Polynomial expansions via embedded Pascal’s triangles

Abstract

An expansion is given for polynomials of the form (ω + λ₁₎ · · ·(ω +λ_n). The coefficients of the resulting polynomials are related to their roots, and a system of equations that enables one to numerically
determine the roots in terms of the coefficients is specified. The case where all the roots λ_i are equal is considered as well. A multinomial extension to polynomials of the form (x₁+ · · · + x_I )ⁿ is then provided. As it turns out, the coefficients of the monomials contained in the resulting polynomial expansion can be determined in terms of the coefficients of the monomials included in the expansion of (x₁+ · · · + x_I-1 )ⁿ  and the rows of embedded Pascal’s triangles of successive orders. An algorithm is provided for generating and concatenating these rows, with the particulars of its implementation by means of the symbolic computation software Mathematica being discussed as well. Potential applications of such expansions to combinatorics and genomics are also suggested.