Infinite Families of Congruences for Overpartitions With Restricted Odd Differences

Abstract

<jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline1.png" /><jats:tex-math>$\overline{t}(n)$</jats:tex-math></jats:alternatives></jats:inline-formula> be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline2.png" /><jats:tex-math>$\overline{t}(n)$</jats:tex-math></jats:alternatives></jats:inline-formula> modulo small powers of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline3.png" /><jats:tex-math>$2$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline4.png" /><jats:tex-math>$3$</jats:tex-math></jats:alternatives></jats:inline-formula> have been established. We present two infinite families of congruences modulo <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline5.png" /><jats:tex-math>$5$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline6.png" /><jats:tex-math>$27$</jats:tex-math></jats:alternatives></jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001254_inline7.png" /><jats:tex-math>$\overline{t}(n)$</jats:tex-math></jats:alternatives></jats:inline-formula>, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, <jats:italic>Proc. Math. Sci.</jats:italic> <jats:bold>129</jats:bold> (2019), Article 31].</jats:p>