# GENERALIZATIONS OF RAMANUJAN’S RANK FUNCTIONS COLLECTED FROM RAMANUJAN’S LOST NOTEBOOK

## Authors

• Nil RatanBhattacharjee Professor, Department of Mathematics, University of Chittagong, BANGLADESH
• Sabuj Das Senior Lecturer, Department of Mathematics, Raozan University College, BANGLADESH

## Keywords:

Congruent To, Dyson’s Rank, Mock Theta, Modulo, Ramanujan’s Lost Notebook

### Abstract [English]

In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers. We prove the Mock Theta Conjectures with the help of Dyson’s rank and S. Ramanujan’s Mock Theta functions. These functions were quoted in Ramanujan’s lost notebook and unpublished papers. In1916, Ramanujan stated the theta series in x like A(x), B(x), C(x), D(x). We discuss the Ramanujan’s functions with the help of Dyson’s rank symbols. These functions are useful to prove the Mock Theta Conjectures. Now first Mock Theta Conjecture is “The number of partitions of 5n with rank congruent to 1 modulo 5 equals the number of partitions of 5n with rank congruent to 0 modulo 5 plus the number of partitions of n with unique smallest part and all other parts   the double of the smallest part”, and Second Mock Theta Conjecture is “The double of the number of partitions of   with rank congruent to 2 modulo 5 equals the sum of the number of partitions of   with rank congruent to 0 and congruent to1 modulo 5, and the sum of one and the number of partitions of n with unique smallest part and all other parts  one plus the double of the smallest part”. This paper shows how to prove the Theorem 1.3 with the help of Dyson’s rank symbols N(0,5,5n+1), N(2,5, 5n+1) and shows how to prove the Theorem 1.4 with the help of Ramanujan’s theta series and Dyson’s rank symbols N(1,5, 5n+2), N(2,5, 5n+2) respectively.

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2016-03-31

## How to Cite

Bhattacharjee, N. R., & Das, S. (2016). GENERALIZATIONS OF RAMANUJAN’S RANK FUNCTIONS COLLECTED FROM RAMANUJAN’S LOST NOTEBOOK. International Journal of Research -GRANTHAALAYAH, 4(3), 1–20. https://doi.org/10.29121/granthaalayah.v4.i3.2016.2780

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