Local asymptotics of the ratio of orthogonal polynomials in the neighbourhood of an end-point of the support of the orthogonality measure

A new method of studying the asymptotic behaviour of orthogonal polynomials in the neighbourhood of an end-point of the support of the orthogonality measure is put forward. The asymptotics in the neighbourhood of the end-point is found in terms of the asymptotics of the values of the polynomials at the point itself and the asymptotic behaviour of the coefficients of the recurrence relations. Applications of the result obtained are considered.