Transformation Formulas: 10th Order Mock Theta Functions

Ramanujan’s lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan’s tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan’s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell’s transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

S. Ramanujan introduced the mock theta functions in his last letter to G.H. Hardy. In his letter and Lost Notebook, he presented mock theta functions and their identities. The expression of each mock theta function relies on the hypergeometric series. G.N. Watson studied the relations between Ramanujan’s fifth order mock theta functions and the function introduced by M. Lerch in 1893. In this manuscript, we develop the relations between the basic bilateral hypergeometric series and the functions deduced from all mock theta functions introduced by G.E. Andrews, B.C. Berndt, S.H. Chan, R. McIntosh, Ramanujan, and Watson.

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction F ( q ) F(q) . In order to prove some of these claims, G. N. Watson established two important theorems about F ( q ) F(q) that he found in Ramanujan’s notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

We obtain four Hecke-type double sums for three of Ramanujan's third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann, Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.

By examining the transformation formulae between unilateral series and bilateral ones derived from the bilateral Bailey lemma, we establish numerous identities of false theta functions, including most of the known ones discovered mainly by Rogers [40] and Ramanujan in his Lost Notebook [39].

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e2πir (r rational), there is a theta function Fr(q) with F(q) − Fr(q) = O(1). In this paper we establish the relationship between two families of mock theta functions.

In this paper we express two new mock theta functions with one parameter as the Appell-Lerch sums using the Bailey mechanism. Meanwhile, we also obtain some Hecke-type double sums for some new $q$-series. In addition, we establish the relationships between the mock theta functions and the classical sixth and eighth order mock theta functions.Furthermore, we give the Hecke-type double sums for the second order mock theta function $D_5(q)$.

<abstract><p>Using the three-term Weierstrass relation for theta functions and the properties of Hecke-type double sums and Appell-Lerch sums, we give new and simple proofs for the seventh order mock theta conjectures.</p></abstract>

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$\phi(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=\phi(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of $q$-products. As corollaries, we deduce the congruences $a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv0~(\textup{mod}~25)$ as well as $a(1250n+250r+219)\equiv 0~(\textup{mod}~125)$, where $r=1$, $3$, and $4$. The first three congruences were conjectured by Chan in 2012, whereas the congruences modulo 125 are new. We also prove two more conjectural congruences of Chan for the coefficients of two Appell-Lerch sums.

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.

Congruences for the second order mock theta function B(q)

Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan’s two-parameter mock theta functions are the special cases of these functions.

We take a new approach to the generation of Jacobi theta function identities. It is complementary to the procedure which makes use of the evaluation of Parseval-like identities for elementary cylindrically-symmetric functions on computer holograms. Our method is more simple and explicit than this one, which was an outcome of the construction of neurocomputer architectures through the Heisenberg model.

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities.

For every partition and its conjugation , there is an important invariant , which denotes the number of different parts. That is , . We will derive a series of symmetric q-identities from the invariant in partition conjugation by studying modified Durfee rectangles. The extensive applications of the several symmetric q-identities in q-series [1] will also be discussed. Without too much effort one can obtain much well-known knowledge as well as new formulas by proper substitutions and elementary calculations, such as symmetric identities, mock theta functions, a two-variable reciprocity theorem, identities from Ramanujan’s Lost Notebook and so on.