The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the ‘holes’ in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two differe...

In this article, we have obtained some novel results on bilateral generating functions of the polynomials, YYnn+rr αα−nnnn(xx;kk), a modified form of Konhauser biorthogonal polynomials, YYnn(xx; kk) by group-theoretic method. As special cases, we obtain the corresponding results on Laguerre polynomials, LLnn αα(xx). Some applications of our results are also discussed.

The author studies the uniform convergence of extended Lagrange interpolation processes based on the zeros of Generalized Laguerre polynomials. 2009 Elsevier Inc. All rights reserved.

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.

In this article, the quasi-Laguerre iteration is established in the spirit of Laguerre's iteration for solving polynomial f with all real zeros. The new algorithm, which maintains the monotonicity and global convergence of the Laguerre iteration, no longer needs to evaluate f". The ultimate convergence rate is + 1. When applied to approximate the eigenvalues of a symmetric tridiagonal matrix, t...

Abstract. Through an algebraic method using the Dunkl–Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the AN−1and BN Calogero models with bosonic, fermionic and distinguishable particles are investigated. The Rodrigues formulas of column type that algebraically generate the monic nonsymmetric multivariable Hermite and Laguerre polynomials corresponding to...

Abstract In this article, we obtain explicit solutions of a linear PDE subject to a class of radial square integrable functions with a monotonically increasing weight function |x|e 2/2, β ≥ 0, x ∈ R. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n > 1, the solution is expressed in terms of a family of weighted ge...

Using Casorati determinants of Meixner polynomials (m n )n , we construct for each pair F = (F1, F2) of finite sets of positive integers a sequence of polynomials ma,c;F n , n ∈ σF , which are eigenfunctions of a second order difference operator, where σF is certain infinite set of nonnegative integers, σF N. When c and F satisfy a suitable admissibility condition, we prove that the polynomials...

In this paper we give new proofs of some elementary properties of the Hermite and Laguerre orthogonal polynomials. We establish Rodriguestype formulae and other properties of these special functions, using suitable operators defined on the Lie algebra of endomorphisms to the vector space of infinitely many differentiable functions.

We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as Wall did for little q-Laguerre polynomials.