2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165059We introduce signed q-analogs of Tornheim's double series, and evaluate them in terms of double q-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series, and correct some mistakes in the literature.12 pages, AMSLaTeX. The multinomial notation introduced on page 3 just before Theorem 1 is insufficiently general in version 1, since it may happen that the upper number is negative. This is corrected in version 2, which allows for a negative or even complex upper argumentNumber TheoryClassical Analysis and ODEs11M41; 05A30; 30B50text