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3 questions
-2
votes
1
answer
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Two-variable continuous function which results in an integer if and only if arguments are integer
I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...
8
votes
0
answers
237
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Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial
This is mostly a reference request.
I'm working with complex coefficients, although all I have in mind have integer coefficients.
Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
16
votes
2
answers
1k
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are these polynomials or rationals functions?
Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by
$$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$
...