1 Introduction

In 2008, Andrews introduced a weighted count on partitions given by counting in a partition the number of times the smallest part appears. This function, denoted by $\operatorname {spt}(n)$ and defined as the number of smallest parts among all the partitions of n, is known to possess an intriguing modular structure and to satisfy the Ramanujan-type congruences

$$ \begin{align*} \operatorname{spt}(5n + 4) & \equiv 0\, \pmod{5}, \\\operatorname{spt}(7n+5) & \equiv 0\, \pmod{7}, \\\operatorname{spt}(13n+6) & \equiv 0\, \pmod{13}, \end{align*} $$

among other beautiful combinatorial and arithmetic properties. The origin of these arithmetic properties is attributed to the connection between its generating function and harmonic Maass forms.

Let $N,m,n$ be nonnegative integers. Throughout, we use the following standard notation:

$$ \begin{align*} (a)_N = (a;q)_N &:= \prod_{k=0}^{N-1}(1-aq^k), \\(a)_{\infty} = (a;q)_{\infty} &:= \lim_{N \rightarrow \infty}(a)_N\quad\text{where}\ \lvert q\rvert<1. \end{align*} $$

The basic hypergeometric series ${}_{r+1}\phi _r$ is defined by

$$ \begin{align*} {}_{r+1}{}\phi_{r} \bigg( \genfrac{}{}{0pt}{}{a_1,a_2,\ldots,a_{r+1}} {b_1,b_2,\ldots,b_{r}}; {q,z} \bigg):=\sum_{n=0}^\infty\frac{(a_1;q)_n(a_2;q)_n\cdots(a_{r+1};q)_n}{(q;q)_n(b_1;q)_n(b_2;q)_n\cdots(b_{r};q)_n}z^n. \end{align*} $$

When we do not specify the base as in $(a)_n$ , the base is q.

By standard combinatorial arguments, the generating function of the smallest parts function (spt-function) due to Andrews is given by

$$ \begin{align*}\sum_{n=1}^\infty\mathrm{spt}(n)q^n=\sum_{n=1}^\infty\frac{q^{n}}{(1-q^{n})^2(q^{n+1};q)_\infty}.\end{align*} $$

Starting from this, Andrews’ identity for the spt-function, which relates the spt-function to the partition function and the second Atkin–Garvan rank moment, is obtained using Watson’s q-analogue of Whipple’s theorem ( ${}_{8}\phi _{7}$ -transformation):

$$ \begin{align*} & {}_8\phi_7\bigg[ \begin{matrix} a,\; q\sqrt{a},\; -q\sqrt{a},\; b,\; c,\; d,\; e,\; q^{-N} \\ \sqrt{a},\; -\sqrt{a},\; aq/b,\; aq/c,\; aq/d,\; aq/e,\; aq^{N+1} \end{matrix} ;\, q,\; \frac{a^2 q^{N+1}}{bcde} \bigg] \\ &\quad = \frac{(aq,\; aq/de;q)_N}{(aq/d,\; aq/e;q)_N} \; {}_4\phi_3\bigg[ \begin{matrix} aq/bc,\; d,\; e,\; q^{-N} \\ aq/b,\; aq/c,\; de q^{-N}/a \end{matrix} ;\, q,\; q \bigg]. \end{align*} $$

The spt-function identity of Andrews [Reference Andrews3, Theorem 4] is

(1.1) $$ \begin{align} \sum_{n=1}^\infty\mathrm{spt}(n)q^n&=\frac{1}{(q;q)_\infty}\sum_{n=1}^\infty\frac{nq^{n}}{1-q^{n}} +\frac{1}{(q;q)_\infty}\sum_{n=1}^\infty\frac{(-1)^n(1+q^n)}{(1-q^{n})^2}q^{{3n}(n+1)/2}. \end{align} $$

The first sum on the right-hand side is the generating function for $np(n)$ , where $p(n)$ denotes the number of partitions of n and the second is the generating function for $\tfrac 12N_2(n)$ , where

$$ \begin{align*}N_2(n)=\sum\limits_{m=-\infty}^{\infty}m^2N(m,n)\end{align*} $$

is the second Atkin–Garvan rank moment. Here, $N(m,n)$ denotes the number of partitions of n with rank m. The rank of a partition is defined to be the largest part minus the number of parts. For instance, the rank of the partition $5+3+3+1$ is $5-4=1$ . See [Reference Atkin and Swinnerton-Dyer9, Reference Dyson11] for more on the rank and [Reference Atkin and Garvan8] for more on rank moments.

