Fast and efficient methods of evaluation of the connection coefficients between shifted Jacobi and Bernstein polynomials are proposed. The complexity of the algorithms is O(n), where n denotes the degree of the Bernstein basis. Given results can be helpful in a computer aided geometric design, e.g., in the optimization of some methods of the degree reduction of Bézier curves.

We construct Jacobi-weighted orthogonal polynomials (α,β,γ) n,r (u,v,w), α,β,γ > −1, α+ β + γ = 0, on the triangular domain T . We show that these polynomials (α,β,γ) n,r (u, v,w) over the triangular domain T satisfy the following properties: (α,β,γ) n,r (u,v,w) ∈ n, n≥ 1, r = 0,1, . . . ,n, and (α,β,γ) n,r (u,v,w) ⊥ (α,β,γ) n,s (u,v,w) for r =s. Hence, (α,β,γ) n,r (u,v,w), n= 0,1,2, . . ., r =...

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another spe...

First we discuss the description of inverse polynomial images of [−1, 1], which consists of two Jordan arcs, by the endpoints of the arcs only. The polynomial which generates the two Jordan arcs is given explicitly in terms of Jacobi’s theta functions. Then the main emphasis is put on the case where the two arcs are symmetric with respect to the real line. For instance it is demonstrated that t...

our aim in this paper is to prove an analog of younis's theorem on the image under the jacobi transform of a class functions satisfying a generalized dini-lipschitz condition in the space $mathrm{l}_{(alpha,beta)}^{p}(mathbb{r}^{+})$, $(1< pleq 2)$. it is a version of titchmarsh's theorem on the description of the image under the fourier transform of a class of functions satisfying the dini-lip...

We shall give bounds on the spacing of zeros of certain functions belonging to the LaguerrePólya class and satisfying a second order linear differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polynomials, which are uniform in all the parameters.

Let α, β ≥ −1/2, and for k = 0, 1, · · ·, pk (α,β) denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H , α, and β such that

Recently the author derived a Laplace integral representation, a product formula and an addition formula for Jacobi polynomials Ppfi). The results are (1) (2) and (3) * j, ,I " (~0~2 0-f-2 sin2 0 + ir cos #I sin 28)n * ' P$ycos 281) PFycos 282) 2&x + 1) Pgq 1) Pfq 1)-= l/22 qa-p)r(/l+#. . .

In this note we have obtained some novel result on mixed trilateral relations involving extended Jacobi polynomials by group theoretic method which inturn yields the corresponding results involving Hermite, Laguerre and Jacobi polynomials.

In this paper we show how one can obtain simultaneous rational approximants for ζq(1) and ζq(2) with a common denominator by means of Hermite-Padé approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, ζq(1), ζq(2) are linearly independent over Q. In particular this implies that ζq(1) and ζq(2) are irrational. Furthermor...