Let \(X=\{X(t), t\geq 0\}\), \(X(0)=0\), be a time homogeneous process with independent increments, \(\varphi^X(z) =E\exp (izX(1))\), \(\Psi^X_t (dx)=P \{X(t)\in dx\}\). Assuming that \(\varphi^X(z)\) is analytic in the neighbourhood of \(z=0\), there are found all processes \(X\) and analytic in the neighbourhood of \(z=0\) functions \(u(z)\), \(u(0)=0\), \(u'(0)\neq 0\), such that a polynomial set \(\{Q_m (x,t),\;m\geq 0,\;t\geq 0\}\), defined by the generating function \[ \exp \bigl(xu(z)\bigr) \biggl(\varphi^X \bigl(-iu(z)\bigr) \biggr)^{-t}= \sum^\infty_{m=0} Q_m(x,t) {z^m \over m!}, \] for each fixed \(t\geq 0\) is orthogonal in \(L^2(R^1,\Psi_t^X)\). These processes are expressed in terms of Wiener, Poisson, Gamma, Pascal and Meixner processes. Because for each \(z\) in the neighbourhood of \(0\), \(\{\exp (X(t)u(z))\) \((\varphi^X(-iu(z))^{-t},\;t\geq 0\}\), obviously, are martingales, the polynomials \(\{Q_m (X(t),t),\;t\geq 0\}\), \(m\geq 0\), enjoy the same property.