In 2015, Andrews et al. [Reference Andrews, Dixit and Yee6] obtained new partition theoretic interpretations for $q\omega (q)$ and $\nu (-q)$ , where $\omega (q)$ and $\nu (q)$ are the third-order mock theta functions

$$ \begin{align*}\omega(q)=\sum_{n=0}^{\infty}\frac{q^{2n(n+1)}}{(q;q^{2})_{n+1}} \quad\text{and}\quad \nu(q)=\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{(-q;q^{2})_{n+1}}, \end{align*} $$

among a wide range of other results. Perhaps the most striking of these are the congruences

$$ \begin{align*} \operatorname{spt}_{\omega}(5n+3) & \equiv 0\, \pmod{5}, \\ \operatorname{spt}_{\omega}(10n+7) & \equiv 0\, \pmod{5}, \\ \operatorname{spt}_{\omega}(10n+9) & \equiv 0\, \pmod{5}, \end{align*} $$

where $\operatorname {spt}_{\omega }(n)$ is the total number of smallest parts in all of the partitions of n in which each odd part is less than twice the smallest part. These three divisibility properties are a consequence of a q-series identity obtained using the following limiting case of a ${}_{10}\phi _9$ -transformation due to Bailey (limiting case due to Andrews [Reference Andrews2, (2.10)]):

(1.2) $$ \begin{align} & \lim_{N\to\infty} {}_{10}\phi_{9}\Bigg( \begin{matrix} a,\ q^{2}\sqrt{a},\ -q^{2}\sqrt{a},\ p_{1},\ p_{1}q,\ p_{2},\ p_{2}q,\ f,\ q^{-2N},\ q^{-2N+1} \\ \sqrt{a},\ -\sqrt{a},\ \dfrac{aq^{2}}{p_{1}},\ \dfrac{aq}{p_{1}},\ \dfrac{aq^{2}}{p_{2}},\ \dfrac{aq}{p_{2}},\ \dfrac{aq^{2}}{f},\ aq^{2N+2},\ aq^{2N+1} \end{matrix}; q^{2},\ \dfrac{a^{3}q^{4N+3}}{p_{1}^{2}p_{2}^{2}f} \Bigg) \notag \\& \quad = \frac{(aq;q)_\infty\bigg(\dfrac{aq}{p_{1}p_{2}};q\bigg)_\infty} {\bigg(\dfrac{aq}{p_{1}};q\bigg)_\infty\bigg(\dfrac{aq}{p_{2}};q\bigg)_\infty} \sum_{n=0}^{\infty} \frac{(p_{1};q)_n (p_{2};q)_n \bigg(\dfrac{aq}{f};q^{2}\bigg)_n} {(q;q)_n (aq;q^{2})_n \bigg(\dfrac{aq}{f};q\bigg)_n} \bigg(\dfrac{aq}{p_{1}p_{2}}\bigg)^n. \end{align} $$

The q-series identity due to Andrews et al. [Reference Andrews, Dixit and Yee6, Lemma 6.1] is an analogue of the Andrews spt-function identity (1.1):

(1.3) $$ \begin{align} &\sum_{n=1}^\infty\frac{q^{n}}{(1-q^{n})^2(q^{n+1};q)_n(q^{2n+2};q^2)_\infty} \\ \nonumber &\quad =\frac{1}{(q^2;q^2)_\infty}\sum_{n=1}^\infty\frac{nq^{n}}{1-q^{n}}+\frac{1}{(q^2;q^2)_\infty} \sum_{n=1}^\infty\frac{(-1)^n(1+q^{2n})}{(1-q^{2n})^2}q^{n(3n+1)}. \end{align} $$

