On recurrence coefficients for rapidly decreasing exponential weights

Let, for example,Wx=exp-expk1-x2-α,x∈-1,1,where α>0α>0, k⩾1k⩾1, and expk=exp(exp(…exp()))expk=expexp…exp denotes the k  th iterated exponential. Let {An}An denote the recurrence coefficients in the recurrence relationxpn(x)=Anpn+1(x)+An-1pn-1(x)xpnx=Anpn+1x+An-1pn-1xfor the orthonormal polynomials {pn}pn associated with W2W2. We prove that as n→∞n→∞,12-An=14logkn-1/α1+o1,where logk=log(log(…log()))logk=loglog…log denotes the k  th iterated logarithm. This illustrates the relationship between the rate of convergence to 12 of the recurrence coefficients, and the rate of decay of the exponential weight at ±1±1. More general non-even exponential weights on a non-symmetric interval (a,b)a,b are also considered.