Asymptotic behavior of solutions to polynomial renewal equations (Q1086717)

Many special functions - the examples discussed include the Bessel functions and the incomplete \(\gamma\)-function, arise as the ''renormalized'' limits of sequences satisfying a polynomial renewal equation \[ a_{n+1}(x)=(\lambda +\mu +1)a_ n(x)+\sum^{n- 1}_{k=0}a_ k(x)b_{n-k}x^{n-k}\quad n\geq 1, \] with given scalars \(a_ 0\), \(a_ 1:a_ n(x)=a_ n(x;\lambda;\mu)\), \(a_ 0(x)=a_ 0\), \(a_ 1(x)=a_ 1\) and positive parameters \(\lambda\),\(\mu\). The author determines the asymptotic behavior of these sequences of polynomials for both ordinary and coefficientwise convergence and illustrates it with explicit examples.