Sums of finite products of Chebyshev polynomials of two different types

Taekyun Kim; Dae San Kim; Dmitry V. Dolgy; Jongkyum Kwon; Taekyun Kim; Dae San Kim; Dmitry V. Dolgy; Jongkyum Kwon

doi:10.3934/math.2021722

AIMS Mathematics

2021, Volume 6, Issue 11: 12528-12542. doi: 10.3934/math.2021722

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Research article

Sums of finite products of Chebyshev polynomials of two different types

1.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
2.
Department of Mathematics, Sogang University, Seoul 121-742, Korea
3.
Kwangwoon Glocal Education Center, Kwangwoon University, Seoul 139-701, Korea
4.
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

Received: 31 May 2021 Accepted: 18 August 2021 Published: 31 August 2021
MSC : 11B83, 33C05, 33C45

In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.
- Chebyshev polynomials of the first, second, third and fourth kinds,
- terminating hypergeometric function
Citation: Taekyun Kim, Dae San Kim, Dmitry V. Dolgy, Jongkyum Kwon. Sums of finite products of Chebyshev polynomials of two different types[J]. AIMS Mathematics, 2021, 6(11): 12528-12542. doi: 10.3934/math.2021722

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Abstract

In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.

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