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For $k ge 2$ even, and $ alpha ge -(2k+1)/4 $, we provide a uniform approximation of the ultraspherical polynomials $ P_k^{(alpha,, alpha)}(x) $ in the oscillatory region with a very explicit error term. In fact, our result covers all $alpha$ for which the expression oscillatory region makes sense. We show that there the function $g(x)={c sqrt{b(x)} , (1-x^2)^{(alpha+1)/2} P_k^{(alpha, alpha)}(x)=cos mathcal{B}(x)+ r(x)}$, where $c=c(k, alpha)$ is defined by the normalization, $mathcal{B}(x)=int_{0}^ x b(x) dx$, and the functions $c,, b(x), , mathcal{B}(x)$, as well as bounds on the error term $r(x)$ are given by some rather simple elementary functions.
We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an Abel-type
In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre polynomials
We introduce a new family of orthogonal polynomials on the disk that has emerged in the context of wave propagation in layered media. Unlike known examples, the polynomials are orthogonal with respect to a measure all of whose even moments are infinite.
We shall establish two-side explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of $|H_k(x)| e^{-x^2/2},$ on the real axis, where $H_k$ are the Hermite polynomials.
Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with inequalitie