Approximation theory

Daghestan Electronic Mathematical Reports: Issue 7 (2017)

Recurrence formulas for Chebyshev polynomials orthonormal on uniform grid

UDK: 517.538

Pages: 86 - 93

We consider recurrence relations for the classical Chebyshev polynomials $\left\{ \tau_n^{\alpha, \beta}(x, N) \right\}_{n=0}^{N-1}$, forming a finite orthonormal system on a uniform grid $\Omega_N = \left\{ 0, 1, \ldots, N-1\right\}$ with weight $\mu_N^{\alpha,\beta}(x) = c \, \frac{\Gamma(x+\beta+1)\Gamma(N-x+\alpha)}{ \Gamma(x+1)\Gamma(N-x)}$, where $c = \frac{\Gamma(N)2^{\alpha+\beta+1}}{\Gamma(N+\alpha+\beta+1)}$, $\alpha,\beta>-1$. Special attention is paid to the most commonly used cases: $\alpha=\beta$; $\alpha=\beta=0$; $\alpha=\beta=\pm 1/2$ and several others. In the proof of recurrence formulas we substantially use the well-known properties of the considered Chebyshev polynomials such as the orthogonality property, difference properties and the connection with the generalized hypergeometric function.

Keywords: Chebychev polynomials; recurrence formulas; polynomials orthogonal on grids; uniform grid; function approximation.

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