On polynomials orthogonal with respect to an inner product involving differences

Abstract

In this paper we consider the inner product <f,g>= ∫ R ƒ(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 { Q n λ ( x)} be the sequence of monic orthogonal polynomials with respect to this inner product and { P n ( x)}, { P n c ( x)} the sequences of monic standard orthogonal polynomials ( λ = 0) with respect to d ψ( x) and ( x − c)( x − c − 1)d ψ( x), respectively. We derive an explicit representation for Q n λ ( x) in terms of P n λ ( x) and P n c ( x) and we present some results on the distribution of the zeros of Q n λ ( x) in relation to the zeros of P n ( x). Finally, we treat the special case where P n ( x) are Charlier polynomials and c = 0.