Acceleration of convergence of some infinite sequences {An} whose asymptotic expansions involve fractional powers of n via the d~(m)${\tilde {d}}^{(m)}$ transformation

In this paper, we discuss the application of the author’s $\tilde {d}^{(m)}$ transformation to accelerate the convergence of infinite series ${\sum }^{\infty }_{n=1}a_n$ when the terms an have asymptotic expansions that can be expressed in the form $$ a_n\sim(n!)^{s/m}\exp\left[\sum\limits^{m}_{i=0}q_in^{i/m}\right]\sum\limits^\infty_{i=0}w_i n^{\gamma-i/m}\quad\text{as } n\to\infty,\quad s\ \text{integer.} $$ We discuss the implementation of the $\tilde {d}^{(m)}$ transformation via the recursive W-algorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the $\tilde {d}^{(m)}$ transformation can also be used efficiently to accelerate the convergence of infinite products ${\prod }^{\infty }_{n=1}(1+v_n)$ , where $v_n\sim {\sum }^{\infty }_{i=0}e_in^{-t/m-i/m}$ as $n\to \infty $ , t ≥ m + 1 an integer. Finally, we give several numerical examples that attest the high efficiency of the $\tilde {d}^{(m)}$ transformation for the different cases.