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2
votes
0answers
45 views

q-Series no clue

I see in some table pages sum like $$\sum _{k=1}^{\infty } \frac{1}{\left(q^k+2\right)^2 \left(q^k+1\right)^2 \left(q^k+3\right)}\text{==}$$ $$\frac{2 \log (q) \left(-9 \psi _q^{(0)}\left(-\frac{\log ...
2
votes
0answers
24 views

Find simple proofs of the two $q$-series identities

When I read an article, I found the following two $q$-series identities very interesting $$ \sum_{k=-\infty}^{+\infty}(-1)^k{2n\brack n+2k}q^{2k^2}=(-q;q^2)_n, $$ $$ ...
3
votes
0answers
84 views

Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's ...
4
votes
0answers
76 views

Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
2
votes
0answers
27 views

double integrals on quantum calculus

I need references or book recommendations to find properties of double integrals on quantum calculus. Especially i need analogue of Fubini's theorem on q-calculus.
2
votes
0answers
55 views

improper integrals in q-calculus

In quantum calculus is this equality possible for improper integrals? $\lim_{x\to\infty}\int_0^xf(t)d_qt=\int_0^\infty f(x)d_qx$
2
votes
0answers
43 views

How to prove the $q$-series identity?

How to prove the following identity: $$\sum_{n\ge0}\frac{2q^{n^{2}+n}}{(q)_{n}^{2}(1+q^{n})}=\sum_{n\ge0}\frac{q^{n^{2}+n}}{(q)_{n}^{2}(1-q^{2n+2})}$$
1
vote
0answers
44 views

q-theta function and their properties

I want to compute the residue integral for the $q$-theta function, and derive its properties. First, I'll briefly explain the definition \begin{align} & ...
1
vote
1answer
54 views

octagonal number theorem $q$-Pochhammer symbol expression

Setting the exponents of this analogue of the series in Euler's Pentagonal Number theorem to be the octagonal numbers: $$U(q)= \sum_{n\in\mathbb{Z}} (-1)^{n}q^{n(6n-4)/2}$$ in mpmath: ...
1
vote
0answers
35 views

a question on sum of q_binomials

I was trying to calculate something and at some point I get the following sum: \begin{equation} \sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...
3
votes
0answers
79 views

Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
8
votes
2answers
319 views

An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$

Let $c\in\mathbb{R}\setminus\{ 1\}$, $c>0$. Let $U_i = \left\lbrace U_{i, 0}, U_{i, 1}, \dots \right\rbrace$, $U_i\in\mathbb{R}^\mathbb{N}$. We know that ...
5
votes
0answers
76 views

About $\prod{\left(1-q^n\right)^{5}}$

Is there a result about the non-vanishing of coefficients of $$\prod_{n=1}^{+\infty}{\left(1-q^n\right)^{5}}=1-5q+5q^2+10q^3-15q^4-6q^5-5q^6+25q^7+15q^8-20q^9+\cdots \text{ ?}$$ Thanks !
4
votes
1answer
209 views

A question on a sum of $q$-binomial coefficients

I am trying to enumerate a certain quantity and at some point I get the following sum: \begin{equation} \sum_{i=0}^{m}{m \brack i}_q \sum_{j=0}^{n-m} q^{j(m-i)}{n-m \brack j}_q \sum_{k=0}^{r} ...
3
votes
1answer
90 views

$q$-series identity

I have to prove the following identity: $$\sum_{n\geq 0} (-1)^n(2n+1)q^{\frac{n(n+1)}{2}} = (q;q)_\infty^3$$ where $(a;q)_\infty = \prod_{i\geq 0}1-aq^i$ is the $q$-Pochhammer symbol. In my notes the ...
1
vote
1answer
101 views

Ramanujan Notebook Part 1 (1.16): $\sum q^{n^2} = (-q;q^2)_\infty^2(q^2;q^2)_\infty=\frac{(-q;-q)_\infty}{(q;-q)_\infty}$

I am having trouble with proving a statement in Ramanujan's Lost Notebook Part 1 (1.16). The statement is as follows: $\varphi(q)=f(q,q)=\sum_{n=-\infty}^\infty q^{n^2} = ...
2
votes
2answers
1k views

A problem with the geometric series and matrices?

