Congressus Numerantium (in review).
Ömer Egecioglu and Charles Ryavec
Polynomial Families Satisfying a Riemann Hypothesis
Abstract.
Consider a linear transformation $T : R[x] \rightarrow R [x]$
defined on
basis elements $ 1, x, x^2, \ldots $ by
\[
T [x^k ] = \frac {(x)_k}{k!}
\]
where
$(x)_k = x(x+1)(x+2)\cdots (x+k-1),~ k\ge 0$.
We create infinite families of polynomials of the form
$T [p_n(x)]$, each member of which
satisfies a Riemann hypothesis;
i.e., their zeros lie on the line
$ [s = \half + it: ~ t \, {\mbox{ real}}]. $
These families are indexed by a real parameter $r$, and are of the form
$p_n(x)= (x+r)^n + (1-x+r)^n$ for $ n\geq 2 $.
Our proof uses a positivity argument together with
certain elements of the theory of
3-term polynomial
recursions.
omer@cs.ucsb.edu