In a recent paper, Andrews and Bachraoui [Reference Andrews and Bachraoui4] established new partition theoretic interpretations, showing that $q\omega (q)$ and $\nu (-q)$ are generating functions for two-colour partitions in which the even or the odd parts may occur only in one colour, without appeal to such analogues. Additionally, even more recently, a newly discovered analogue of Andrews spt-function identity by Andrews and Kumar [Reference Andrews and Kumar7, Theorem 1.3] has led them to discover a new partition theoretic interpretation of the other third-order mock theta function

$$ \begin{align*}f(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(-q;q)_n^{2}},\end{align*} $$

which in turn relates partitions with positive odd rank and two-colour partitions [Reference Andrews and Kumar7, Theorems 1.1 and 1.2]. This new analogue is obtained using the limiting case of another ${}_{10}\phi _9$ -transformation due to Bailey (limiting case due to Andrews [Reference Andrews2, (2.9)]),

$$ \begin{align*} &\lim_{N\to\infty} {}_{10}\phi_{9}\Bigg( \begin{matrix} a,\ q\sqrt{a},\ -q\sqrt{a},\ b,\ r_{1},\ -r_{1},\ r_{2},\ -r_{2},\ q^{-N},\ -q^{-N} \\ \sqrt{a},\ -\sqrt{a},\ \dfrac{aq}{b},\ \dfrac{aq}{r_{1}},\ -\dfrac{aq}{r_{1}},\ \dfrac{aq}{r_{2}},\ -\dfrac{aq}{r_{2}},\ aq^{N+1},\ -aq^{N+1} \end{matrix}; q,\ -\dfrac{a^{3}q^{3+2N}}{b r_{1}^{2} r_{2}^{2}} \Bigg) \notag \\& \quad = \frac{(a^{2}q^{2};q^{2})_{\infty} \bigg(\dfrac{a^{2}q^{2}}{r_{1}^{2}r_{2}^{2}};q^{2}\bigg)_{\infty}} {\bigg(\dfrac{a^{2}q^{2}}{r_{1}^{2}};q^{2}\bigg)_{\infty} \bigg(\dfrac{a^{2}q^{2}}{r_{2}^{2}};q^{2}\bigg)_{\infty}} \sum_{n=0}^{\infty} \frac{(r_{1}^{2};q^{2})_{n}(r_{2}^{2};q^{2})_{n}(-aq/b;q)_{2n}} {(q^{2};q^{2})_{n}(a^{2}q^{2}/b^{2};q^{2})_{n}(-aq;q)_{2n}} \bigg(\dfrac{a^{2}q^{2}}{r_{1}^{2}r_{2}^{2}}\bigg)^{n}, \end{align*} $$

and is given by

(1.4) $$ \begin{align} &\sum_{n=1}^\infty\frac{q^{2n}}{(1-q^{2n})^2(-q^{n+1};q)_n(q^{n+1};q)_\infty} \nonumber \\ &\quad =\frac{1}{(q;q)_\infty}\sum_{n=1}^\infty\frac{nq^{2n}}{1-q^{2n}}+\frac{1}{(q;q)_\infty}\sum_{n=1}^\infty \frac{(-1)^n(1+q^n)}{(1-q^{2n})^2}q^{{3n}(n+1)/2}. \end{align} $$

So far, these analogues have been produced using ${}_{10}\phi _9$ -transformations. We derive more such analogues using higher parameter basic hypergeometric transformations.

First, we consider the following ${}_{12}\phi _{11}$ -transformation due to Verma and Jain [Reference Verma and Jain14, (1.5)], [Reference Sills13, (3.5)] and use it to arrive at another analogue of Andrews’ spt-function identity. The identity is

