A conjecture of Baruah and Begum on the smallest parts function of restricted overpartitions

Abstract

In 2017, Andrews, Dixit, Schultz and Yee introduced the function $$\overline{\text {spt}}_\omega (n)$$ , which denotes the number of smallest parts in the overpartitions of n in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of 5 satisfied by $$\overline{\text {spt}}_\omega (n)$$ . Moreover, they conjectured a family of internal congruences modulo all powers of 5 and two families of congruences modulo all even powers of 5. In this paper, we confirm the three families of congruences due to Baruah and Begum.