Congruences for sixth order mock theta functions $\lambda(q)$ and $\rho(q)$

1.   Introduction

In 1920, Ramanujan introduced 17 mock theta functions of odd order in his last letter to Hardy. In addition Ramanujan also gave the mock theta functions of order 6 that are listed in the Lost Notebook. In the study of the arithmetic properties of mock theta functions, many authors have found some congruence properties for their coefficients. For instance, Andrews et al. [1] found several congruences for the partition functions $p_{\omega}(n)$ and $p_{\nu}(n)$ corresponding to the mock theta functions $\omega(q)$ and $\nu(q)$ respectively, defined as:

where

for some positive integer $n$ and

with $|q|<1.$They proved congruences for modulo 2 and some infinite families of congruences for $p_{\omega}(n)$ and $p_{\nu}(n)$. In 2017, Fathima and Pore [4] obtained a number of congruences for $p_{\omega}(n)$ and $p_{\nu}(n)$ modulo 20 and some infinite families of congruences modulo 2. In the sequel, Baruah and Begum [2] in 2019 established many congruences for the same partition functions modulo powers of 5.

Zhang in 2018 proved some congruences for the sixth order mock theta function $\beta(q)$ shown below and also gave some conjectures in [8].

Brietzke, Silva, and Sellers [3] in 2019 found many arithmetic properties satisfied by the coefficients of the eighth order mock theta function $V_0(q)$ given as:

Silva and Sellers in [7] proved some congruence relations for the third order mock theta function $\xi(q)$ given by Gordon and McIntosh given below:

The main purpose of this paper is to study the arithmetic properties of the sixth order mock theta functions $\lambda(q)$ and $\rho(q)$ given by Ramanujan where the two mock theta functions are defined as:

Ramanujan also listed linear relations connecting the sixth order mock theta functions with each other as:

where ${\psi}_6(q)$ is the sixth order mock theta function

The proof technique of all the congruences for mock theta functions involves applying identities on the coefficients in arithmetic progressions, we use the same idea to prove infinite family of congruences modulo certain numbers of the form $2^{\alpha}{\cdot}3^{\beta}$ for $p_{\lambda}(n)$ and $p_{\rho}(n)$. The main results are found in Theorem 3.2–3.7 given in Section 3. Before proceeding towards the main theorems, we need some preliminary results given in Section 2 for proving the results in Section 3.

2.   Preliminaries

To shorten the notations, we define

for some positive integer $l$.

Now, we define Ramanujan's general theta function

Jacobi's triple product identity is defined as:

The special cases for $f(a,b)$ are:

In some of the proofs, we make use of the following identities:

The following lemma exhibits the 3-dissection of $\psi(q)$ and ${1/{\varphi(-q)}}.$

Lemma 2.1. We have

Proof. Identity (12) is equation (14.3.3) of [5]. Identity (13) comes from [6] shown in equation (2).

Lemma 2.2. We have

Proof. The above identity (14) is (22.1.13) in [5]. And (15) follows from (14) by replacing $q$ by $-q$ and using

From the binomial theorem, for any positive integer $l$ and prime $p$, we have

3.   Congruence relations for $\lambda(q)$ and $\rho(q)$

We first prove the 2-dissection of ${\lambda}(q)$.

Theorem 3.1. We have

Proof. To prove the above dissections, consider (5) and replacing $q$ by $-q$ in (5), we have

where the last equality follows from (14). Extracting even and odd terms from above equation, we get

Replace $q^2$ by $q$ in (18) to arrive at (16). Divide (19) by $q$ and replace $q^2$ by $q$ to obtain (17).

Theorem 3.2. We have

Proof. According to (16), we have

Therefore, we complete the proof.

Theorem 3.3. We have

Proof. According to (16), we have

Therefore, we complete the proof.

Corollary 1. Let $p>3$ be a prime and $r$ an integer such that $8r+1$ is a quadratic non-residue modulo $p$. Then for all $n \geq 0$,

Proof. As

Thus, $2r \equiv k^2+k\;({\rm mod}\; p)$ or $8r+1 \equiv {(2k+1)}^2\pmod{p}$ which contradicts the fact that $8r+1$ is a quadratic non-residue modulo $p$. Therefore, from Theorem 3.2,

Similarly,

then $2r \equiv 9k^2-3k\pmod{p}$ or $8r+1 \equiv {(6k-1)}^2\pmod{p}$ which contradicts the fact that $8r+1$ is a quadratic non-residue modulo $p$. Therefore, from Theorem 3.3,

From (22) and (23), we readily arrive at the main result.

