Algebraic theory of degenerate general bivariate Appell polynomials and related interpolation hints

Abstract

The algebraic study of polynomials based on determinant representations is important in many fields of mathematics, ranging from algebraic geometry to optimization. The motivation to introduce determinant expressions of special polynomials comes from the fact that they are useful in scientific computing in solving systems of equations effectively. It is critical for this application to have determinant representations not just for single valued polynomials but also for bivariate polynomials. In this article, a family of degenerate general bivariate Appell polynomials is introduced. Several different explicit representations, recurrence relations, and addition theorems are established for this family. With the aid of different recurrence relations, we establish the determinant expressions for the degenerate general bivariate Appell polynomials. We also establish determinant definitions for degenerate general polynomials. Several examples are framed as the applications of this family and their graphical representations are shown. As concluding remarks, we propose a linear interpolation problem for these polynomials and some hints are provided.