Kummer congruence for the Bernoulli numbers of higher order

Abstract

The authors studied the properties of Bernoulli numbers of higher order [Appl. Math. Comput., in press; Bull. Aust. Math. 65 (2002) 59]. For q=1, we can also find their results [Proc. Jangjeon Math. Soc. 1 (2000) 97; Arch. Math. 76 (2001) 190; Proc. Jangjeon Math. Soc. 1 (2000) 161; Adv. Stud. Contemp. Math. 2 (2000) 9; Proc. Jangjeon Math. Soc. 2 (2001) 23; J. Math. Phys. A 34 (2001) L643; Proc. Jangjeon Math. Soc. 2 (2001) 19; Proc. Jangjeon Math. Soc. 2 (2001) 9; Proc. Jangjeon Math. Soc. 3 (2001) 63]. The authors suggested the question to inquire the proof of Kummer congruence for Bernoulli numbers of higher order [Appl. Math. Comput., in press]. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the question in [Appl. Math. Comput., in press].