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  最后修改时间: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.
基金项目:
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. 
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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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