We follow with Yekum Purkan, Ashrei. We return the Torah scrolls to the Ark and the chazzan says half-Kaddish.
Mussaf for Festivals with mention of Shabbos. We include mention of Shabbos in chazzan’s repetition of the Shemoneh Esreh as well as Hoshanos (see Shacharis – most Ashkenaz congregations say it at the conclusion of Mussaf) and conclude with Kaddish Tiskabbel, Ein KeElokeinu, Aleinu, Shir shel Yom, LeDavid Hashem Ori and their respective Kaddish recitations. Kiddush of Shabbos, we add Leishev basukkah. We wash for the Shabbos Seuda. Birkas HaMazon includes Retzeh and Ya’aleh VeYavo.
Mincha for Shabbos: Ashrei, U’va LeTziyyon, chazzan recites half-Kaddish – we remove a Torah scroll from the Ark (Va’yehi Binso’a). We call up three Aliyos and read Parashas Vezos HaBeracha (Deuteronomy 33:1-17). We return the scroll to the Ark and the chazzan recites half-Kaddish. All say the Shemoneh Esreh of Shabbos with Ya’aleh VeYavo, also in chazzan’s repetition. We do not say Tzidkas’cha and chazzan concludes with Kaddish Tiskabbel, Aleinu and finally the Mourner’s Kaddish. (As it is Shabbos, we wash for the Seuda Shelishis, which we eat in the sukkah. We include Retzeh and Ya’aleh VeYavo in Birkas Hamazon.
Maariv: As usual on Motza’ei Shabbos, we say Ata Chonantanu. We include Ya’aleh VeYavo. After Kaddish Tiskabbel (we do not say Vi’yehi No’am and Ve’ata Kadosh) we say Aleinu, LeDavid Hashem Ori and their respective Kaddish recitations. Havdala is in the sukkah, concluding with the blessing of Leishev basukkah.
Sunday morning, 2nd day of Chol Hamo’ed, Shacharis: some don tefillin and others do not, each according to his custom – but without a beracha – and yet others recite the blessing in a whisper (see Rema, Shulchan Aruch, Orach Chayyim 31:2; also Mishna Berura ad loc. s.v. She’ein mevarchim for advisable procedure.)
Usual weekday Shemoneh Esreh with Ya’aleh VeYavo and Ya’aleh VeYavo again in chazzan’s repetition.
We now take the lulav and esrog (same procedure as on the first two days – recite the blessing Al netilas lulav (see chart for the various minhagim at the waving of the lulav). We say whole Hallel. At the conclusion some congregations (Nusach Sefarad and some Ashkenaz) recite Hoshanos – on the third day it is E’eroch Shu’i. We circle the Bimah, Kaddish Tiskabbel.
We remove a Torah scroll from the Ark and read in Parashas Pinchas (Numbers 29:20-28), “U’vayom HaShelishi,” until “Ve’Niskah.” We call 4 Aliyos. We repeat “U’vayom HaShelishi,” “Uvayom HaRevi’i for the 4th Aliyah). The ba’al keriah says half- Kaddish. We return the scroll to the Ark, Yehallelu, Ashrei U’Va LeTziyyon, half-Kaddish.
Mussaf of Festivals – we add U’vayom haShelishi, Uminchasam… Following the Amidah repetition, Hoshanos (see Shacharis – most Ashkenaz congregations say it at the conclusion of Mussaf), Kaddish Tiskabbel. We conclude with Aleinu, Shir shel Yom, LeDavid HaShem Ori and their respective Kaddish recitations.
Mincha is usual weekday Shemoneh Esreh with inclusion of Ya’aleh VeYavo. At the conclusion of repetition, Kaddish Tiskabbel, Aleinu, Mourner’s Kaddish.
Maariv is usual weekday tefilla with inclusion of Ya’aleh VeYavo in the Shemoneh Esreh. At the conclusion – Kaddish Tiskabbel, Aleinu, Le David Hashem Ori and their respective Kaddish recitations.
Monday morning, 3rd day of Chol Hamo’ed: all tefillos are the same as yesterday, with the exception of Hoshanos – we start with Kel Lemosha’os followed by Torah reading (Numbers 29:23-31) – Mussaf ,U’vayom HaRevi’i.
Tuesday morning, 4th day of Chol Hamo’ed: all tefillos are the same as yesterday with the exception of Hoshanos – we start with Adon Hamoshia followed by Torah reading (Numbers 29:26:34).
Wednesday is Hoshana Rabbah. According to the Shulcham Aruch (Orach Chayyim 664) we say many mizmorim on Hoshana Rabbah. We say Pesukei DeZimra as we do on Shabbos. Rema notes that we do not say Nishmas but we do say the Shabbos and Yom Tov text of Ein kamocha, Shema Yisrael. We add Ribbono Shel Olam at the removal of the Sifrei Torah from the Ark.
We also treat this day with some of the same gravity as Yom Kippur. Some who are scrupulous immerse themselves in a mikveh. The chazzan dons a kittel. The Mishna Berura (ad loc.) explains that donning the kittel indicates that this day is the point at which the seal is affixed on our judgment. Thus some are accustomed to greet one another until Hoshana Rabbah with Gemar Chasima Tova.