Abstract
We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including a representation by hypergeometric series, recurrence relations, and derivative relations.
Highlights
The following two differential equations are known as the Chebyshev differential equations: (1 − t2 )y00 − ty0 + n2 y = 0, (1)
We demonstrate the difference equation that the discrete Chebyshev polynomials of the second kind solve
We prove the discrete analogue of the three-term recurrence for the discrete Chebyshev polynomials of the second kind
Summary
The following two differential equations are known as the Chebyshev differential equations:. A Bessel difference equation was investigated in [4], whose solutions were shown to be generalized hypergeometric series with variable parameters. Such “discrete Bessel functions” were applied in [5] to solve discrete wave and diffusion equations. Discrete special functions defined by an instance of Equation (13) import the same parameter set a1 , . . .}, it is not possible to always naively map what appears in the independent variable arguments of functions defined by Equation (4) to 1−t their discrete analogues (13) in general, explaining the extra parameters. The final argument appearing in Equations (5) and (6) acts as a barrier to a simple importation of Chebyshev polynomials to the discrete case from the continuous case, but we resolve this dilemma in the sequel
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