Consider the following series function of the two variables $x$ and $y$ expressed in terms of the hypergeometric function: $$ f(x,y) = \sum_{n=0}^\infty \frac{x^{2n+1}}{n!^2} {}_2F_1 \left( -n,-\frac{1}{2} - n; 2; y \right) , $$ wherein $0 < x,y < A$. Since a series expansion of the hypergeometric function as $m\to\infty$ does not seem possible, I was wondering what would be the best approach to follow so as to compute $f(x,y)$ numerically by truncating the infinite series at some very large number $N$ beyond which the error is less than a certain $\epsilon$. Any hint is highly appreciated. Thanks you!
2026-03-30 01:52:08.1774835528
Truncation error in evaluating $\sum_{n=0}^\infty \frac{x^{2n+1}}{n!^2} {}_2F_1 \left( -n,-\frac{1}{2} - n; 2; y \right)$
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