 Thomé obtained partial results, but the final word on the region in which this continued fraction converges was not given until 1901, by Edward Burr Van Vleck.Let formula_1 be a sequence of analytic functions so that
for all formula_3, where each formula_4 is a constant.Then

Setting formula_6
So

Repeating this ad infinitum produces the continued fraction expression

In Gauss's continued fraction, the functions formula_10 are hypergeometric functions of the form formula_11, formula_12, and formula_13, and the equations formula_14 arise as identities between functions where the parameters differ by integer amounts.