https://www.geni.com/people/Hananya-ben-Haninai-haKohen-al-Nahr-Paqod-Dayan-of-the-Gate-Gaon-of-Sura/6000000001263102021

david:

You are absolutely correct in the Halachic sense.

However, the Talmud does highlight the fact that the House of the Nasi in Israel at the time of the second Temple, all descendants from Hillel the elder, was considered to be "of the House of David", even though it was through a maternal and not a paternal link. In fact, the Talmud further states that the Nesiim were of a 'lower' geneological stature than the House of the exilarch of Babylonia, because the latter was a direct patrilineal line as opposed to the former.
Considering that the geonim of Israel were seeking to establish their authority at the time, it is entirely understandable that they would highlight their descent from King David, even if there was a mother in the line, because it directly establishes them as the true successors to the nesiim of Hillel's family, who likewise had a maternal component of their lineage from King David.
It is obvious, of course, that one cannot append the title 'hakohen' to Bustenai, who was a direct male descendant of David; his son in law, though, is a different matter.
As an interesting historical aside, Bustenai's MOTHER was the daughter of Chaina, Gaon of Sura from 610-614. No surprise, that the families of the Exilarch and the families of the geonim intermarried; they were considered to be the 'elite' of Jewry at the time. This Chanina also had a son, Huna, who succeeded his father as Gaon of Sura, 614-650; it was HIS son, Chaninai (the suffix -;ai' refers to 'son of', or, 'descendant of' much like 'ov' does in Russian, or "itz' in other eastern european languages; so he was likely named for his grandfather), who married Adda (or Adoi), daughter of his first cousin Bustenai, and who was in turn the father of Chamamia hakohen of Nehar pekod..

Thank you Rabbi Shraga for your enlightment. If we have the case of what you state, that a ga'on who is a kohen states he is from the House of David because of a maternal link, on the profile please can we include your note and the source in the Talmud which gives this precedence (although what is in the Talmud is obviously referring to an earlier period in history as you mention).
David

Talmud bavli Horayot 11b - we (the house of the nasi in Israel) are subservient to them (the exilarch); though the house of the nasi is also credited to davidic descent, they are from benjamin on the patrelineal side. see, too, Yerushalmi taanit 4:2, and Yerushalmi kilayin 9:3.
Talmud bavli Ketubot 62b says that Hillel the elder, founder of the house of nesiim in israel, was a descendant of shefatyah the son of dovid's wife avital.
so: exilarch was direct patrilineal from david.
Nasi was from the tribe of benjamin, but had a maternal ancestor of direct davidic descent, and was therefore considered 'davidic', albeit in an inferior manner to the exilarch.
it is therefore NOT a contradiction to say that a kohain can be of davidic descent.

Agreed and thanks for the citation. We need a word that means Davidic Descent via a son-to-son pedigree and another word that means Davidic via a female ancestor in the pedigree. Are there such words? Just to be clear, Bustenai is considered a Davidic (son-to-son) descendant, correct? This would mean that any son-to-son descendants of his are also Davidic (son-to-son) descendants. This I believe is where the problem resided in this branch as currently configured on GENI. At one end there is Bustenai and along the way are several son-to-son descendants of his who are also designated as HaCohen (two such persons are already mentioned in this thread: Natronai Yosef "Mari" HaCohen and Masliah ben Shlomo HaCohen. The big question in my mind is whether Bustenai is really a son-to-son Davidic descendant, or was this something that was invented in the early Middle Ages?