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The (infinite) q-Pochhammer symbols are defined as

$$(a;q)_{\infty} := \prod_{k=0}^{\infty}(1-aq^{k})$$

I am interested a natural generalization of this symbol to multiple q's. In specific the 2 D and 3D cases which are defined as,

For 2 symbols q,r

$$(a;q,r)_{\infty} := \prod_{k=0}^{\infty}\prod_{l=0}^{\infty}(1-aq^{k}r^{l})$$

For 3 symbols q,r,s

$$(a;q,r,s)_{\infty} := \prod_{k=0}^{\infty}\prod_{l=0}^{\infty}\prod_{m=0}^{\infty}(1-aq^{k}r^{l}s^{m})$$

We can assume that $$|q|<1,|r|<1,|s|<1$$ for cases of interest.

I searched if they are discussed in the literature, but I could not find any leads. These generalizations naturally appear in the Grand Canonical partition function for fermions in a harmonic trap

Prathyush
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  • What is your question?? – Rounak Sarkar Sep 19 '21 at 07:48
  • It seems to me that the generalization would be $\textstyle\displaystyle{(a;q_1,\dots,q_n){\infty}:=\prod{k_1,\dots,k_n\geq 0}\left(1-a\prod_{i=1}^{n}q_i^{k_i}\right)}$ – Rounak Sarkar Sep 19 '21 at 07:52
  • @RounakSarkar I am interested in simplifications and insights into the said formulas and pointers to standard results from literature.

    That is a valid way to generalize, but I am not interested in that for my problems of interest.

     – Prathyush Sep 20 '21 at 10:00
  • Correction I misread and Did not notice the sub script i in $q_i$. Yes that is exactly what I am thinking about. – Prathyush Sep 20 '21 at 11:45
  • So that is a part of your answer. The other thing you want is basically some identities about it right?? – Rounak Sarkar Sep 20 '21 at 11:57
  • Yes, I am looking for Identities – Prathyush Sep 20 '21 at 12:01
  • Borcherds Products may be relevant: https://mathoverflow.net/questions/290794/wherefore-art-thou-a-borcherds-product https://arxiv.org/pdf/math/0404427.pdf – graveolensa Jan 25 '22 at 04:11

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