The Troparion from the Byzantine liturgy for January 1st:
O Jesus, enthroned on high together with Your eternal Father and Divine Spirit, You willed to be born on earth of a maiden and virgin mother and on the eighth day you were circumcised. Glory to Your most noble decision. Glory to plan of salvation. Glory to Your condescension; You alone love mankind.
As an interesting aside for those interested in 20th century liturgical reforms, within the Roman missal prior to 1960, today's feast was denoted as the "Feast of the Circumcision of the Our Lord and Octave Day of the Nativity". From the 1960 editions on, it is simply denoted as the Octave Day of the Nativity. However, whether pre or post 1960, today's Mass in the usus antiquior continues to have the same propers inclusive of a Gospel reading from St. Luke 11:2, which is about Our Lord's circumcision eight days after his birth.
Within the 1965 Missale Ambrosianum, January 1st remained denoted as both the Octave Day of the Nativity and Circumcision of Our Lord. The Circumcision of the Lord also appears in the modern Missale Hispano-Mozarabicum.
Within the modern Roman liturgy, today is both the Octave Day of the Nativity, as has already been mentioned, and the Solemnity of Mary, Mother of God.
Introit for today's feast in the modern Roman calendar from the Graduale:
Hail Holy Mother, the Child-Bearer who has brought forth the King, the ruler of heaven and earth forever. V. My heart overflows with a goodly theme; I address my works to the King.
(Or the Introit for today may also be the Lux fulgebit from the Mass at Dawn of Christmas.)
Detail of Icon of the Mother of God by Iconographer Kenneth Dowdy
