Jason and Amy are my only friends that probably would be interested in this but...just in case I am wrong, here it is.
The Beracha of Shehehiyanu on the Second Night of Rosh Hashanah
There is considerable discussion among the Halachic authorities regarding the recitation of the Beracha of Shehehiyanu on the second night of Rosh Hashanah. It is clear according to all views that one must recite the Beracha in Kiddush on the first night of Rosh Hashanah, before drinking the wine, just as we do on the first night of every other Yom Tob. On the second night of Rosh Hashanah, however, some Rishonim (Medieval Halachic scholars) maintained that one should not recite Shehehiyanu. In their view, the two days of Rosh Hashanah differ from other Yamim Tobim in that they constitute a "Yama Arichta" - a prolonged day of Yom Tob. The two days of Rosh Hashanah, according to this position, are not to be viewed as two separate festive occasions, each of which requiring its own recitation of Shehehiyanu, but rather as a single occasion. As such, the Beracha of Shehehiyanu, which celebrates the onset of the festival, is recited only on the first night, and not on the second night.
Other Rishonim, however, including Rashi (Rabbi Shlomo Yishaki of Troyes, France, 1040-1105) and the Rashba (Rabbi Shlomo Ben Aderet of Barcelona, Spain, 1235-1310), disagreed. They held that Rosh Hashanah is no different from other festivals in this regard, and the two days of the holiday constitute independent festive occasions, both of which require the recitation of Shehehiyanu. This also appears to be the view of the Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204).
The Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327) suggested a compromise position, advising that one should place a new fruit - meaning, a fruit he had not eaten since the beginning of the fruit's season - on the table during Kiddush on the second night of Rosh Hashanah. He should have in mind while reciting Kiddush that if Halacha follows the view that Shehehiyanu is not required on the second night of Rosh Hashanah, then the Shehehiyanu which he recites should refer to the new fruit, which he then eats with his meal. This way, one may recite Shehehiyanu without running the risk of reciting a Beracha Le'batala (meaningless Beracha) according to all opinions.
As for the final Halacha, the Shulhan Aruch rules in accordance with the view of Rashi and the Rashba, that one must recite Shehehiyanu on both nights of Rosh Hashanah. He adds, however, that it is preferable to place a new fruit on the table during Kiddush, as the Rosh recommended, in order to satisfy all opinions, though according to the strict Halacha this is not necessary.
Indeed, the widespread practice is to place a new fruit on the table during Kiddush on the second night of Rosh Hashanah, and then to eat the fruit as part of the meal. Although Rav Haim Vital (Safed, Israel, 1543-1620) wrote in his Sha'ar Ha'kavanot that a new fruit is unnecessary, as Halacha accepts the view requiring the recitation of Shehehiyanu, this is nevertheless the widespread custom, in accordance with the Shulhan Aruch's ruling. It should be emphasized, however, that one who does not have a new fruit on the second night of Rosh Hashanah should certainly recite Shehehiyanu nonetheless. This is the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909).
Many later scholars addressed the interesting question of how placing a new fruit on the table resolves the dilemma and enables a person to satisfy all views. Let us assume, for argument's sake, that it is acceptable to recite Shehehiyanu upon seeing the new fruit, and then eat the fruit later, as part of the meal. Still, reciting this Beracha after reciting Kiddush and before drinking would appear to constitute a Hefsek (improper disruption) in between Kiddush and drinking. As discussed, we place the fruit on the table so that the Shehehiyanu recited as part of Kiddush will refer to the fruit according to the view that Shehehiyanu is not otherwise warranted on the second night of Rosh Hashanah. But if so, then this Beracha is entirely irrelevant to Kiddush, and thus constitutes a Hefsek in between Kiddush and drinking, which should, seemingly, disqualify the Kiddush. It would thus seem to emerge that while endeavoring to resolve one problem, we bring ourselves into another, far more serious, predicament!
Rav Haim Palachi (Izmir, Turkey, 1788-1869) suggested that the Beracha of Shehehiyanu would not constitute a Hefsek because, as we saw, Halacha follows the view that one in any event should recite Shehehiyanu on the second night of Rosh Hashanah. This explanation, however, seems insufficient to answer the question, as it essentially concedes that placing a new fruit on the table will not achieve anything according to the view that Shehehiyanu is not recited on the second night of Rosh Hashanah. If so, then why do we place the fruit at all?
Another answer is suggested by Rabbi Moshe Halevi (Israel, 1961-2001), in his work Birkat Hashem, where he explains that the Beracha of Shehehiyanu is not inconsistent with the spirit of Kiddush. This Beracha simply expresses gratitude to God for bringing us to this special occasion; it does not contain any specific reference that would be in contrast with the theme of Kiddush. For example, if in the middle of a Kiddush a person recited the Beracha of She'hakol, this would certainly constitute a Hefsek as this Beracha is entirely out of place in Kiddush. This is not the case with regard to Shehehiyanu, and therefore this Beracha would not constitute a disruption.
In any event, one should preferably follow the widespread custom to place a new fruit on the table during Kiddush on the second night of Rosh Hashanah, if possible. One should then eat the fruit during the meal, without reciting an additional Beracha of Shehehiyanu, as it was covered by the Shehehiyanu recited at Kiddush.
Summary: According to some authorities, one does not recite Shehehiyanu at Kiddush on the second night of Rosh Hashanah. It is therefore customary to place a new fruit on the table during Kiddush on this night, and have in mind while reciting Shehehiyanu that if Halacha follows the view that this Beracha is not recited, then the Beracha he recites applies to the fruit. He should eat the fruit during the meal without repeating Shehehiyanu. Although this is the widespread custom, according to the strict Halacha on recites Shehehiyanu on the second night of Rosh Hashanah even if he does not have a new fruit.
Source: Sephardic Daily Halacha by Rabbi Eli Mansour