Weighted $$L^2$$-norms of Gegenbauer polynomials

Information entropy of Gegenbauer polynomials

The information entropy of Gegenbauer polynomials is relevant since this is related to the angular part of the information entropies of certain quantum mechanical systems such as the harmonic oscillator and the hydrogen atom in D dimensions. We give an effective method to compute the entropy for Gegenbauer polynomials with an integer parameter and obtain the first few terms in the asymptotic ex...

ON l2 NORMS OF SOME WEIGHTED MEAN MATRICES

k=1 |ak|, in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ ...

A NOTE ON l2 NORMS OF WEIGHTED MEAN MATRICES

k=1 |ak| , in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤...

Gegenbauer Polynomials and Semiseparable Matrices

In this paper, we develop a new O(n logn) algorithm for converting coefficients between expansions in different families of Gegenbauer polynomials up to a finite degree n. To this end, we show that the corresponding linear mapping is represented by the eigenvector matrix of an explicitly known diagonal plus upper triangular semiseparable matrix. The method is based on a new efficient algorithm ...

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials