Questions tagged [q-series]
Questions that are based on, use, or include the q-series in their content or solutions.
118
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Integral representation of continued fractions [closed]
In $q$-series and allied areas, one try to express a continued fraction in terms of definite integrals. Ramanujan given integral representation of Rogers-Ramanujan continued fraction. In Ramanujan's ...
- 21
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How to find the Lambert series expansion of this function
Let's consider the function
\begin{align}
\frac{\eta ^m\left( q \right)}{\eta \left( q^m \right)}
\end{align}
here η is the Dedekind eta function $
\eta \left( q \right) =q^{\frac{1}{24}}\prod_{n=1}^{\...
- 69
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Prove that $\lim_{q\to 1} \frac{ (tq;q)_{\infty} }{ (tq^y;q)_{\infty} }=(1-t)^{y-1}$
I want to show that \begin{equation}
\lim_{q\to 1^{-}}\frac{(1-tq)(1-tq^2)\ldots }{(1-tq^y)(1-tq^{y+1})\ldots}=(1-t)^{y-1}\end{equation}
For the case where $y$ is a natural number is evident, I could ...
- 574
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1
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proof of $\frac{(1-q)(1-q^2)(1-q^3)..}{(1+q)(1+q^2)(1+q^3)..} = 1 - 2 q + 2 q^4 - 2 q^9 +~..$ [closed]
I came across the following fascinating identity involving infinite products and series:
\begin{eqnarray}
\frac{(1-q)(1-q^2)(1-q^3)..}{(1+q)(1+q^2)(1+q^3)..} &=&(1-2q+2q^2-2q^3+~..)(1-2q^2+2q^...
- 261
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35
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Need help in implementing q-SeriesToq-Product in Mathematica
In Mathematica Guidebook for symbolic computations (https://www.amazon.com/dp/0387950206/wolframresearch-20), in the Exercises, 30 (c)(p. 359), there is a question:
I have no clue how to implement ...
- 21
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votes
1
answer
87
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Proving Gauss’s summation theorem for the $q$-binomial coefficients
I am following Warren P. Johnson's "An Introduction to q-analysis". We are supposed to prove $$\binom{n+1}{k+1}_q=\sum_{m=k}^{n}q^{m-k}\binom{m}{k}_q$$ Here is my attempt.
We start off by ...
- 23
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An identity related to the series $\sum_{n\geq 0}p(5n+4)x^n$ in Ramanujan's lost notebook
While browsing through Ramanujan's original manuscript titled "The Lost Notebook" (the link is a PDF file with 379 scanned pages, so instead of a click it is preferable to download) I found ...
- 88.3k
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votes
1
answer
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Ramanujan's identity concerning a quotient of Dedekind's eta functions
In his paper On certain Arithmetical Functions (published in Transactions of the Cambridge Philosophical Society, XXII, No. 9, 1916, pp. 159-184) Ramanujan presents the following identities (as if ...
- 88.3k
4
votes
2
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189
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An identity related to $q$-series
While studying Ramanujan's theta functions, I encountered a q-series $(q;q)_\infty^2\phi(q)$. I calculated the first few terms of $(q;q)_\infty^2\phi(q)$ and observed that it seems to have the ...
- 907
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100
views
Closed form on variation of q-binomial theorem
The q-binomial theorem states that $$\sum_{n=0}^{\infty}\frac{\left(a;q\right)_n}{\left(q;q\right)_n}z^n =\frac{\left(az;q\right)_\infty}{\left(z;q\right)_\infty}$$. Is there a similar closed-form ...
3
votes
1
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325
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Numbers with a unique partition as a sum of two squares
The well known Ramanujan tau function $\tau(n)$ is defined as the nth Fourier coefficient of the modular discriminant
$\displaystyle \Delta(q)=q\prod_{m=1}^\infty (1-q^m)^{24} = \sum_{n=1}^\infty \tau(...
- 2,813
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vote
1
answer
288
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A certain conjectured criterion for restricted partitions
Given the number of partitions of $n$ into distinct parts $q(n)$, with the following generating function
$\displaystyle\prod_{m=1}^\infty (1+x^m) = \sum_{n=0}^\infty q(n)\,x^n\tag{1a}$
Which may be ...
- 2,813
3
votes
1
answer
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Exercise in the book "Basic Hypergeometric Series" of Gasper and Rahman
I am trying to solve Exercise 1.12.(iii) from the book "Basic Hypergeometric Series" of Gasper and Rahman (see the picture below). I am especially interested in the case where $c=ab$ and $n=...
- 1,538
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Ramanujan Identity Proof
Ramanujan defined a now famous q-series as
$$\sum_{n=-\infty}^{\infty}q^{n^2} = \left(-q;q^2\right)^2_{\infty}\left(q^2;q^2\right)_{\infty}$$
I wanted to prove this identity but I wasn't sure where to ...
- 166
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votes
2
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326
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Conjectured identity for the ratio of Ramanujan theta functions
Following Ramanujan, we define theta functions as follows $$\chi(q):=\prod_{n = 1}^{\infty}\left(1+q^{2n-1}\right),\\\phi(q)=\sum_{n=-\infty}^{\infty}q^{n^2},\\\displaystyle \psi(q)=\sum_{n = 0}^{\...
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