Let $n$ be a positive integer. Let $A$ be a square matrix. Let $I$ be the identity matrix with the same size as $A$. I want to simplify $f_n(A) = I + A + A^2 + A^3 + A^4 + \cdots + A^n$ Now I know ...
4
votes
1answer
182 views

Combinatorial Identity

I have to validate the following identity which is defined: $$ \sum_{k=1}^n (-1)^{k-1}*q^{\frac{k(k-1)}{2}} *\frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1 $$ where $0<q<1$. I ...
2
votes
1answer
48 views

Q-series identities #2

Prove the following $$\frac{1}{(z;q)_{\infty}} = \sum^{\infty}_{k=0} \frac{z^k}{(q;q)_k}$$ I am looking for a proof that doesn't involve the q-binomial theorem . where $$(a;q)_k = ...
3
votes
1answer
176 views

q-series identities

Here is my first question in this site Prove the following $$\lim_{q \to 1 }\frac{(a)_{\infty}}{(aq^x)_{\infty}}=(1-a)^x$$ for $x$ an integer it was an easy task but it was generally for any ...
1
vote
2answers
60 views

A q-series related to adjoint representation of lie group

What is the infinite sum expansion in the degree of q? $$ \exp \left[\sum_{n=1}^\infty \frac{1}{n}\frac{2q^n }{1-q^n } \right] = \prod_{n=1}^\infty \exp \left[ \frac{1}{n}\frac{2q^n }{1-q^n } \right] ...
4
votes
1answer
405 views

How to evaluate this infinite product

How to evaluate this one $$\prod\limits_{n=1}^{\infty }{\left( 1-\frac{1}{{{2}^{n}}} \right)}$$
2
votes
2answers
215 views

$\log P(e^{-2\pi t}) - \log P(e^{-2\pi /t}) = \frac{\pi}{12} \Bigl( \frac{1}{t} - t \Bigr)+ \frac{1}{2} \log t$

Let $t$ be a real $>0$. Let $P(x) = \prod_{n=1}^\infty \dfrac{1}{1-x^n}$ $\log P(e^{-2\pi t}) - \log P(e^{-2\pi /t}) = \frac{\pi}{12} \Bigl( \frac{1}{t} - t \Bigr) + \frac{1}{2} \log t$ How to ...
4
votes
4answers
2k views

Proving q-binomial identities

I was wondering if anyone could show me how to prove q-binomial identities? I do not have a single example in my notes, and I can't seem to find any online. For example, consider: ${a + 1 + b \brack ...
13
votes
2answers
339 views

Combinatorial interpretation of this identity of Gauss?

Gauss came up with some bizarre identities, namely $$ \sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}=\prod_{k\geq 1}\frac{1-q^k}{1+q^k}. $$ How can this be interpreted combinatorially? It strikes me as being ...
3
votes
0answers
104 views

factoring infinite products of $q$-series with constant term equal to 1

I was thinking about the following infinite product: $$\prod_{n=0}^{\infty} \frac{ae^{-2n}+be^{-n}+c}{c}$$ The right way of generalizing it is to think in terms of $q$-Pochhammer symbols. If $r_{1}$ ...
2
votes
1answer
129 views

Closed form for $\sum_{m \geq 1} (-1)^m q^{m(m+1)/2 + m \Delta}$?

Is there a useful closed form for the following series ($|\Delta|$ is a small integer)? $$f(q,\Delta) =\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 + m \Delta}$$ It is a large-$n$ approximation of the ...
7
votes
1answer
95 views

arithmetic/category theoretic information encoded in $q$-series reciprocals

According to the pentagonal number theorem: $$\prod_{n=1}^{\infty} (1-q^{n}) = \sum_{k=-\infty}^{\infty} (-1)^{k}q^{k(3k-1)/2}$$ Now the reciprocal of this has the partition numbers $p(k)$ in its ...
6
votes
1answer
165 views

How to prove the q-series identity?

How could I prove that $$(-q;q^2)_\infty (q;q)_\infty = 1 + 2 \sum_{i=1}^\infty (-1)^i q^{2 i^2}?$$ If that is too difficult is there a way to show $$(-q;q^2)_\infty (q;q)_\infty \equiv 1 \pmod 2?$$ ...
4
votes
1answer
108 views

Validity of a q-series theorem

Define the $q$-analog $(a;q)_n = \prod_{k=0}^n \left(1 - aq^k\right)$. I want to prove the identity $\frac{(q^2;q^2)_\infty}{(q;q)_\infty}=\frac{1}{(q;q^2)_\infty}$. I viewed the LHS this way: ...