Since gcd(6, $p$) = 1, among the $p-1$ residues modulo $p$, there are $(p-1)/2$ residues $r$ for which $8r+1$ is a quadratic non-residue modulo $p$. So the above result leads us to a number of congruences for different primes $p>3$ as shown below:

Theorem 3.4. We have

Proof. From (16), we get

Using (9) and (10), we have

Using Lemma 2.1, we obtain

Extracting the terms involving $q^{3n}, q^{3n+1},q^{3n+2}$ from above equation, we have

To prove (25), dividing (31) by $q^2$ and replacing $q^3$ by $q$, we have

which can also be written as

The above equation readily implies (25). Consider (30), dividing by $q$ and replacing $q^3$ by $q$, we have

which proves (24). Now using (13) in above equation,

Extracting the terms involving $q^{3n},q^{3n+1},q^{3n+2}$ from (33), we have

Dividing (35) and (36) by $q$ and $q^2$ respectively, replacing $q^3$ by $q$, we arrive at

From the above equations, we get (26) and (27).

Corollary 2. We have

Now we present the infinite families of congruences modulo 12.

Theorem 3.5. For prime $p \geq 5$, we have

where $i = 1,2,\cdots ,p-1.$

Proof. From (29), replacing $q^3$ by $q$, we have

Reducing modulo 4,

By (21), we have

Extracting the terms involving $q^{3n}$ and replacing $q^3$ by $q$, we have

From (39) and (40), we have

This implies that $p_{\lambda}(6k)\equiv 0 \pmod{12}$ unless $k$ is a pentagonal number, or equivalently, unless $24k+1$ is a square. Now letting $k = p^2n+pi+(p^2-1)/24$ where $i = 1,2,\cdots,(p-1)$ and $p$ is a prime, we have that $24k+1 = 24p^2n+24pi+p^2$, and this is evidently not a square since $p^2$ divides the first and third terms but not the middle term. Thus $p_{\lambda}(6k) = p_{\lambda}(6p^2n+6pi+(p^2-1)/4)\equiv 0\pmod{12}.$

Theorem 3.6. For $m \geq 1$, we have

Theorem 3.7. For $m \geq 1$, we have

Proof of Theorems 3.6 and 3.7. The proof for the above theorems follows by induction. Let us first prove the first step of induction, for $m = 1.$

From (32), we have

Reducing modulo 12, we have

which proves (41) for $m = 1.$ Using (13), we get

or

Extracting the terms involving $q^{3n},q^{3n+1},q^{3n+2}$ from above, we have

Dividing (52) and (53) by $q$ and $q^2$ respectively then replacing $q^3$ by $q$, we have

Here (54) and (55) proves (46) and (49), respectively for $m = 1.$ Replacing $q^3$ by $q$ in (51), we have

The above equation proves (43) for $m = 1$ and it can also be written as:

Using Lemma 2.1, we have

Extracting the terms involving $q^{3n},q^{3n+1},q^{3n+2}$ from above, we get

Replacing $q^3$ by $q$ in (57), we have

which proves (47) for $m = 1$. Now, dividing (58) and (59) by $q$ and $q^2$ respectively then replacing $q^3$ by $q$,

The above congruences imply (42) and (45) for $m = 1.$ Consider

or

Extracting the terms involving $q^{3n},q^{3n+1},q^{3n+2}$ from above, we arrive at

Replacing $q^3$ by $q$ in (60), we obtain

which proves (44) for $m = 1$.

Similar to (56), extracting the terms involving $q^{3n},q^{3n+1},q^{3n+2}$, we ultimately get

Here (63) is the case when $m = 1$ in (48). For the next step of induction, let us suppose that (41)-(49) holds true for $m = k$. Then, for $m = (k+1)$, we prove the relations in similar manner mentioned above starting from (50) (taking $m = k$) and obtain (64) ($m = k+1$). Same process follows for other parts.

Corollary 3. For $m \geq 1$, we have

Now, we prove the 2-dissection of $\rho(q)$.

Theorem 3.8. We have

Proof. From (6), we have

Substituting the value from (15), we have

Extracting even and odd terms from above equation, we easily arrive at (69) and (70).

Finally on comparing (16) and (69), we arrive at the following theorem.

Theorem 3.9. We have

The above theorem yields that all the results shown above for $\lambda(q)$ also hold for ${\rho}(q)$.

4.   Conclusion and discussions

We have provided elementary proofs of numerous infinite family of congruences satisfied by $p_{\lambda}(n)$ and $p_{\rho}(n)$. We did not carry out a computer search for congruences, and so we are unaware whether other congruences hold beyond the ones we prove in this paper, but certainly there is a possibility to explore more in this direction.

Acknowledgments

The first author is supported by UGC, under grant Ref No. 971/ (CSIR-UGC NET JUNE 2018) and the second author is supported by SERB-MATRICS grant MTR/2019/000123. We would like to thank the two referees for carefully reading our paper and offering corrections and helpful suggestions.