x\right)$ is irreducible or $G\left(x\right)$ is linear factor times irreducible polynomial. This is a consequence of the estimate $s>1.9k$ whenever $G\left(x\right)$ has a factor of degree $k\ge 2$ and $(n,k,s)\ne (10,5,4)$. This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.     

Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

We consider the polynomials orthogonal with respect to the Sobolev type inner product

where and is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order if is a nonnegative integer and

     

Laguerre polynomials as Jensen polynomials of Laguerre-Pólya entire functions

We prove that the only Jensen polynomials associated with an entire function in the Laguerre-Pólya class that are orthogonal are the Laguerre polynomials.     

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on (0,∞) from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution. Error estimate is given. Numerical results are produced.     

New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

This article proposes a new unconditionally stable scheme to solve one‐dimensional telegraph equation using weighted Laguerre polynomials. Unlike other numerical schemes, the time derivatives in the equation can be expanded analytically based on the Laguerre polynomials and basis functions. By applying a Galerkin temporal testing procedure and using the orthogonal property of weighted Laguerre polynomials, the time variable can be eliminated from computations, which results in an implicit equation. After solving the equation recursively one can obtain the numerical results of telegraph equation by using the expanded coefficients. Some numerical examples are considered to validate the accuracy and stability of this proposed scheme, and the results are compared with some existing numerical schemes.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1603–1615, 2017     

Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials

For each the nth Laguerre polynomial has an m-fold zero at the origin when α=−m. As the real variable α→−m, it has m simple complex zeros which approach 0 in a symmetric way. This symmetry leads to a finite value for the limit of the sum of the reciprocals of these zeros. There is a similar property for the zeros of the q-Laguerre polynomials and of the Jacobi polynomials and similar results hold for sums of other negative integer powers.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

Orthogonal polynomials on the unit circle associated with the Laguerre polynomials

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.

     

Divergent Laguerre series

We prove failure of a.e. convergence of partial sums of Laguerre expansions of functions for 4$">. The idea which is used goes back to Stanton and Tomas. We follow Meaney's paper (1983), where divergence results were proved in the Jacobi polynomial case.

     

A note on summability of multiple Laguerre expansions

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an -radial function is shown to be an -radial function that coincides with the Laguerre polynomial expansion of , which allows us to settle the problem of summability below the critical index for the -radial functions.

     

A note on Sobolev orthogonality for Laguerre matrix polynomials

Let {Ln(A,λ)(x)}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by Ln(A,λ)(x)=n!/(-λ)n∑nk=0(-λ)κ/k!(n-1)! (A I)n[(A I)k]-1 xk,where A ∈ Cr×r. It is known that {Ln(A,λ)(x)}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) ＞ - 1 for every z ∈σ(A).In this note we show that forA such that σ(A) does not contain negative integers, the Laguerre matrix polynomials Ln(A,λ) (x) are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.     

Accurate computations with Laguerre matrices

This paper provides an accurate method to obtain the bidiagonal factorization of collocation matrices of generalized Laguerre polynomials and of Lah matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values, and inverses of these matrices. Numerical examples are included.     

基于Laguerre函数的Landau型不等式

基于经典正交的广义 Laguerre多项式 Lsn( x)建立了 Landau型不等式 ,并且指出其不等式的系数在某种意义上是最好可能的 ,加强并推广了 Varma和 Bojanov的结果 .