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Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 797–811 | Cite as

Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts

  • B. Hemanthkumar
  • H. S. Sumanth Bharadwaj
  • M. S. Mahadeva NaikaEmail author
Article
  • 44 Downloads

Abstract

Let \(\text {pod}_{9}(n)\), \(\text {ped}_{9}(n)\), and \(\overline {A}_{9}(n)\) denote the number of 9-regular partitions of n wherein odd parts are distinct, even parts are distinct, and the number of 9-regular overpartitions of n, respectively. By considering \(\text {pod}_{9}(n)\) from an arithmetic point of view, we establish a number of infinite families of congruences modulo 16 and 32, and some internal congruences modulo small powers of 3. A relation connecting above partition functions in arithmetic progressions is obtained as follows. For any \(n\geq 0\),
$$ 6 \text {pod}_{9}(2n + 1) = 2 \text {ped}_{9}(2n + 3) = 3 \overline {A}_{9}(n + 1).$$

Keywords

Regular partitions Overpartitions Congruences Theta function Distinct odd parts 

Mathematics Subject Classification (2010)

11P83 05A17 

Notes

Acknowledgements

The authors would like to thank anonymous referee for his/her valuable comments.

Funding Information

The second author was supported by the Council of Scientific and Industrial Research, India through SRF (No. 09/039(0111)/2014-EMR-I).

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • B. Hemanthkumar
    • 1
  • H. S. Sumanth Bharadwaj
    • 2
  • M. S. Mahadeva Naika
    • 2
    Email author
  1. 1.Department of MathematicsM. S. Ramaiah University of Applied SciencesBangaloreIndia
  2. 2.Department of MathematicsBangalore University, Central College CampusBengaluruIndia

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