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 Daghestan Electronic Mathematical Reports, 2017, Issue 7, Pages 86–93 (Mi demr41)

Recurrence formulas for Chebyshev polynomials orthonormal on uniform grid

M. S. Sultanakhmedov

Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala

Abstract: We consider recurrence relations for the classical Chebyshev polynomials $\{ \tau_n^{\alpha, \beta}(x, N) \}_{n=0}^{N-1}$, forming a finite orthonormal system on a uniform grid $\Omega_N = \{ 0, 1, \ldots, N-1\}$ 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

DOI: https://doi.org/10.31029/demr.7.10

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UDC: 517.538
Revised: 31.01.2017
Accepted:03.02.2017

Citation: M. S. Sultanakhmedov, “Recurrence formulas for Chebyshev polynomials orthonormal on uniform grid”, Daghestan Electronic Mathematical Reports, 2017, no. 7, 86–93

Citation in format AMSBIB
\Bibitem{Sul17} \by M.~S.~Sultanakhmedov \paper Recurrence formulas for Chebyshev polynomials orthonormal on uniform grid \jour Daghestan Electronic Mathematical Reports \yr 2017 \issue 7 \pages 86--93 \mathnet{http://mi.mathnet.ru/demr41} \crossref{https://doi.org/10.31029/demr.7.10} \elib{http://elibrary.ru/item.asp?id=35033753}