  The duality relation of one-variable multiple polylogarithms was proved by
Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper,
we give a new proof using the method of connected sums, which was recently
invented by Seki and the author. Interestingly, the connected sum involves the
hypergeometric function in its connector. Moreover, we introduce two kinds of
$q$-analogues of the one-variable multiple polylogarithms and generalize the
duality to them.
