Bernoulli numbers with level 2

Stirling numbers with higher level may be considered to have been introduced by Tweedie (Proc Edinb Math Soc 37:2–25, 1918). These numbers have been recently rediscovered and studied more deeply, in particular, from combinatorial aspects. When \(s=2\), by connecting with Stirling numbers with level 2, poly-Bernoulli numbers with level 2 may be naturally introduced as analogous to poly-Benroulli numbers. As a special case, Bernoulli numbers with level 2 are introduced and behave as an analogue of classical Bernoulli numbers. In this paper, we study Bernoulli numbers with level 2. With the help of some numbers introduced by Glaisher as well as Euler and complementary Euler numbers, we show some identities, relations and expressions for Bernoulli numbers with level 2.