Sixth Order Mock Theta Functions Research Articles

Congruences for the coefficients of a pair of third and sixth order mock theta functions

Congruences for the coefficients of a pair of third and sixth order mock theta functions

Combinatorics of some fifth and sixth order mock theta functions

<p style='text-indent:20px;'>The goal of this paper is to provide a new combinatorial meaning to two fifth order and four sixth order mock theta functions. Lattice paths of Agarwal and Bressoud with certain modifications are used as a tool to study these functions.</p>

Proofs of some conjectures of Chan on Appell–Lerch sums

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.

Congruences for partition functions related to mock theta functions

Partitions associated with mock theta functions have received a great deal of attention in the literature. Recently, Choi and Kim derived several partition identities from the third and sixth order mock theta functions. In addition, three Ramanujan-type congruences were established by them. In this paper, we present some new congruences for these partition functions.

Partition identities from third and sixth order mock theta functions

In a recent paper, the first author gave a connection between bilateral basic hypergeometric series and mock theta functions, which led to many new identities involving mock theta functions. This paper is a sequel, and our goal is to provide partition theoretic properties of new identities involving third or sixth order mock theta functions.

On identities involving the sixth order mock theta functions

We present q q -series proofs of four identities involving sixth order mock theta functions from Ramanujan’s lost notebook. We also show how Ramanujan’s identities can be used to give a quick proof of four sixth order identities of Berndt and Chan.

PARTIAL SECOND ORDER MOCK THETA FUNCTIONS, THEIR EXPANSIONS AND PADE APPROXIMANTS

By proving a summation formula, we enumerate the expansions for the mock theta functions of order 2 in terms of partial mock theta functions of order 2, 3 and 6. We show a relation between Ramanujan`s -function and his sixth order mock theta functions. In addition, we also give the continued fraction representation for and 2nd order mock theta functions and approximants.

Sixth order mock theta functions

Two new mock theta functions of the sixth order are defined. The main theorem in this paper (Theorem 1.1) provides four transformation formulas relating the new mock theta functions with Ramanujan's mock theta functions of the sixth order. Two further representations of the new mock theta functions are established. Lastly, a hitherto unproved entry from Ramanujan's lost notebook related to sixth order mock theta functions is proved.

Identities for Ramanujan's Sixth-Order Mock Theta Functions

Identities for Ramanujan's Sixth-Order Mock Theta Functions

On some relations for the rank moduli 9 and 12

In this paper we prove some identities, conjectured by Lewis, concerning the rank moduli 9 and 12, which are similar to Dyson's identities for the rank moduli 5 and 7 which give a combinatorial explanation to Ramanujan's partition congruences. For this we use multisection of series and some identities for the third and sixth order Mock theta functions, in such a way that all the identities for a given modulus reduce to a single theta identity.

Ramanujan's “lost” notebook VII: The sixth order mock theta functions

Ramanujan's “lost” notebook VII: The sixth order mock theta functions