Bell Polynomials for Beltrami Operators

The Bell polynomials, which known in name of Eric Temple Bell, are widely used in combinatorial mathematics to explore set partitions. The Bell polynomials have a relation to Bell numbers and Stirling numbers. They can also be found in a various applications, like the Faà di Bruno formula. In this paper, we define generalized Bell polynomials, and investigate basic properties of these polynomials. We find Bell polynomials for the Beltrami operator. Also, we obtain explicit formulas for the powers of the Beltrami operator and a generalized Beltrami operator. We introduce an application of the obtained Bell polynomial for the Beltrami operator, namely, a Beltrami- Faà di Bruno formula is established. Illustrative examples are given.