Journal of the Mathematical Society of Japan

Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations

Masanori SAWA and Yukihiro UCHIDA

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Abstract

We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky (2005). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert's solution (1909) of Waring's problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermite polynomials, and show a nonexistence theorem for solutions of Hausdorff-type equations by applying our discriminant formula.

Note

The first author was supported in part by Grant-in-Aid for Young Scientists (B) 26870259 and Grant-in-Aid for Scientific Research (B) 15H03636 by the Japan Society for the Promotion of Science (JSPS). The second author was also supported by Grant-in-Aid for Young Scientists (B) 25800023 by JSPS.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 831-860.

Dates
Received: 14 February 2018
First available in Project Euclid: 17 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1558080017

Digital Object Identifier
doi:10.2969/jmsj/79877987

Mathematical Reviews number (MathSciNet)
MR3984244

Subjects
Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 05E99: None of the above, but in this section 65D32: Quadrature and cubature formulas
Secondary: 12E10: Special polynomials 11E76: Forms of degree higher than two

Keywords
classical quasi-orthogonal polynomial compact formula discriminant Gaussian design Hausdorff-type equation quadrature formula

Citation

SAWA, Masanori; UCHIDA, Yukihiro. Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations. J. Math. Soc. Japan 71 (2019), no. 3, 831--860. doi:10.2969/jmsj/79877987. https://projecteuclid.org/euclid.jmsj/1558080017


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