| 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 |
| 二项式展开的抵消系数以及$q$-摸拟 |
| Annihilation Coefficients, Binomial Expansions and $q$-Analogs |
| 投稿时间：2009-02-06 修订日期：2009-07-06 |
| DOI：10.3770/j.issn:1000-341X.2010.02.001 |
| 中文关键词: Annihilation coefficient Binomial expansion stirling number of the first kind stirling number of the second kind vadermonde convolution. |
| 英文关键词:Annihilation coefficient Binomial expansion stirling number of the first kind stirling number of the second kind vadermonde convolution. |
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| 中文摘要: |
| 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). |
| 英文摘要: |
| 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). |
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