Estimation of the Remainder Terms of Certain Horn Hypergeometric Series

We construct integral representations and asymptotic estimates for remainder terms arising in the summation of the Appell hypergeometric series $${{F}_{1}}$$ and the related series $${{G}_{2}}$$ , which are included in Horn’s list of hypergeometric series of two variables. The obtained formulas can be used to develop algorithms for calculating $${{F}_{1}}$$ with the help of analytic continuation formulas to the entire space $${{\mathbb{C}}^{2}}$$ . The results can find application in problems of mathematical physics and computational function theory, including in the construction of conformal mappings of complicated polygons based on the Schwarz–Christoffel integral.