On polynomials orthogonal with respect to an inner product involving differences

In this paper we consider the inner product <f,g>=∫Rƒ(x)g(x)dψ(x)+⋋Δƒ(c)Δg(c) where c ∈ R and ψ is a distribution function with infinite spectrum such that ψ has no points of increase in the interval (c,c + 1). Furthermore λ ⩾ 0, ƒ and g are functions on R and Δƒ(c) = ƒ(c + 1) − ƒ(c). Let {Qnλ(x)} be the sequence of monic orthogonal polynomials with respect to this inner product and {Pn(x)}, {Pnc(x)} the sequences of monic standard orthogonal polynomials (λ = 0) with respect to dψ(x) and (x − c)(x − c − 1)dψ(x), respectively. ive an explicit representation for Qnλ(x) in terms of Pnλ(x) and Pnc(x) and we present some results on the distribution of the zeros of Qnλ(x) in relation to the zeros of Pn(x). Finally, we treat the special case where Pn(x) are Charlier polynomials and c = 0.