On truncation error bounds of branched continued fraction expansions of some ratios of Lauricella--Saran's hypergeometric functions $F_K$
Abstract
The paper considers the problem of approximating
Lauricella--Saran's hypergeometric func\-tions $F_K$ in special
cases by bran\-ched continued fractions as a special family of
functions. Under the certain conditions on the elements of
bran\-ched continued fraction expansions of some ratios of these
functions, it is proven that every expansion converges to the
function that is analytic in the domain of analytic continuation
\[\mathfrak{D}_\eta=\{\mathbf{z}\in\mathbb{R}^3\colon z_1\le\eta,\;z_2\le\eta,\;z_3\le0\},\quad0<\eta<1,\]
at least as fast as a geometric series with a ratio less then
unity. For this purpose, the method based on the formula for the
difference of two approximants of a branched continued fraction
and the PF method (based on the so-called property of fork for a
branched continued fraction with positive elements) was used.
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