Toeplitz arrays, linear sequence transformations and orthogonal polynomials

Summary

Lets _n be a given sequence,$U = \left[ {\mu _{i,j} } \right]_{i,j = 0}^\infty$ an infinite array of complex numbers$\mu _{i,j} = 0,i< j,\sum\limits_{k = 0}^i {\mu _{i,k} = 1}$ and$s_{n,m} = \sum\limits_{k = 0}^m {\mu _{m,k} s_{n + k} }$. We develop and discuss linear sequence transformations defined by

$$s_{n,m + 1} = a_m s_{n + 1,m} + b_m s_{n,m} $$

or

$$s_{n,m + 1} = a_m s_{n,m} + b_m s_{n + 1,m} + c_m s_{n,m - 1} ,$$

n, m=0, 1, 2,... We ask, ifs _n →α asn→∞, doess _n,m →α asm→∞? This convergence and the rapidity of it are seen to depend on the location of the zeros of the polynomial$P_m \left( \lambda \right) = \sum\limits_{k = 0}^m {\mu _{m,k} \lambda ^k }$. When theP _m are chosen to be certain polynomials encountered in special function theory, such as Bessel polynomials or Legendre polynomials, the result is an elegant and powerful set of algorithms for improving the convergence of large classes of sequences.