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二项式展开的抵消系数以及$q$-摸拟
作者姓名:H. W. GOULD  J. QUAINTANCE
作者单位:Department of Mathematics, West Virginia University, Morgantown, W. Va. 26506, U. S. A.;Department of Mathematics, West Virginia University, Morgantown, W. Va. 26506, U. S. A.
摘    要:Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A(n,k;A) are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n k=0 A(n,k;A)(x - Ak)n-k = xn. (0.1) Similary, we define B(n,k;A) to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n k=0 B(n,k;A)(x - Ak)k = xn. (0.2) The main result of this paper is an explicit formula for B(n,k;A), which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations (0.1) and (0.2).

关 键 词:类似物  二项展开式  NK细胞  任意序列  自然数  二项式  乘积  方程
收稿时间:2/6/2009 12:00:00 AM
修稿时间:7/6/2009 12:00:00 AM

Annihilation Coefficients, Binomial Expansions and $q$-Analogs
H. W. GOULD,J. QUAINTANCE.Annihilation Coefficients, Binomial Expansions and $q$-Analogs[J].Journal of Mathematical Research and Exposition,2010,30(2):191-204.
Authors:H W GOULD and J QUAINTANCE
Institution:Department of Mathematics,West Virginia University,Morgantown,W.Va.26506,U.S.A.
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).
Keywords:Annihilation coefficient  Binomial expansion  stirling number of the first kind  stirling number of the second kind  vadermonde convolution  
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