Annihilation Coefficients, Binomial Expansions and $q$-Analogs
Received:February 06, 2009  Revised:July 06, 2009
Key Word: Annihilation coefficient   Binomial expansion   stirling number of the first kind   stirling number of the second kind   vadermonde convolution.
Fund ProjectL:
 Author Name Affiliation 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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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).