Polynomials biorthogonal to dilations of measures, and their asymptotics

We analyze polynomials P n that are biorthogonal to dilates of a positive measure μ , supported on ( 0 , ∞ ) : ∫ 0 ∞ P n ( x ) d μ ( σ n , j x ) = 0 , 1 ≤ j ≤ n . We establish representations for P n in terms of the associated dilation polynomial R n ( y ) = ∏ j = 1 n ( y + 1 / σ n , j ) . In the case where d μ ( t ) = t α e − t β d t on  ( 0 , ∞ ) , we show that strong asymptotics for R n in the complex plane, as n → ∞ , lead to strong asymptotics for P n , via the method of steepest descent.