Heat polynomial analogues for equations with higher order time derivatives (Q596756)

The authors present analogues of heat polynomials for an equation involving a higher order time derivative \[ \frac{\partial^\ell u}{\partial t^\ell}=\sum_\alpha a_\alpha\partial^\alpha_x u(\alpha,t),\tag{1} \] where \(\ell\) may be any positive integer. Cauchy data for this equation involves \(\ell\) initial conditions \[ \frac{\partial^ku(x,0)}{\partial t^k}=f_k(x),\quad 0\leq k<\ell,\tag{2} \] and correspondingly the authors require \(k\) families of polynomials \(\{P_{\beta,k}(x,t)\}\), \(0\leq k<\ell\). The \(k\)th family of polynomial solutions of (3) solves the Cauchy conditions \[ \frac{\partial^j}{\partial t^j}\,P_{\beta,k}(x,0)=\partial_{j,k}x^\beta=\begin{cases} x^\beta,\;& \text{if }j=k,\\ 0,\;& \text{if }j\neq k\text{ and }0\leq j<\ell.\end{cases}. \]