We use the well-known observation that solutions of Jacobi's differential equation can be represented via non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial transforms. More explicitly, it follows from this matrix corresponding discrete transform is Hadamard product numerically low-rank Fourier (DFT) matrix. The applicatio...

Jacobi Matrices (real symmetric tridiagonal matrices) have a wide range of applications in physics and engineering, and are closely and non-trivially linked with many other mathematical objects, such as orthogonal polynomial, one dimensional Schrödinger operators and the Sturm-Liouville problem. In the past couple of decades, constructing Jacobi matrices from different types of data was studied...

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of Akn, which generalizes the well known estimates both classical Jacobi and one-variable polynomial sums.

A commuting family of symmetric matrices are called the block Jacobi matrices, if they are block tridiagonal. They are related to multivariate orthogonal polynomials. We study their eigenvalues and joint eigenvectors. The joint eigenvalues of the truncated block Jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials.

Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), is well known. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. As examples, the Jacobi, Laguerre and Charlier polynomials are considered.  2005 Elsevier Inc. All...

The one-parameter family of polynomials Jn(x, y) = ∑n j=0 ( y+j j ) x is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n ≥ 6, the polynomial Jn(x, y0) is irreducible over Q for all but finitely many y0 ∈ Q. If n is odd, then with the exception of a finite set of y0, the Galois group of Jn(x, y0) is Sn; if n is even, then the exceptional set is thin.

— In this paper we show that the invariant polynomial ring of the associated Clifford-Weil group can be embedded into the ring of Jacobi modular forms over the totally real field, so, therefore, that of Hilbert modular forms over the totally real field. Résumé (Anneau des invariants du groupe de Clifford-Weil, et forme de Jacobi sur un corps totallement réel) Dans cet article nous démontrons qu...