Sequence transformations accomplish an acceleration of convergence or asummation in the case of divergence by detecting and utilizing regularities ofthe elements of the sequence to be transformed. For sufficiently large indices,certain asymptotic regularities normally do exist, but the leading elements ofa sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1(a, b; c; z) is well suited to illuminate problems of that kind. Sequencetransformations perform quite well for most parameters and arguments. If,however, the third parameter $c$ of a nonterminating hypergeometric series 2F1is a negative real number, the terms initially grow in magnitude like the termsof a mildly divergent series. The use of the leading terms of such a series asinput data leads to unreliable and even completely nonsensical results. Incontrast, sequence transformations produce good results if the leadingirregular terms are excluded from the transformation process. Similar problemsoccur also in perturbation expansions. For example, summation results for theinfinite coupling limit k_3 of the sextic anharmonic oscillator can be improvedconsiderably by excluding the leading terms from the transformation process.Finally, numerous new recurrence formulas for the 2F1 (a, b; c; z) are derived.
