Dedicated to the memory of Germund Dahlquist the author describes seminal work of G. Dahlquist [Math. Scand. 4, 33–53 (1956; Zbl 0071.11803) and Proc. Symp. Appl. Math. 15, 147–158 (1963; Zbl 0123.32405)] in which he introduced criteria for convergence and stability of difference approximations to solutions of ordinary differential equations. This opened a new field of research.
The first paper considered a \(p\)-step approximation (i) \(\sum^p_{j=0}\alpha_jy_{j+n} =h\sum^p_{j=0}\beta_jf_{j+n}\), to an ordinary differential equation \(y' =f(t,y)\), \(t,y\) in \(\mathbb R\), associated polynomials (ii) \(\rho(\zeta)= \sum^p_{j=0}\alpha_j \zeta^j\), \(\sigma(\zeta)=\sum^p_{j=0}\beta_j \zeta^j\), and (iii) \(R(z)\), \(S(z)\) obtained from (ii) by the transformation \(\zeta=z+1/z-1\). The solution of (i) is called stable if the roots of \(R(z)= 0\) are in the left half plane or with multiplicity one on the imaginary axis. It was proved that for convergence of a stable system of order \(k\) that the degree \(p\) of (ii) could not exceed \(p=k+2\). If \(R(z)\) is an odd function then \(p=k+2\) with all roots on \(\text{Im}\,z=0\). If \(k\) is odd then \(p=k+1\). If \(k\) is even then \(p=k+2\) when \(\alpha_v=\alpha_{k-v}\), \(\beta_v=\beta_{k-v}\) and all roots of \(R(z)\) are on \(\text{Im}\, z=0\).
Interest in stiff systems of differential equations led to the introduction of \(A\)-stability in the second paper. The system (i) is \(A\)-stable if all solutions of (i) applied to \(y'=qy\), \(\text{Re}\,q<0\), tend to zero \(n\to\infty\). This led to the result that the order \(p\) of an \(A\)-stable method (i) could not exceed \(p=2\)., The smallest error constant occurs with the trapezoid rule. Later Dahlquist introduced \(G\)-stability which is discussed in a companion paper by J. C. Butcher [BIT 46, No. 3, 479–489 (2006; Zbl 1105.65085)]. Following the introduction of \(A\)-stability the idea of analysing stability of various discrete systems was successfully followed by many researchers, introducing appropriate polynomials. The author sketches many important conjectures and theorems in this direction. The paper includes warm reminiscences of this distinguished scientist by the author.