A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey–Wilson to Wilson polynomials and from q-Racah to Racah polynomials ar...

We discuss the relationships among Jacobi matrices, orthogonal polynomials, spectral measure, moments, minors, Gaussian quadrature, resolvents and continued fractions in the simplest setting, namely the finite-dimensional one. The formal structure is essentially the same as that in the infinite-dimensional setting, where it leads into the rich analytic world of orthogonal polynomials on the rea...

We introduce two kinds of multiple little q-Jacobi polynomials p~n with multi-index ~n = (n1, n2, . . . , nr) and degree |~n| = n1 + n2 + · · · + nr by imposing orthogonality conditions with respect to r discrete little q-Jacobi measures on the exponential lattice {qk, k = 0, 1, 2, 3, . . .}, where 0 < q < 1. We show that these multiple little qJacobi polynomials have useful q-difference proper...

Article history: Received 9 March 2007 Received in revised form 16 June 2008 Accepted 20 October 2008 Available online 1 November 2008

In the present paper, an effort has been made to give Expansion of The Polynomial Set Sn(x1, x2, x3) in Terms Jacobi Polynomials. Many interesting new results may be obtained on specializing respective parameters which some them are believed new.AMS Subject Classification: 33c.

The paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so–called Bernstein–Bézier form of a polynomial.

This paper surveys numerical methods for general sparse nonlinear eigenvalue problems with special emphasis on iterative projection methods like Jacobi–Davidson, Arnoldi or rational Krylov methods and the automated multi–level substructuring. We do not review the rich literature on polynomial eigenproblems which take advantage of a linearization of the problem.

A representation of the Jacobi algebra h1 ⋊ su(1, 1) by first order differential operators with polynomial coefficients on the manifold C×D1 is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with polynomials coefficients act is constructed.

It is proved in this short note that for any polynomial p of d variables and degree at most n we have the sharp Bernstein–Markov type inequality ∫Bd(1−|x|2)μ+1|∂p|2≤M∫Bd(1−|x|2)μ|p|2,μ>−1, with M=n(n+d+2μ) M=n(n+d+2μ)−d+1 if even or odd, respectively. Here Bd unit ball Rd ∂p stands gradient polynomial. The upper bounds are attained certain Jacobi polynomials.

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence Sn = ( 3n)( 3n 2n) 2( n )(2n+1) , and the binomial coefficients ( 3n n )