  The aim of this paper is to introduce truncated degenerate Bell polynomials
and numbers and to investigate some of their properties. In more detail, we
obtain explicit expressions, identities involving other special polynomials,
integral representations, Dobinski-like formula and expressions of the
generating function in terms of differential operators and linear incomplete
gamma function. In addition, we introduce truncated degenerate modified Bell
polynomials and numbers and get similar results for those polynomials. As an
application of our results, we show that the truncated degenerate Bell numbers
can be expressed as a finite sum involving moments of a beta random variable
with certain parameters.
