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Exponential sums with Catalan numbers and middle binomial coefficients. (English) Zbl 1147.11044

The authors estimate some exponential sums involving products of two Catalan numbers and two middle binomial coefficients – it may be useful to recall that the \(n\)th Catalan number \(c_n\) is \(\frac{1}{n+1}\binom{2n}{n}\), so these are closely related. The bounds are derived through estimates for the number of solutions to certain congruences modulo a prime.
The paper seems to be largely about methods as there are many results. However, one which stands out as among the neatest to state is the following. Suppose that \(p\) is an odd prime and write
\[ N_{p}:=\{n+n'p: n,n' \in \{1,\dots,(p-1)/2\}\} \quad\text{and}\quad T(a):=\sum_{n \in N_p}{\exp(2\pi i a c_n/p)}. \]
Then trivially \(| T(a)| = O(p^2)\) and the authors manage to get a power saving away from the trivial modes: \(| T(a)| = O(p^{2-1/24})\) whenever \((a,p)=1\).

MSC:

11L07 Estimates on exponential sums
11B65 Binomial coefficients; factorials; \(q\)-identities
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References:

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