Bikkurim, Chapter 1, Mishnah 4. Our mishnah continues the list of "Eilu mevi'in v'lo korin" - those who are permitted to bring bikkurim to the Beis HaMikdash, but are not permitted to recite the passage of mikra bikkurim that accompanies the bringing.
"Hager mevi v'eino korei": the convert brings but does not recite.
One who converted to Judaism may bring bikkurim, since he grew his fruits in his own field, on land that belongs to him. But he does not recite, and the mishnah gives the reason: "she'eino yachol lomar asher nishba Hashem la'avoseinu lases lanu" - because he cannot say "that Hashem swore to our forefathers to give to us." The convert did indeed acquire land in Eretz Yisrael, but his forefathers were not promised a portion in the Land, and he has no inherent share in it, and therefore he cannot say this text.
"Im haysah imo miYisrael": if his mother was Jewish.
If his mother is Jewish, even though his father is not, then he is "mevi v'korei" - he brings the bikkurim and also recites the passage, since he is able to say "that Hashem swore to our forefathers," even though technically he cannot say "to give to us."
Some explain this differently: Tosafos in Bava Basra writes that the reference is to the children of Moshe's father-in-law, who received a portion near Yericho; and since they received a portion in the Land, they were able to recite as well. So it is according to some of the opinions.
The convert's wording in prayer:
From here the mishnah moves on to discuss the wording that a convert says in his prayers:
"Ukh'shehu mispallel beino l'vein atzmo" - when he prays by himself, he says "the God of the fathers of Israel," and not "the God of our fathers," since they are not his fathers.
"Ukh'shehu b'veis haknesses" - when he is in the synagogue, passing before the ark, he says "the God of your fathers," referring to the congregation.
"Im haysah imo miYisrael" - if his mother was Jewish, he says "the God of our fathers," the same wording that the entire congregation says.
The view of the Rambam and the Yerushalmi:
The Rambam writes that the halachah does not follow this mishnah; rather, a convert brings bikkurim and also recites the passage. The basis for this is in the Yerushalmi, which says in the name of Rabbi Yochanan that a convert may say the verses, because it is stated that Avram's name was changed to Avraham, on account of his being destined to be "the father of a multitude of nations" - that is, the father of all converts. For this reason a convert is called "the son of Avraham Avinu" or "the daughter of Avraham Avinu." And since Avraham is their father, and the promise was given to him, all the verses that a Jew says using the phrase "our fathers" - they too may say them.
In summary: in this mishnah we learned that a convert brings bikkurim but does not recite, because he cannot say "that Hashem swore to our forefathers to give to us"; that if his mother was Jewish he brings and recites; we saw the explanation of Tosafos regarding the children of Moshe's father-in-law, the wording of the convert's prayers when praying alone and when in the synagogue, and the view of the Rambam and the Yerushalmi that a convert does recite, since Avraham Avinu is "the father of a multitude of nations" and the father of all converts.