Heat polynomial analogs for higher order evolution equations (Q2718937)

Polynomial solutions to the higher order linear evolution equation NEWLINE\[NEWLINE\frac{\partial }{\partial t}u(x,t)=\sum_{| \alpha| \leq l}a_{\alpha}(t)\frac{\partial^{| \alpha| }}{\partial x^{\alpha}}u(x,t),\tag{*}NEWLINE\]NEWLINE where \(a_{\alpha}(t)\) are continuous functions on an interval containing the original are considered (for classical heat polynomials see \textit{P.C.Rosenbloom} and \textit{D.V.Widder} [Trans. Am. Math. Soc. 92, 220-266; (1959, Zbl 0086.27203)]. The explicit formula for polynomial in \(x\) solution \(p_\beta(x,t)\) to the initial value problem for the equation (*) with initial condition \(p_\beta(x,0)=x^\beta\) is given. The differentiation and recursion formulas for \(p_\beta(x,t)\) are obtained. In case of parabolic equation (*) having constant coefficients and involving space derivatives only of the highest order \(l\) the heat polynomials \(p_\beta(x,t)\) and associated functions \(q_\beta(x,t)\) are investigated.