Congruences for sequences analogous to Euler numbers

2013/07/28 by Sun, Zhi-Hong, Wang, Hai-Yan
#11A07 #11B68 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1307.7370

Abstract

For a given real number a we define the sequence \E_n,a\ by E_0,a=1 and E_n,a=-a∑_k=1^[n/2] \binom n2kE_n-2k,a (n≥ 1), where [x] is the greatest integer not exceeding x. Since E_n,1=E_n is the n-th Euler number, E_n,a can be viewed as a natural generalization of Euler numbers. In this paper we deduce some identities and an inversion formula involving \E_n,a\, and establish congruences for E_2n,a\mod2^\rm ord₂n+8, E_2n,a\pmod3^\rm ord₃n+5 and E_2n,a\pmod5^\rm ord₅n+4 provided that a is a nonzero integer, where \rm ord_pn is the least nonnegative integer α such that p^\a| n but p^\a+1\nmid n.

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