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In this paper, by considering higher-order degenerate Bernoulli and Euler polynomials which were introduced by Carlitz, we investigate some properties of mixed-type of those polynomials. In particular, we give some identities of mixed-type degenerate special polynomials which are derived from the fermionic integrals on Zp and the bosonic integrals on Zp.

term. By contrast, there is a surface term for the bosonic variables [ p [ T,j. (20.18b)-dxd8= fn...1(x) Jvc>x Jdu The ^-integral is analogous to a bosonic integral always extended from —oo to +00 for functions that vanish fast enough at infinity. There is no concept of boundary of the ^-integration range. 486 Chapter Twenty The purpose of this section is to investigate how (20.17) transforms under changes of variables and to show that there is a natural measure in any superphase space, the "Liouville measure." 20.2.2. Supertrace—Superdeterminant A matrix M=(£ BD