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Congruences modulo powers of 11 for some eta-quotients

  • Shashika Petta MestrigeEmail author
Research
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Abstract

The partition function \( p_{[1^c11^d]}(n)\) can be defined using the generating function,
$$\begin{aligned} \sum _{n=0}^{\infty }p_{[1^c{11}^d]}(n)q^n=\prod _{n=1}^{\infty }\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}. \end{aligned}$$
In this paper, we prove infinite families of congruences for the partition function \( p_{[1^c11^d]}(n)\) modulo powers of 11 for any integers c and d, which generalizes Atkin and Gordon’s congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup \(\Gamma _0(11)\).

Mathematics Subject Classification

Primary 11P83 Secondary 05A17 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentLouisiana State UniversityBaton RougeUSA

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