The first eight days of the Easter season ("Bright Week") make up the Octave of Easter and celebrated as Solemnities of the Lord. Singing the Easter sequence, Victimae Paschali Laudes, at Mass throughout the Octave is recommended, though not required. At the end of Masses during the Easter Octave, the dismissal, as at Easter, is sung by the deacon or priest; to which the people respond in song, “Thanks be to God, Alleluia! Alleluia!”
“Let us keep the holy feast of Pascha and then, adding day by day
the holy Pentecost, which we regard as feast upon feast,
we shall keep the festival of the Spirit.” ~ St. Athanasius (4th c)
"We are an Easter people
and 'Allelujah' is our song."
~ St Augustine
"... we are given two liturgical seasons, one before Easter and the other after. The season before Easter signifies the troubles in which we live here and now, while the time after Easter which we are celebrating at present signifies the happiness that will be ours in the future. What we commemorate before Easter is what we experience in this life; what we celebrate after Easter points to something we do not yet possess. This is why we keep the first season with fasting and prayer; but now the fast is over and we devote the present season to praise. Such is the meaning of the Alleluia we sing. ...
Now therefore, brethren, we urge you to praise God. That is what we are all telling each other when we say Alleluia. You say to your neighbor, “Praise the Lord!” and he says the same to you. We are all urging one another to praise the Lord, and all thereby doing what each of us urges the other to do. But see that your praise comes from your whole being; in other words, see that you praise God not with your lips and voices alone, but with your minds, your lives and all your actions."
Excerpt from St. Augustine's discourse
on the Psalms (Ps. 148, 1-2: CCL 40, 2165-2166)