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

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