Abstract
In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers.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 These functions were quoted in Ramanujan’s lost notebook and unpublished papers.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.
Highlights
In this paper we give the definitions of, rank of partition, N m,n, N m,t,n, z, x, zx, xn m, xk ; x5 m, 0 n, 1 n, 1 n and 2 n which are collected from Ramanujan’s Lost Notebook VI and generate the generating functions of Dyson’s Rank
We prove the Theorem, which is known as First Mock Theta Conjecture
Theorem 1: 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, i.e. N 1,5,5n N 0,5,5n 0 n, where 0 n is the number of partitions of n with unique smallest part and all other parts the double of the smallest part
Summary
In this paper we give the definitions of , rank of partition, N m,n , N m,t,n , z, x , zx , xn m , xk ; x5 m , 0 n , 1 n , 1 n and 2 n which are collected from Ramanujan’s Lost Notebook VI and generate the generating functions of Dyson’s Rank. A(x), D(x), x , (x), 0 n and 1 n and prove the two Theorems first and second mock theta. We give two numerical examples which are related to first and second mock theta conjectures respectively when n = 1. We generate the generating functions for 1 n and 2 n collected from Ramanujan’s Lost notebook V1. We prove the Theorem 3 in terms of 1 n with the help of Ramanujan’s identities and prove the Theorem 4 in terms of 2 n with the help of Dyson’s rank symbols and Ramanujan’s identities
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