GENERALIZATIONS OF RAMANUJAN'S RANK FUNCTIONS COLLECTED FROM RAMANUJAN'S LOST NOTEBOOK

In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers.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 These functions were quoted in Ramanujan’s lost notebook and unpublished papers.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 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) respectivel

The Rogers-Ramanujan Identities

In 1894, Rogers found the two identities for the first time. In 1913, Ramanujan found the two identities later and then the two identities are known as The Rogers-Ramanujan Identities. In 1982, Baxter used the two identities in solving the Hard Hexagon Model in Statistical Mechanics. In 1829 Jacobi proved his triple product identity; it is used in proving The Rogers-Ramanujan Identities. In 1921, Ramanujan used Jacobi’s triple product identity in proving his famous partition congruences. This paper shows how to generate the generating function for , , and , and shows how to prove the Corollaries 1 and 2 with the help of Jacobi’s triple product identity. This paper shows how to prove the Remark 3 with the help of various auxiliary functions and shows how to prove The Rogers-Ramanujan Identities with help of Ramanujan’s device of the introduction of a second parameter a

Generating Functions for X(n) and Y(n)

This paper shows how to prove the Theorem , i.e., the number of partitions of n with no part repeated more than twice is equal to the number of partitions of n with no part is divisible by 3

Mock Theta Conjectures

This paper shows how to prove the two Theorems first and second mock theta conjectures respectively

The Rogers-Ramanujan Identities

In 1894, Rogers found the two identities for the first time. In 1913, Ramanujan found the two identities later and then the two identities are known as The Rogers-Ramanujan Identities. In 1982, Baxter used the two identities in solving the Hard Hexagon Model in Statistical Mechanics. In 1829 Jacobi proved his triple product identity; it is used in proving The Rogers-Ramanujan Identities. In 1921, Ramanujan used Jacobi’s triple product identity in proving his famous partition congruences. This paper shows how to generate the generating function for , , and , and shows how to prove the Corollaries 1 and 2 with the help of Jacobi’s triple product identity. This paper shows how to prove the Remark 3 with the help of various auxiliary functions and shows how to prove The Rogers-Ramanujan Identities with help of Ramanujan’s device of the introduction of a second parameter a

Development of Partition Functions of Ramanujan’s Works

In 1986, Dyson defined the rank of a partition as the largest part of a partition minus the number of parts of . In 1988, Garvan discussed the theta series in x like A(x), B(x), C(x), D(x) and also discussed Jacobi’s triple product Identity (1829). Both of the authors have worked on Ramanujan’s seminal works “Ramanujan’s Lost Notebooks”. This paper proves the Theorem 1 with the help of Dyson’s rank conjectures N(0,5,5n +1), N(2,5, 5n +1) and proves the Theorem 2 with the help of Garvan’s theta series and Dyson’s rank conjectures N(1,5, 5n+2), N(2,5, 5n+2), respectively. An attempt has been taken here to the development of the Ramanujan’s works with the contributions of Dyson and Garvan. Definitions and simple mathematical calculations are presented here to make the paper easier to the common readers

Generating Functions for β1(n) and β2(n)

This paper shows how to prove the two Theorems, which are related to the terms β1(n) and β2(n) respectively Theorem: N(0,5,5n+1)= β1(n)+N (5,5,5n+1) and Theorem: N(1,5,5n+1)= β2(n)+ N(2,5,5n+2)

Mock Theta Conjectures

This paper shows how to prove the two Theorems first and second mock theta conjectures respectively

Generating Functions for and

This paper shows how to prove the Theorem = , i.e., the number of partitions of n into p-parts is equal to the number of partitions of n having largest part p

Generating Functions for P1r (n) and P2r (n)

In 1970 George E. Andrews defined the generating functions for P1r (n) and P2r (n). In this article these generating functions are discussed elaborately. This paper shows how to prove the theorem P2r (n) = P3r (n) with a numerical example when n = 9 and r = 2. In 1966 Andrews defined the terms A/(n) and B/(n), but this paper proves the remark A/(n) = B/(n) with the help of an example when n = 10. In 1961, N. Bourbaki defined the term P(n, m). This paper shows how to prove a Remark in terms of P(n, m), where P(n, m) is the number of partitions of the type of enumerated by P3r (n ) with the further restrictions that b1< 2m