Decompositions of differential equations from medical imaging

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.