  Studying medical imaging, Peter Kuchment and Sergey Lvin encountered an
countable family of differential identities for sine, cosine, and the
exponential function. Specifically if for a smooth function $u$ and a complex
number $\lambda$ the minimal differential equation $u' = \lambda u$ held, then
$u$ satisfied all of their identities (i). If $u" = \lambda^2 u$, then $u$
satisfied all odd-indexed identities (ii). They were unable to determine (iii)
if there were some other pattern as well. We realize their $n$-th identity as a
polynomial $f_{n,\lambda}(u)$ in the variable $u$ that turns out to have
integer coefficients. We construct combinatorial relations on the coefficients
to provide an alternate proof of one of Kuchment and Lvin's results. We also
isolate the part of $f_{n,\lambda}(u)$ that is linear in the variable $u$ to
answer (iii) negatively, and describe how the analysis of the linear polynomial
may connect to the analysis of the whole polynomial.
