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Basis Functions Generated by Subdivision Matrices
Published in Charles Chui, Johan de Villiers, Wavelet Subdivision Methods, 2010
Charles Chui, Johan de Villiers
where β0 := 0, and w(x) > 0,x ∊ (a,b), is an arbitrarily given continuous function. Prove that {Pj} is an orthogonal polynomial sequence with respect to the weight function w, in the sense that ∫abw(x)Pj(x)Pk(x)dx=0,j≠k.
Summation Kernels for Orthogonal Polynomial Systems
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
Frank Filbir, Rupert Lasser, Josef Obermaier
By recurrence relation (15.21) we get an orthogonal polynomial sequence (Jn(α,β))n∈ℕ0 of Jacobi polynomials.
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
We next prove a general result about the generating function of a polynomial sequence, and exhibit it for the simple example n2. The much harder Example 2.6 gives the generating function for the rational sequence 12n+1.
Degenerate 2D bivariate Appell polynomials: properties and applications
Published in Applied Mathematics in Science and Engineering, 2023
Shahid Ahmad Wani, Arundhati Warke, Javid Gani Dar
Polynomial sequences are of interest in enumerative combinatorics, algebraic combinatorics, and applied mathematics. The Laguerre polynomials, Chebyshev polynomials, Legendre polynomials, and Jacobi polynomials are a few polynomial sequences that appear as solutions to particular ordinary differential equations in physics and approximation theory. The most significant polynomial sequences is a class of Appell polynomial sequences [1]. Many applications of the Appell polynomial sequence may be found in theoretical physics, approximation theory, mathematics, and related fields of mathematics. The set of all Appell sequences is closed as a result of umbral polynomial sequence composition. This process turns the collection of all Appell sequences into an abelian group.