The Jewish concept of GILGULIM (REINCARNATION) was introduced by Rabbi Akivah. “The Bahir (Sefer HaBahir)” explains how Rabbi Akivah came to this conclusion that must be reincarnated souls, and I will explain it in a later article.
I think it was Rabbi Aryeh Kaplan who stated the following:
Rabbi Saadia Gaon rejected reincarnations because he didn’t have access to secret kabbalistic teachings.
In those days, not even the most famous and respected Rabbis had access to all the existing writings. Neither Rabbi Saadia Gaon nor the Rambam (Maimonides) ever saw “The Bahir”.
The concept of reincarnation is further developed in the Zohar (Book of Splendor) and especially in the Book of Reincarnations (Sefer HaGilgulim) by Rabbi Yitzchak Luria.