(1.5) $$ \begin{align} &{}_{12}{}\phi_{11} \Bigg( \genfrac{}{}{0pt}{}{a,q\sqrt{a},-q\sqrt{a},x,\omega x,\omega^2 x,y,\omega y,\omega^2 y, q^{-N},\omega q^{-N},\omega^2 q^{-N}} {\sqrt{a},-\sqrt{a},\frac{aq}{x},\frac{aq}{\omega x},\frac{aq}{\omega^2 x},\frac{aq}{y},\frac{aq}{\omega y},\frac{aq}{\omega^2 y},aq^{N+1},\omega^{2}aq^{N+1},\omega aq^{N+1}}; {q;\frac{a^4q^{4+3N}}{x^3y^3}} \Bigg) \nonumber \\& \quad =\frac{(a^3q^3;q^3)_N(a^3q^3/x^3y^3;q^3)_N}{(a^3q^3/x^3;q^3)_N(a^3q^3/y^3;q^3)_N} \nonumber \\&\qquad \cdot {}_{6}{}\phi_{5} \Bigg( \genfrac{}{}{0pt}{}{q^{-3N},x^3,y^3,aq,aq^2,aq^3} {(aq)^{{3}/{2}},-(aq)^{{3}/{2}},a^{{3}/{2}}q^3,-a^{{3}/{2}}q^3,{x^3y^3q^{-3N}}/{a^3}}; {q^3;q^3}\Bigg). \end{align} $$

This identity is one of several such identities between basic hypergeometric functions on different bases developed by the authors in [Reference Verma and Jain14] leading to a number of new identities of the Rogers–Ramanujan type.

Theorem 1.2 (First analogue of Andrews’ spt-function identity)

We have

$$ \begin{align*} &\sum_{n=1}^\infty\frac{q^{3n}}{(1-q^{3n})^2(q^{3(n+1)};q^3)_{n}(q^{3n+1};q)_{\infty}} \\ &\quad =\frac{1}{(q;q)_{\infty}}\sum_{n=1}^\infty\frac{nq^{3n}}{1-q^{3n}}+ \frac{1}{(q;q)_{\infty}}\sum_{n=1}^\infty\frac{(-1)^n(1+q^n)}{(1-q^{3n})^2}q^{{n(3n+5)}/{2}}. \end{align*} $$

Next, we consider the other ${}_{12}\phi _{11}$ -transformation due to Verma and Jain [Reference Verma and Jain14, (1.6)], [Reference Sills13, (3.6)], to arrive at yet another analogue of Andrews’ spt-function identity. This identity is

(1.6) $$ \begin{align} &{}_{12}{}\phi_{11} \Bigg( \genfrac{}{}{0pt}{}{a,q^3\sqrt{a},-q^3\sqrt{a},x,xq,xq^2,y,yq,yq^2, q^{-N},q^{-N+1},q^{-N+2}} {\sqrt{a},-\sqrt{a},\frac{aq^3}{x},\frac{aq^2}{x},\frac{aq}{x},\frac{aq^3}{y},\frac{aq^2}{y},\frac{aq}{y},aq^{N+3},aq^{N+2},aq^{N+1}}; {q^3;\frac{a^4q^{3+3N}}{x^3y^3}}\Bigg) \nonumber \\ &\quad =\frac{(aq;q)_N(aq/xy;q)_N}{(aq/x;q)_N(aq/y;q)_N}{}_{6}{}\phi_{5} \Bigg( \genfrac{}{}{0pt}{}{\sqrt[3]{a},\omega \sqrt[3]{a},\omega^2 \sqrt[3]{a},x,y,q^{-n}} {\sqrt{a},-\sqrt{a},\sqrt{aq},-\sqrt{aq},\frac{xy}{a}q^{-n}}; {q;q}\Bigg). \end{align} $$

Theorem 1.3 (Second analogue of Andrews’ spt-function identity)

We have

$$ \begin{align*} &\sum_{n=1}^\infty\frac{q^n}{(1-q^n)(q^{3n};q^3)_{\infty}(-q;q)_{n-1}(q;q^2)_n} \\ &\quad =\frac{1}{(q^3;q^3)_{\infty}}\sum_{n=1}^\infty\frac{nq^{n}}{1-q^{n}}+ \frac{1}{(q^3;q^3)_{\infty}}\sum_{n=1}^\infty\frac{(-1)^n(1+q^{3n})}{(1-q^{3n})^2}q^{{3n(3n+1)}/{2}}. \end{align*} $$

The paper contains two more sections. Section 2 is devoted to proofs of the two theorems stated above. In the final section, we discuss possible questions arising from our work.