Analyzing, developing and exploiting results obtained by Laplace in 1785 on the Fourier series expansion of the function $(1-2\alpha \cos \theta+\alpha^2)^{-s}$, we obtain a family of new expansions and generating functions for Chebyshev polynomials. A relation between the generating functions of the Chebyshev polynomials $T_n$ and the Gegenbauer polynomials $C_n^{(2)}$ is given.
A Family of New Generating Functions for the Chebyshev Polynomials, Based on Works by Laplace, Lagrange and Euler
Redivo-Zaglia, Michela
Writing – Review & Editing
2024
Abstract
Analyzing, developing and exploiting results obtained by Laplace in 1785 on the Fourier series expansion of the function $(1-2\alpha \cos \theta+\alpha^2)^{-s}$, we obtain a family of new expansions and generating functions for Chebyshev polynomials. A relation between the generating functions of the Chebyshev polynomials $T_n$ and the Gegenbauer polynomials $C_n^{(2)}$ is given.
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