Tuilier, André. "La Didachè Et Le Problème Synoptique." in Jefford, Clayton (editor). The Didache in Context: essays on its text, history, and transmission. Leiden: Brill, 1995, 110-130.
Tuilier considers the Didache to be a possible source for some harmonization of the synoptic problem, as it appears to use material common to the Synoptists but not to John (Tuilier 1995, 110). He proceeds to detail synoptic references from Didache 1.2, 1.3, 1.4, 1.5a, 1.5b, 3.7, 6.2, 7.1 & 3, 8.2, 9.5, 10.5, 10.6, 11.7, 13.1, 16.1a, 16.1b, 16.3, 16.4a, 16.4b, 16.5a, and 16.6, 8 (Tuilier 1995, 111-117). Though the ideas are present, he observes that there is not identical word usage. Therefore, he does not consider the Didache to how a clear literary dependence. Tuilier concludes then that elements of the Didache, Matthew, and Luke depend on a Q source which the Didachist identifies as "the Gospel of the Lord" (Tuilier 1995, 118). Based on its content, this would seem to be a collection of the sayings of Jesus.
Tuilier further considers Luke to be a more literary derivative of Matthew, possibly, however, created with consultation of this Q source, which served as a primary source of the sayings of Jesus (Tuilier 1995, 120-121). The Didache does not appear to take notice of Mark, but does possibly recognize Matthew and Luke. Likewise, Mark appears to have a heavier influence from oral catechesis rather than a strong literary connection to other sources (Tuilier 1995, 121).
In subsequent studies, Tuilier notes that some authors considered The Gospel of the Hebrews or the Ebionite Gospel to represent authentic sources used by Matthew. Yet it is unreasonable to consider these as documents which were used in creation of the Synoptic Gospels, though they may have accurately conveyed some of the Jesus traditions from the early part of the 1st century (Tuilier 1995, 123).
Tuilier turns his attention to later works. Theodoret considered that Matthew took an Aramaic text which he may have used as a foundation for his Greek Gospel, and which may have been at least related to the Gospel of the Hebrews (Tuilier 1995, 125). Tuilier considers that it may be sensible to take this Aramaic text as the "Gospel of the Lord" referred to by the Didachist.