This is what I told you while I was still with you: 'Everything must be fulfilled that is written about Me in the Law of Moses, the Prophets, and the Psalms. (Luke 24:44)
Why not just say, "the Tanakh" or "the Word of God."? Or the "Oral Law"? Or a word like the "Talmud." What is fulfilled that is found in all three mentioned by Jesus?
Apparently, the acronym Tanakh (תנ"ך) wasn't in use by the first century.
I added a related question here:
“Law … Prophets … Psalms” expands “Moses and all the Prophets” in v. 27 by adding the Psalms as a major component of the third division of the OT, the so-called Writings.
Liefeld, W. L. (1984). Luke. In F. E. Gaebelein (Ed.), The Expositor’s Bible Commentary: Matthew, Mark, Luke (Vol. 8, p. 1057). Zondervan Publishing House.
that all things must be fulfilled which were written in the law of Moses, and in the Prophets, and in the Psalms—the three current Jewish divisions of the Old Testament Scriptures, concerning me
Brown, D., Fausset, A. R., & Jamieson, R. (n.d.). A Commentary, Critical, Experimental, and Practical, on the Old and New Testaments: Matthew–John: Vol. V (p. 343). William Collins, Sons, & Company, Limited.
[Luke 24:]44–45 The Torah of Moshe, the Prophets (N˒vi˒im) and the Psalms (standing for the Wisdom books or K˒tuvim), in other words, the entire Tanakh (see vv. 25–27&N). Note especially Isaiah 52:13–53:12 and Hosea 6:2.
46 See vv. 25–27&N, 1C 15:3–4&N.
Stern, D. H. (1996). Jewish New Testament Commentary : a companion volume to the Jewish New Testament (electronic ed., Luke 24:44–46). Jewish New Testament Publications.
“Moses … Prophets … Psalms” These represent the three divisions of the Hebrew Canon: Law, Prophets, and Writings. This context says something of the Christocentric unity of the Old Testament.
Utley, R. J. (2004). The Gospel according to Luke: Vol. Volume 3A (Lk 24:44). Bible Lessons International.
"Why not just say, "the Tanakh""
Sefaria Library says:
The Tanakh, or Hebrew Bible, is Judaism’s foundational text. “Tanakh” is an acronym for the three sections of the canon, the Torah (the Five Books of Moses), Nevi’im (Prophets), and Ketuvim (Writings).
"Torah, Nevi'm, and Ketuvim" are abbreviated as TNK. When written as TN"K, the '"' indicates that this is an acronym and should be pronounced with an "a" between the consonants: TaNaK. It's what Christians refer to as The Old Testament or The Hebrew Scriptures.
In Hebrew, this is (read right to left):
תּוֹרָה + נְבִיאִים + כְּתוּבִים ⇐ תָּנָ״ךְ
This acronym wasn't created until the middle ages, as were many other similar acronyms, such as RMB"M ("Rambam", "Rabbi Moisha Ben Maimon"), which is Maimonides in Greek and Professor Moses Maimonson in English, or the modern Orthodox organization Chabad (חב״ד).
In Jesus's time, people generally referred to the names of the three sets of books individually. as in Luke 24:44, or informally used the term Mikra ("that which is read").
The Tanakh - whether the term was widely used at the time or not - includes other works beside the Law, the Prophets and the Psalms. In Jesus' day his audience would have definitely accepted the Law (Torah) and the Prophets as scripture, as well as the Book of Psalms. Other writings were not as universally accepted or as widely read. More important, some of the other writings did not testify so clearly to the point Jesus was making in this particular story.
This saying of Jesus comes during his dialog with the unnamed disciples on the Road to Emaus. The episode includes the following:
Luke 24
21 "But we had hoped that he was the one who was going to redeem Israel."...25 He said to them, “How foolish you are, and how slow to believe all that the prophets have spoken! 26 Did not the Messiah have to suffer these things and then enter his glory?” 27 And beginning with Moses and all the Prophets, he explained to them what was said in all the Scriptures concerning himself.
Jesus helps the disciples to revise their expectations of him (they hoped he would act at the Jewish Messiah to redeem Israel from foreign occupation and restore the throne of David) by showing them that there were passages in the scriptures that predicted his suffering. Later in the chapter, he adds the Psalms to the list. This is probably because the Psalms include passages, formerly overlooked by Jews in terms of their messianic import, that seem to foreshadow Jesus' death (e.g. Ps. 22).
Jesus did not say 'Oral Law' here because, assuming he considered it authoritative, it was still unwritten. He did not say 'Talmud' because the Talmud (a written compilation of Oral Law) was still in the future, not to mention that it does not testify to Jesus.
Finally, the OP asks: "What is fulfilled that is found in all three mentioned by Jesus?" Given the context, the answer is almost certainly that the Messiah would suffer, an idea that was apparently new to those to whom he spoke.
Conclusion: Jesus mentioned all three types of scripture: Law, Prophets, and Psalms because he wanted to show his disciples how each type predicted his suffering, which his followers did not expect. Over the coming decades, the church would use these scriptures effectively to present the gospel message - that God sent Jesus to die for humanity's sins - rather than to restore the throne of David in his lifetime.
