Integral Transforms and Special Functions Vol. 17, No. 5, May 2006, 329–353 Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials E. H. DOHA*† and H. M. AHMED‡ †Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt ‡Department of Mathematics, Faculty of Industrial Education, Amiria, Cairo, Egypt (Received 21 November 2004) Two formulae expressing explicitly the difference derivatives and the moments of a discrete orthogonal polynomials {P n (x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of a general-order difference derivatives ∇ q f(x), and for the moments x  ∇ q f (x), of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica), in order to build and solve recursively for the connection coefficients between two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed. Keywords: Hahn, Meixner, Kravchuk and Charlier polynomials; Expansion coefficients; Recurrence relations; Linear difference equations; Connection coefficients AMS Classification: 33C25; 65Q05 1. Introduction Models of systems in the physical sciences provide important insight into the working of the nature world. The construction of the associated discrete models, in general, relies on the discrete nature of how the various properties of these systems are measured and analyzed. Thus, the discrete mathematical formulation is often an exact reflection of the actual experimental procedures used to define the system of interest. Many models problems of atomic, molecular and nuclear physics, electrodynamics and acoustics may be reduced to differential equation of hypergeometric type [1, equation (0.1)]. Some solutions of this differential equation are functions extensively used in mathematical *Corresponding author. Email: eiddoha@frcu.eun.eg Integral Transforms and Special Functions ISSN 1065-2469 print/ISSN 1476-8291 online © 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10652460500422270