Generalized binomial transform applied to the divergent series

TL;DR

This paper explores the generalized binomial transform's ability to extract meaningful information from divergent series related to Laplace integrals and anharmonic oscillators, enabling calculations at strong coupling.

Contribution

It demonstrates the effectiveness of the generalized binomial transform in evaluating functions and energies from divergent series, extending the delta expansion method.

Findings

Approximate computation of functions at zero argument from series at infinity.

Calculation of strong coupling limit of ground state energy.

Comparison showing the generalized binomial transform's advantages over linear delta expansion.

Abstract

The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of $\delta$-expansion proposed recently. We show that, by the use of the generalized binomial transform, the values of functions in the limit of zero of an argument is approximately computable from the series expansion around the infinity of the same argument. In the Laplace integral, we investigate the subject in detail with the aid of Mellin transform. In the anharmonic oscillator, we compute the strong coupling limit of the ground state energy and also the expansion coefficients at strong coupling from the weak coupling perturbation series. The obtained result is compared with that of the linear delta expansion.