Stochastic construction of new orthogonal polynomials

Abstract:

For $u > - 1,t \geqslant 0,x \in R$, let ${(1 + u)^{Q(t) + t}}{e^{ - ut}}$ satisfies $dy(t) = uy(t)\;dQ(t)$ where $Q(t) = P(t) - t$ is the centered Poisson process with parameter $\lambda = 1$, (ii) ${K_n}(t,x),n \geqslant 0$, is a system of complete orthogonal polynomials in ${L^2}(R,dF)$ where $F(x)$ is the distribution function of $P(t)$, (iii) $(n + 1){K_{n + 1}}(t,x) - (x - t - n){K_n}(t,x) + t{K_{n - 1}}(t,x) = 0,n \geqslant 1$, (iv) $t{\Delta ^2}{K_n}(t,x) + (t - x)\Delta {K_n}(t,x) + n{K_n}(t,x) = 0,n \geqslant 0$.