Abstract
The aim of this paper is to study the complete and incomplete degenerate Bell polynomials, which are degenerate versions of the complete and incomplete Bell polynomials, and to derive some properties and identities for those polynomials. In particular, we introduce some new polynomials associated with the incomplete degenerate Bell polynomials. In fact, they are the coefficients of the reciprocal of the power series given by 1 plus the one appearing as the exponent of the generating function of the complete degenerate Bell polynomials.
Highlights
In recent years, we have seen that studying degenerate versions of many special polynomials and numbers, which was initiated in [2] by Carlitz, yielded very fruitful and interesting results
We consider the problem of finding the reciprocal power series of the invertible formal power series which is equal to 1 plus the power series appearing as the exponent of the generating function of the complete degenerate Bell polynomials in (10)
This leads us to introduce the new polynomials Tn,λ(a1, a2, . . . , an) (see (21)) that are associated with the incomplete degenerate Bell polynomials
Summary
We have seen that studying degenerate versions of many special polynomials and numbers, which was initiated in [2] by Carlitz, yielded very fruitful and interesting results. We get a recurrence relation for the degenerate Stirling numbers of the second kind. We consider the problem of finding the reciprocal power series of the invertible formal power series which is equal to 1 plus the power series appearing as the exponent of the generating function of the complete degenerate Bell polynomials in (10). This leads us to introduce the new polynomials Tn,λ(a1, a2, . An) (see (21)) that are associated with the incomplete degenerate Bell polynomials As a corollary, this gives us an expression for the reciprocal of the degenerate exponential function eλ(at). For the rest of this section, we recall the facts that are needed throughout this paper
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