Every classical discrete orthogonal polynomial family \(p_n(x)\) is a solution of a difference equation of the type \[ \sigma(x) \Delta \nabla p_n(x) +\tau(x) \Delta p_n(x)+ \lambda_n p_n(x)=0 \] where \(\sigma\) and \(\tau\) are polynomials of degree \(\leq 2\) and 1, respectively, and \(\lambda_n\) is constant; it satisfies a Rodrigues type formula \(p_n(x)= {B_n\over \rho(x)} \nabla^n \rho_n (x)\) with \(\rho_n(x)= \rho(x+n) \prod^n_{k=1} \sigma (x+k)\) and \(\rho(x)\) being a solution of the Pearson type difference equation \(\Delta (\sigma (x) \rho(x)) =\tau(x) \rho(x)\); its orthogonality relation reads as \(\sum^{b-1}_{x=a} p_m(x) p_n(x) \rho(x) =\delta_{mn} d^2_n\); see [A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable (1991; Zbl 0743.33001)]. The authors prove the following theorem: Assume that \(r_m(x) q_j(x) =\sum^{m+j}_{n=0} c_{jmn} p_n(x)\), where \(p_n\) is a system as above, and \(c_m,q_j\) are polynomials of degree \(m\) and \(j\), respectively; then \[ c_{jmn} ={(-1)^n B_n\over d^2_n} \sum^{b-1}_{x=a} \rho_n(x-n) \nabla^n \bigl\{r_m (x)q_j (x)\bigr\}; \] a more explicit version is given if \(q_j(x)\) is also a system of classical discrete orthogonal polynomials. Some specific situations are considered in more detail. In particular, the connection coefficients \(c_{mn}\) between any two particular systems of discrete orthogonal polynomials (Charlier, Meixner, Krawtchouk, Hahn) are given in terms of generalized hypergeometric functions.