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MSRI-UP 2014 Research Topic

MSRI-UP 2014: Arithmetic Aspects of Elementary Functions

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The question of evaluation of finite sums with entries in a reasonable large class (of hypergeometric type) has been settled by the algorithms developed by H. Wilf, D. Zeilberger and collaborators.  On the other hand, arithmetic properties of these sums offer interesting challenges. For instance, it is an elementary result that the central binomial coefficient is always even. This motivates the natural question: what is the exact power of \( 2 \) that divides these coefficients?  Is there a closed-form formula for this?

The fact that binomial coefficients satisfy certain recurrences, for example in the formation of Pascal's triangle, has been used to analyze their arithmetic properties. What can be said about sequences generated by similar recurrences? For example, factorials \( n! \) satisfy \( x_{n} = n\, x_{n-1} \). Is it possible to describe arithmetic properties for \( y_{n} = P(n)\,y_{n-1} \) with a polynomial \(P\)? Very few results are known.

Graphical representations offer some indication of the complexity involved. For example, there is a marked difference between the power of two that divides \( n^{2}+1\) and \( n^{2}+7 \). What is the reason behind this? The second graph looks almost random compared to the first. Is there a way to quantify this phenomena?

Some sequences with surprising arithmetical properties include Stirling numbers, Catalan numbers that count legal typing words using parenthesis, the ASM numbers that count the number of matrices with entries from \( \{ 0, \, \pm 1 \} \) satisfying an ordering condition and many other coming from Combinatorics. Recent symbolic experiments include sequences such as the harmonic numbers \(H_{n} = 1 + \tfrac{1}{2} + \cdots + \tfrac{1}{n} \) and the sequence of formed by partial sums of the exponential function.

These type of problems are ideal for introduction to undergraduates: they can be explained with a minimal amount of background, data can be obtained by using symbolic languages and partial results are available in the literature. Thus, this REU is accessible to students who have had three semesters of calculus, linear algebra, and a course in which they have had to write proofs.