During Kiddush on the first night of Rosh Hashanah, one must according to all authorities add the Beracha of She'hecheyanu over the occasion of this holiday. The Beracha of She'hecheyanu is recited immediately following the Beracha of "Mekadesh Yisrael Ve'ha'zmanim."
Regarding the recitation of this Beracha on the second night of Rosh Hashanah, however, we find considerable discussion and controversy among the Rishonim (Medieval scholars). Some Rishonim ruled that one should not recite the Beracha of She'hecheyanu on the second night of Rosh Hashanah, because the two days of Rosh Hashanah essentially constitute a "Yoma Arichta" one extended day. Thus, for example, although normally an egg that was laid on the first day of Yom Tov is permissible for consumption on the second day, on the second day of Rosh Hashanah one may not partake of an egg laid on the first day. This Halacha reflects the notion that the second day of Rosh Hashanah is looked upon as part of the same day as the first; hence, once a person recites She'hecheyanu on the first night of Rosh Hashanah, he has no need to repeat it on the second night.
The Shulchan Aruch, however, as well as the majority of the Halachic authorities, ruled otherwise, and required the recitation of She'hecheyanu on the second night of Rosh Hashanah. This is, indeed, the prevalent practice. Nevertheless, the Rosh (Rabbenu Asher Ben Yechiel, Germany-Spain, 1250-1327) writes that since this issue is, after all, subject to some controversy, one should endeavor to satisfy all opinions by placing a new fruit on the table during Kiddush on the second night of Rosh Hashanah. As one recites the Beracha of She'hecheyanu, he should have in mind that the Beracha refers both to the occasion of Rosh Hashanah as well as to the new fruit, and he should then partake of the fruit after completing Kiddush and drinking some wine. In this way, one avoids the possibility of reciting the Beracha of She'hecheyanu unnecessarily. The Shulchan Aruch records this view and writes that it is proper for a person to follow this practice.
The Arizal, however, in "Sha'ar Ha'kavanot," writes that one need not have a new fruit on his table during Kiddush on the second night of Rosh Hashanah, because the teachings of Kabbalah indeed require the recitation of She'hecheyanu on this night. Accordingly, the Chid"a (Rabbi Chayim Yosef David Azulai, Israel, 1724-1806) and the Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1833-1909) ruled that if one cannot obtain a new fruit for the second night of Rosh Hashanah, he may nevertheless recite She'hecheyanu without concern.
If one forgot to recite the Beracha of She'hecheyanu during Kiddush on the first or second night of Rosh Hashanah, may he recite it at a later point?
According to Halacha, if a person forgot to add She'hecheyanu during Kiddush he may recite the Beracha anytime before sundown of the following day. He may recite the Beracha at any point; even if he remembers while walking in the street, he may stop and recite the Beracha right then.
Summary: One must add the Beracha of She'hecheyanu during Kiddush on both nights of Rosh Hashanah. On the second night, it is preferable to place a new fruit on the table and have in mind for the recitation of She'hecheyanu during Kiddush to refer to that new fruit, as well. One should then partake of the fruit after Kiddush. One who does not have access to a new fruit on the second night nevertheless recites She'hecheyanu during Kiddush. If one forgot to recite She'hecheyanu during Kiddush on one of the nights of Yom Tov, he can recite it at any point until sundown the following day.
