The exact analytical expressions for any Lanczos couplings {α n, βn }

This chapter deals with the exact analytical expressions for any Lanczos couplings {α_n, β_n}. The expansion coefficients q_n,n^-r are given in terms of the Lanczos coupling constants {α_n, β_n}, whereas the parameters {a_n} of the continued fractions determine the coefficients q^~n,n^-r. Recursive numerical computations of the Hankel determinants H_m(c₀) and H_m(c₁) can be carried out by Gordon’s PD algorithm which is accurate, efficient and robust with only ~ n² multiplications relative to some formidable n! multiplications in the Cramer rule via direct evaluations. Moreover, the same PD recursion can also be used for the CF coefficients {a_n}.