The 𝑑₂-transformation for infinite double series and the 𝐷₂-transformation for infinite double integrals

Abstract

New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate d d - and D D -transformations. The D 2 D_2 -transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the d 2 d_2 -transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given.