Transformation formulas for complete symmetric polynomials and identities for generalized binomial coefficients and -Stirling numbers

We obtain explicit formulas that express the complete homogeneous symmetric polynomials of the sequence of partial sums s k of a sequence x k as polynomials in the x k , and conversely, the complete homogeneous symmetric polynomials of the x k as polynomials in the s k . Taking particular sequences x k , we obtain combinatorial identities for several types of generalized binomial coefficients, including Gaussian coefficients, Stirling numbers of the second kind, q -Stirling numbers of the second kind, and Legendre–Stirling numbers. We also introduce some new families of generalized binomial coefficients and Stirling numbers.