Relations among derangement type numbers of higher order, Bernoulli, Stirling and combinatorial numbers

We study various types of generating functions for higher-order derangement numbers. We present a generating function for a new class of binomial-type polynomials with coefficients given by higher-order derangement numbers. Using this generating function, we derive explicit formulas for these polynomials, as well as derivative and integral formulas involving Stirling numbers and Cauchy numbers of the first kind. By employing this generating function along with other special formulas, we obtain several new results involving higher-order derangement numbers, Bernoulli numbers, Stirling numbers, combinatorial numbers associated with Daehee numbers, Peters-type Simsek numbers, and special finite sums.