  Polycosecant numbers and polycotangent numbers are introduced as level two
analogues of poly-Bernoulli numbers. It is shown that polycosecant numbers and
polycotangent numbers satisfy many formulas similar to those of poly-Bernoulli
numbers. However, there is much unknown about polycotangent numbers. For
example, the zeta function interpolating them at non-positive integers has not
yet been constructed. In this paper, we show some algebraic properties of
polycosecant numbers and polycotangent numbers. Also, we generalize duality
formulas for polycosecant numbers which essentially include those for
polycotangent numbers.
