To appear in Int. J. Number Theory. LUCAS-TYPE CONGRUENCES FOR CYCLOTOMIC ψ-COEFFICIENTS.pdf

To appear in Int. J. Number Theory. LUCAS-TYPE CONGRUENCES FOR CYCLOTOMIC -COEFFICIENTS Zhi-Wei Sun1 and Daqing Wan2 1Department of Mathematics, Nanjing University Nanjing 210093, People’s Republic of China zwsun@ /zwsun 2Department of Mathematics, University of California Irvine, CA 92697-3875, USA dwan@ /∼dwan Abstract. Let p be any prime and a be a positive integer. For l, n ∈ {0, 1, . . . } and r ∈ Z, the normalized cyclotomic -coefficient n − n−pa−1 −lpa k−r := p pa−1 (p −1) (−1)k n pa r l,pa a k l k≡r (mod p ) is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congru- ences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true. 1. Introduction As usual, the binomial coefficient x is regarded as 1. For k ∈ Z+ = 0 {1, 2, . . . }, we define x x(x − 1) · · · (x − k + 1) = k k! and adopt the convention x = 0. −k 2000 Mathematics Subject Classification. Primary 11B65; Secondary 05A10, 11A07, 11R18, 11R23, 11S05. The first author is supported by the National Science Fund fo