Annihilation Coefficients, Binomial Expansions and $q$-Analogs

Abstract:

      Let $\{A_n\}^\infty_{n=0}$ be an arbitary sequence of natural numbers. We say $A(n,k;A)$ are the Convolution Annihilation Coefficients for $\{A_n\}^\infty_{n=0}$ if and only if $$\sum^n_{k=0}A(n,k;A)(x-A_k)^{n-k}=x^n.\tag 0.1$$ Similary, we define $B(n,k;A)$ to be the Dot Product Annihilation Coefficients for $\{A_n\}^\infty_{n=0}$ if and only if $$\sum^n_{k=0}B(n,k;A)(x-A_k)^k=x^n.\tag 0.2$$ The main result of this paper is an explicit formula for $B(n,k;A)$, which depends on both $k$ and $\{A_n\}^\infty_{n=0}$. This paper also discusses binomial and $q$-analogs of Equations (0.1) and (0.2).