Euler tangent numbers modulo 720 and Genocchi numbers modulo 45. (English) Zbl 1510.11077

Summary: We establish congruences for higher order Euler polynomials modulo 720. We apply this result for constructing analogues of Stern congruences for Euler secant numbers \(E_{4n}\equiv 5\pmod{60}, E_{4n+2}\equiv -1\pmod{60}\) to Euler tangent numbers and Genocchi numbers. We prove that Euler tangent numbers satisfy the following congruences \(E_{4n+1}\equiv 16\pmod{720}\), and \(E_{4n+3}\equiv -272\pmod{720}\). We establish 12-periodic property of Genocchi numbers modulo 45.


11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
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[1] B. C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. · Zbl 0555.10001
[2] L. Carlitz and R. Scoville, Enumeration of up-down permutations by upper records, Monatsh. Math. 79 (1975), 3-12. · Zbl 0315.05004
[3] G. Liu, Congruences for higher-order Euler numbers, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 3, 30-33. · Zbl 1120.11011
[4] N. Nielsen, Traité Elémentaire des Nombres de Bernoulli, Gauthier-Villars, Paris, 1923.
[5] P. Yuan, A conjecture on Euler numbers, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 9, 180-181. · Zbl 1080.11018 · doi:10.3792/pjaa.80.180
[6] W. Zhang, Some identities involving the Euler and the central factorial numbers, Fibonacci Quart. 36 (1998), no. 2, 154-157. · Zbl 0919.11018
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