H. W. GOULD,J. QUAINTANCE.二项式展开的抵消系数以及$q$-摸拟[J].数学研究及应用,2010,30(2):191~204

Annihilation Coefficients, Binomial Expansions and $q$-Analogs

DOI：10.3770/j.issn:1000-341X.2010.02.001

 作者 单位 H. W. GOULD Department of Mathematics, West Virginia University, Morgantown, W. Va. 26506, U. S. A. J. QUAINTANCE Department of Mathematics, West Virginia University, Morgantown, W. Va. 26506, U. S. A.

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).

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).