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On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

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Abstract

We shall study the differential equation

$${y'^2} = {T_n}\left( y \right) - \left( {1 - 2{\mu ^2}} \right)$$

where μ 2 is a constant, T n (x) are the Chebyshev polynomials with n = 3, 4, 6.

The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on 2 F 1(1/4, 3/4; 1; z), 2 F 1(1/3, 2/3; 1; z), 2 F 1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, FL, 32611–8105, USA

    Li Chien Shen

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  1. Li Chien Shen
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Correspondence to Li Chien Shen.

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Shen, L.C. On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6. Acta. Math. Sin.-English Ser. 33, 21–36 (2017). https://doi.org/10.1007/s10114-016-6180-1

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  • DOI: https://doi.org/10.1007/s10114-016-6180-1

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