Order and stability of generalized Padé approximations

Given a sequence of integers [n0,n1,…,nr], where n0,nr⩾0 and ni⩾−1,i=1,2,…,r−1, a sequence of r polynomials (P0,P1,…,Pr) is a generalized Padé approximation to the exponential function if , where the order of the approximation p is given by . The main result of this paper is that if 2n0>p+2, then is not the stability polynomial of an A-stable numerical method. This result, known as the Butcher–Chipman conjecture, generalizes the corresponding result for rational Padé approximations. The special case, formerly known as the Ehle conjecture [B.L. Ehle, A-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973) 671–680], was subsequently proved by Hairer, Nørsett and Wanner [G. Wanner, E. Hairer, S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475–489].