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Estimating the Sequence of Real Binomial Coefficients  
 
  Authors: Vito Lampret,  
  Keywords: Approximation, Binomial, Coefficient, Convergence, Estimate, Error, DeMoivre-Laplace, Euler-Maclaurin, Remainder, Sequence.  
  Date Received: 02/03/06  
  Date Accepted: 17/03/06  
  Subject Codes:

05A10, 26D07, 26D15, 33F05, 40A05, 40A25

  
  Editors: Jozsef Sandor,  
 
  Abstract:

The sequence $ n \mapsto \binom{a}{n}$ of real binomial coefficients is studied in two main cases: $ a\gg n$ and $ n\gg a$. In the first case a uniform approximation with high accuracy is obtained, in contrast to DeMoivre-Laplace approximation, which has essentially local character and is good only for $ n \approx \frac{a}{2}$. In the second case, for every $ a\in\mathbb{R} \;$$ \setminus$$ \left( \mathbb{N}\cup\{-1,0\} \right)$, the functions $ A(a,m)$ and $ B(a,m)$ are determined, such that $ \underset{m\rightarrow\infty} {\lim}\frac{A(a,m)}{B(a,m)}=1$, and

$\displaystyle A(a,m)\cdot\left(n-a\right)^{-(a+1)}<\left\vert\binom{a}{n}\right\vert < B(a,m)\cdot\left(n-a\right)^{-(a+1)}, $

for integers $ m$ and $ n$, obeying $ n> m > \vert a\vert$.;


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