Orthogonal polynomials, bi-confluent Heun equations and semi-classical weights

In this paper, we focus on four weights omega(z, s) = z(lambda)e(-N(z+s(z2-z))), where z is an element of (0, infinity), lambda > -1, 0 <= s <= 1, N > 0; omega(z, t) = z(lambda)e(-z2+tz), where z is an element of (0, infinity), lambda > -1, t is an element of R; omega(z, t(1)) = e(-z2) (A + B theta(z - t(1))), with z is an element of R, A >= 0, A + B >= 0, B not equal 0, where theta(z) is the Heaviside step function; and omega(z) = vertical bar z vertical bar(alpha)e(-N(z2+s(z4-z2))), with z is an element of R, alpha > -1, N > 0, 0 <= s <= 1. The second-order differential equations satisfied by P-n(z), the degree-n polynomials orthogonal with respect to each of these weights, are shown to be asymptotically equivalent to the bi-confluent Heun equations as n -> infinity. In most cases, a parameter other than n must simultaneously be sent to a limiting value.