Minkowski's question mark function

In mathematics, the Jacobi triple product is the mathematical identity:

$\prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+x^{2m-1}y^{-2}\right)=\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n},$

for complex numbers x and y, with |x| < 1 and y ≠ 0.

It was introduced by Template:Harvs in his work Fundamenta Nova Theoriae Functionum Ellipticarum.

The Jacobi triple product identity is the Macdonald identity for the affine root system of type A₁, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra.

Properties

The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity.

Let $x=q^{3/2}$ and $y^{2}=-{\sqrt {q}}$. Then we have

$\phi (q)=\prod _{m=1}^{\infty }\left(1-q^{m}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(3n^{2}-n)/2}.\,$

The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:

Let $x=e^{i\pi \tau }$ and $y=e^{i\pi z}.$

Then the Jacobi theta function

$\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz)$

can be written in the form

$\sum _{n=-\infty }^{\infty }y^{2n}x^{n^{2}}.$

Using the Jacobi Triple Product Identity we can then write the theta function as the product

$\vartheta (z;\tau )=\prod _{m=1}^{\infty }\left(1-\exp(2m\pi i\tau )\right)\left(1+\exp((2m-1)\pi i\tau +2\pi iz)\right)\left(1+\exp((2m-1)\pi i\tau -2\pi iz)\right).$

There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of q-Pochhammer symbols:

$\sum _{n=-\infty }^{\infty }q^{n(n+1)/2}z^{n}=(q;q)_{\infty }\;(-1/z;q)_{\infty }\;(-zq;q)_{\infty }.$

Where $(a;q)_{\infty }$ is the infinite q-Pochhammer symbol.

It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For $|ab|<1.$ it can be written as

$\sum _{n=-\infty }^{\infty }a^{n(n+1)/2}\;b^{n(n-1)/2}=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.$

Proof

This proof uses a simplified model of the Dirac sea and follows the proof in Cameron (13.3) which is attributed to Richard Borcherds. It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as

$\prod _{n>0}(1+q^{n-{\frac {1}{2}}}z)(1+q^{n-{\frac {1}{2}}}z^{-1})=\left(\sum _{l\in \mathbb {Z} }q^{l^{2}/2}z^{l}\right)\left(\prod _{n>0}(1-q^{n})^{-1}\right).$

A level is a half-integer. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite. The energy of the state $S$ is

$\sum \{v\colon v>0,v\in S\}-\sum \{v\colon v<0,v\not \in S\}$

and the particle number of $S$ is

$|\{v\colon v>0,v\in S\}|-|\{v\colon v<0,v\not \in S\}|.$

An unordered choice of the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function $\textstyle \sum _{m,l}s(m,l)q^{m}z^{l}$ for the number $s(m,l)$ of states of energy $m$ with $l$ particles can be expressed as

$\prod _{n>0}(1+q^{n-{\frac {1}{2}}}z)(1+q^{n-{\frac {1}{2}}}z^{-1}).$

On the other hand, any state with $l$ particles can be obtained from the lowest energy $l-$particle state, $\{v\colon v<l\}$, by rearranging particles: take a partition $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{j}$ of $m'$ and move the top particle up by $\lambda _{1}$ levels, the next highest particle up by $\lambda _{2}$ levels, etc.... The resulting state has energy $m'+{\frac {l^{2}}{2}}$, so the generating function can also be written as

$\left(\sum _{l\in \mathbb {Z} }q^{l^{2}/2}z^{l}\right)\left(\sum _{n\geq 0}p(n)q^{n}\right)=\left(\sum _{l\in \mathbb {Z} }q^{l^{2}/2}z^{l}\right)\left(\prod _{n>0}(1-q^{n})^{-1}\right)$

where $p(n)$ is the partition function. The uses of random partitions by Andrei Okounkov contains a picture of a partition exciting the vacuum.

References

• See chapter 14, theorem 14.6 of Template:Apostol IANT
• Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press, ISBN 0-521-45761-0
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010