The Ramanujan Journal

, Volume 33, Issue 1, pp 23–53 | Cite as

The power series expansion of certain infinite products \(q^{r}\prod_{n=1}^{\infty}(1-q^{n})^{a_{1}}(1-q^{2n})^{a_{2}}\cdots(1-q^{mn})^{a_{m}}\)

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Abstract

Let ℕ, \(\mathbb{N}_{0}\), ℤ, ℚ, and ℂ denote the sets of positive integers, nonnegative integers, integers, rational numbers, and complex numbers, respectively. If \(f(q)\) is a complex-valued function with
$$f(q)= \sum_{n=0}^{\infty} f_{n} q^{n} \quad\bigl(q \in\mathbb{C}, \vert q \vert <1 \bigr) $$
we define
$$\bigl[f(q)\bigr]_{n}:=f_{n} \quad(n \in\mathbb{N}_{0}). $$
For \(k \in\mathbb{N}\) we define
$$E_{k}:= \prod_{n=1}^{\infty} \bigl(1-q^{kn}\bigr) \quad\bigl(q \in\mathbb{C}, \vert q \vert <1 \bigr). $$
We show how modular equations of a special form can be used in conjunction with the representation numbers of certain quadratic forms to determine
$$\bigl[q^{r}E_{1}^{a_{1}}\cdots E_{m}^{a_{m}} \bigr]_{n} \quad(r \in\mathbb{N}_{0},m \in \mathbb{N},a_{1}, \ldots,a_{m} \in\mathbb{Z}) $$
for certain products \(q^{r}E_{1}^{a_{1}}\cdots E_{m}^{a_{m}}\). For example, we show that
$$\biggl[q^{2}\frac{E_{1}^{4}E_{16}^{4}}{E_{2}^{2}E_{8}^{2}} \biggr]_{n} = \begin{cases} 0 & \mbox{if}\ n \equiv1\ (\mbox{mod}\ 4),\\ - \sigma(N) & \mbox{if}\ n \equiv3\ (\mbox{mod}\ 4),\\ \sigma(N) & \mbox{if}\ n \equiv2\ (\mbox{mod}\ 4),\\ 4\sigma(N) & \mbox{if}\ n \equiv4\ (\mbox{mod}\ 8),\\ 0 & \mbox{if}\ n \equiv0\ (\mbox{mod}\ 8), \end{cases} $$
where \(N\) denotes the odd part of the positive integer \(n\) and
$$\sigma(n): =\sum_{ \begin{array}{c} \scriptstyle d \in\mathbb{N}\\[-3pt] \scriptstyle d \mid n \end{array}} d. $$

Keywords

Infinite products Quadratic forms Representations Theta functions Modular equations 

Mathematics Subject Classification

11E25 11F27 11B65 05A19 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityNorth YorkCanada
  2. 2.Centre for Research in Algebra and Number Theory, School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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