Let $f(x)=(1-x)^b (1+x)^{(n-b)}= \sum_{i=0}^n a_ix^i$, where $n$ is a positive integer and $b$ is a non-negative integer less than $n$. I want to find an upper bound on $\sum_{i=0}^n |a_i|$ other than the trivial upper bound $2^n$. Also for $b=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach for this problem?
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$\begingroup$ a. Typeset your question. b. the symbol $e$ stands for a frequently used constant, and using it in this context is very confusing. $\endgroup$– Igor RivinCommented Nov 21, 2012 at 17:59
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7$\begingroup$ The coefficient $a_i$ is a value of a Krawtchouk polynomial. A lot is known about their asymptotics. See for instance www6.cityu.edu.hk/rcms/publications/preprint21.pdf and references [11] and [15] therein. $\endgroup$– Richard StanleyCommented Nov 21, 2012 at 18:51
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$\begingroup$ @Stanley: Thank you very much for your kind help. I have tried to understand main results from these papers. But there is no such closed form. I need closed form as a function of $b,n$ so that I can calculate upper bound on $\sum_{i=0}^n |a_i|$. Thank you again for your help. $\endgroup$– user29295Commented Nov 22, 2012 at 1:00
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$\begingroup$ You DO NOT need a closed form to get estimates. See my answer. $\endgroup$– Igor RivinCommented Nov 22, 2012 at 4:04
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2 Answers
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Assume without loss of generality that $b\leq n/2$. Writing $f(x)=(1-x^2)^b(1+x)^{n-2b}$ shows that an upper bound is $2^{n-b}$, but this is very crude.
answered Nov 22, 2012 at 3:44
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$\begingroup$ @Stanley: Thank you very much. For $b\leq n/3$, I have tried similar approach but do not get anything better than $2^{n-b}$. $\endgroup$ Commented Nov 22, 2012 at 16:32
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This is analyzed exhaustively (also exhaustingly) by Domenici in this 2005 preprint.
answered Nov 22, 2012 at 4:03
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$\begingroup$ @Rivin: Thank you very much for the paper. In the paper author tried to estimate different $a_i$. From these estimates $\sum_{i=0}^n|a_i|$ is not easily followed for general $b$. Thank you again for your kind help. $\endgroup$ Commented Nov 22, 2012 at 16:42