Summary: Let \(\{Q^\lambda_n(x)\}\) be the sequence of monic orthogonal polynomials with respect to the inner product \[ \langle f, g\rangle= \int_{\mathbb{R}} f(x)g(x) d\psi(x)+ \lambda\Delta f(c) \Delta g(c), \] where \(c\in \mathbb{R}\), \(\psi\) is a distribution function with infinite spectrum, \(\lambda\geq 0\), \(f\) and \(g\) are real functions and \(\Delta f(c)= {f(c+ h)- f(c)\over h}\) for some \(h> 0\).
For \(Q^\lambda_n(x)\) we derive an explicit representation and a five-term recurrence relation. Moreover, an analogue of the Christoffel-Darboux formula is given and some results on the distribution of the zeros are presented.