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In this paper, the following result is proved.

For any prime $p$, all the Fourier coefficients of

$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$

are non-negative.

As seeing this sequence, I conjectured

all the Fourier coefficients of

$$\eta(q^m)^m / \eta(q) = q^{\frac{m^2-1}{12}} \prod_{n=1}^\infty (1 - q^{mn})^m (1 - q^{n})^{-1}$$

are non-negative for any positive integer $m$, .

Is this conjecture true?

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Yes this is true and these Fourier coefficients actually enumerate some combinatorial objects called $m$-cores. These are partitions with no hooklength divisible by $m$. In fact for $m\geq 4$ these coefficients are positive. This last result was proved in "Defect zero $p-$blocks for finite simple groups", by Granville and Ono.

I want to also point out that Saito's conjecture on nonnegativity of particular eta quotients (which was the subject of the paper you link to) has been fully proven in "K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products" by Berkovich and Garvan.

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  • $\begingroup$ By the way, "$t$-core" is a more common term when $t$ isn't specified. $\endgroup$
    – Douglas Zare
    Commented May 20, 2017 at 0:44
  • $\begingroup$ @DouglasZare I thought it largely depends on the field. For example I almost always see "t-core" in number theory/hypergeometric literature, and "n-core" in algebraic geometry/representation theory literature. Either way, here I was just trying to match the OP's choice. $\endgroup$
    – Gjergji Zaimi
    Commented May 20, 2017 at 1:14
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    $\begingroup$ Yes, I just wanted to give a term that will produce more hits than $m$-core in a search. $\endgroup$
    – Douglas Zare
    Commented May 20, 2017 at 1:21

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