Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-19T07:44:09.711Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  22 April 2024

Joint Graduate School of Mathematics for Innovation, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA e-mail:


Lehmer [‘On certain character matrices’, Pacific J. Math. 6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math. 10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Research Article
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


The first author was supported by JSPS KAKENHI Grant Number JP22KJ2477 and WISE program (MEXT) at Kyushu University. The second author thanks Ken Ono for providing research support with the Thomas Jefferson Fund and the NSF Grant (DMS-2002265 and DMS-2055118).


Beukers, F., Cohen, H. and Mellit, A., ‘Finite hypergeometric functions’, Pure Appl. Math. Q. 11(4) (2015), 559589.CrossRefGoogle Scholar
Carlitz, L., ‘Some cyclotomic matrices’, Acta Arith. 5 (1959), 293308.CrossRefGoogle Scholar
Greene, J., ‘Hypergeometric functions over finite fields’, Trans. Amer. Math. Soc. 301(1) (1987), 77101.CrossRefGoogle Scholar
Griffin, M. and Rolen, L., ‘On matrices arising in the finite field analogue of Euler’s integral transform’, Mathematics 1(1) (2013), 38.CrossRefGoogle Scholar
Ireland, K. F. and Rosen, M. I., A Classical Introduction to Modern Number Theory, revised edn, Graduate Texts in Mathematics, 84 (Springer-Verlag, New York–Berlin, 1982).CrossRefGoogle Scholar
Katz, N. M., Exponential Sums and Differential Equations, Annals of Mathematics Studies, 124 (Princeton University Press, Princeton, NJ, 1990).CrossRefGoogle Scholar
Koike, M., ‘Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields’, Hiroshima Math. J. 25(1) (1995), 4352.CrossRefGoogle Scholar
Lehmer, D. H., ‘On certain character matrices’, Pacific J. Math. 6 (1956), 491499.CrossRefGoogle Scholar
Lehmer, D. H., ‘Power character matrices’, Pacific J. Math. 10 (1960), 895907.CrossRefGoogle Scholar
McCarthy, D., ‘The number of ${F}_p$ -points on Dwork hypersurfaces and hypergeometric functions’, Res. Math. Sci. 4 (2017), Article no. 4.CrossRefGoogle Scholar
Ono, K., ‘Values of Gaussian hypergeometric series’, Trans. Amer. Math. Soc. 350(3) (1998), 12051223.CrossRefGoogle Scholar
Otsubo, N., ‘Hypergeometric functions over finite fields’, Ramanujan J. 63(1) (2024), 55104.CrossRefGoogle Scholar
Slater, L. J., Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).Google Scholar