ON THE POLYNOMIAL xd + ax + b OVER 𝔽q AND GAUSSIAN HYPERGEOMETRIC SERIES

descriptionPublicationkeyboard_double_arrow_right Article 28 Oct 2013 English Publisher:World Scientific Pub Co Pte LtJournal:International Journal of Number Theory, volume 9, pages 1,753-1,763 (issn: 1793-0421, eissn: 1793-7310,

Authors: RUPAM BARMAN; GAUTAM KALITA;

doi: 10.1142/s1793042113500553

ON THE POLYNOMIAL xd + ax + b OVER 𝔽q AND GAUSSIAN HYPERGEOMETRIC SERIES

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Abstract

For d ≥ 2, denote by Pd(x) the polynomial over 𝔽q given by [Formula: see text]. We explicitly find the number of solutions in 𝔽q of the polynomial equation Pd(x) = 0 in terms of special values of dFd-1 and d-1Fd-2 Gaussian hypergeometric series with characters of orders d and d - 1 as parameters. This solves a problem posed by K. Ono (see p. 204 in [Web of Modularity : Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, No. 102 (American Mathematical Society, Providence, RI, 2004)]) on special values of n+1Fn Gaussian hypergeometric series for n > 2.

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