The complete expansion for Twersky’s formula

Abstract First published in 1961, Twersky’s formula provides an alternative representation for a class of Schlömilch series that is important in wave scattering theory. When Twersky’s formula is used in practical applications, its rate of convergence is often accelerated using Kummer’s transformation. This amounts to expanding the summand in negative powers of the index, subtracting these terms and adding back their exact sum. The first few terms can easily be obtained using a computer algebra system, but they become increasingly complicated as the expansion progresses. In this paper, we derive a general expression for the coefficients in the expansion of the summand. We also show that these coefficients can be calculated using a stable, linear recurrence relation. Using these results it is possible to create simple and very rapidly convergent representations for the Schlömilch series.