Poly-Cauchy and poly-Bernoulli numbers with level 2 associated with higher level Stirling numbers

Poly-Cauchy and poly-Bernoulli numbers with level 2 may be introduced from the aspect of higher level Stirling numbers. In this lecture, first we show several expressions, relations, and properties about poly-Cauchy numbers with level $2$. Poly-Cauchy numbers with level $2$ can be expressed in terms of multinomial coefficients with combinatorial summation, Stirling numbers of the first kind, or iterated integrals. We also give some recurrence relations for poly-Cauchy numbers with level 2. When the index is negative, the double summation may be formulated as a closed form.

As closely related numbers with Cauchy numbers, we may introduce poly-Bernoulli numbers with level 2 and give analogous properties in the respect of the classical Bernoulli numbers.