New congruences modulo 5 for partition related to mock theta function $$\omega (q)$$ ω ( q )

Bernard L. S. Lin () and Jiejuan Xiao ()
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Bernard L. S. Lin: Jimei University
Jiejuan Xiao: Jimei University

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 1, 141-148

Abstract: Abstract Let $$p_{\omega }(n)$$ p ω ( n ) be the number of partitions of n in which each odd part is less than twice the smallest part, and assume that $$p_{\omega }(0)=0$$ p ω ( 0 ) = 0 . Andrews, Dixit, and Yee proved that the generating function of $$p_{\omega }(n)$$ p ω ( n ) equals $$q\omega (q)$$ q ω ( q ) , where $$\omega (q)$$ ω ( q ) is one of the third order mock theta functions. Many scholars have studied the arithmetic properties of $$p_{\omega }(n)$$ p ω ( n ) . For example, Waldherr proved that $$p_{\omega }(40n+28)\equiv p_{\omega }(40n+36)\equiv 0 \pmod 5$$ p ω ( 40 n + 28 ) ≡ p ω ( 40 n + 36 ) ≡ 0 ( mod 5 ) by using the theory of Maass forms, which was later proved once again in an elementary method by Andrews, Passary, Sellers, and Yee. In this paper, we shall establish four new congruences modulo 5 for $$p_{\omega }(n)$$ p ω ( n ) .

Keywords: Partition; Mock theta functions; congruence; Ramanujan’s identity; 11P83; 05A17 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00078-9

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