Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients

Let {p"(x);rz = 0,1,2 } be orthogonal polynomials with recurrence relationWe want to investigate the case where the recurrence coefficients a" and b n are unbounded.We will use the notion of regular variation to specify the behavior of a n and b" as n tends to infinity.DEFINITION. (SENETA [20, p. 46]) A sequence {c" : n = 0,1,2,... } is regularly varying at infinity if there exists a sequence {X" : n = 0,1,2,...} suchthat andThe number a is called the index of regular variation.One can show that a regularly varying sequence {c n : n = 0,1,2 } with index a can be written as c 1} = n°L(r?)where L is a positive and measurable function on [0, oc) such that, for every y > 0, hm -y^ -1 .