New recurrence relationships between orthogonal polynomials which lead to new lanczos-type algorithms

Lanczos methods for solving \(Ax = b\) consist in constructing a sequence of vectors \((x_k)\), \(k = 1, …\) such that \(r_k = b − Ax_k = P_k(A)r_0\), where \(P_k\) is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(ξ^i) = (y, A^ir_0)\). Let \(P^(1)_k\) be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to \(c^(1)\) defined as c^(1)(ξ ^{i}) = c^{(ξi+1)}\). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for \(P_k\) and one for \(P^{(1)}_k\). We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all.