Triple work of Jacobi

The triple product of Jacobi is a mathematical identity:

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

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

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

for complex numbers x and y with $|x|<1$ ${\ displaystyle | x | <1}$ ${\ displaystyle | x | <1}$ and $y\neq 0$ ${\ displaystyle y \ neq 0}$ ${\ displaystyle y \ neq 0}$ .

Identity was proposed by Carl Gustav Jacob Jacobi ^[1] in Fundamenta Nova Theoriae Functionum Ellipticarum (New Principles in the Theory of Elliptic Functions).

The Jacobi triple product is for the affine roots of a system of type A ₁ and is the for the corresponding affine .

Content

Properties

Jacobi's proof is based on Euler , which itself is a common case of the Jacobi triple product identity.

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

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

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

The Jacobi triple product also allows us to rewrite the Jacobi theta function as an infinite product:

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

Then the Jacobi theta function

\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }e^{\pi {\rm {i}}n^{2}\tau +2\pi {\rm {i}}nz}

{\ displaystyle \ vartheta (z; \ tau) = \ sum _ {n = - \ infty} ^ {\ infty} e ^ {\ pi {\ rm {i}} n ^ {2} \ tau +2 \ pi {\ rm {i}} nz}}

can be rewritten as

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

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

Using the identity of the Jacobi triple product, we can write the theta function as the product

\vartheta (z;\tau )=\prod _{m=1}^{\infty }\left(1-e^{2m\pi {\rm {i}}\tau }\right)\left[1+e^{(2m-1)\pi {\rm {i}}\tau +2\pi {\rm {i}}z}\right]\left[1+e^{(2m-1)\pi {\rm {i}}\tau -2\pi {\rm {i}}z}\right].

{\ displaystyle \ vartheta (z; \ tau) = \ prod _ {m = 1} ^ {\ infty} \ left (1-e ^ {2m \ pi {\ rm {i}} \ tau} \ right) \ left [1 + e ^ {(2m-1) \ pi {\ rm {i}} \ tau +2 \ pi {\ rm {i}} z} \ right] \ left [1 + e ^ {(2m- 1) \ pi {\ rm {i}} \ tau -2 \ pi {\ rm {i}} z} \ right].}

There are many different notations used to express the Jacobi triple product. It takes a short form, if expressed in terms of q- symbols of Pohhammer :

\sum _{n=-\infty }^{\infty }q^{\frac {n(n+1)}{2}}z^{n}=(q;q)_{\infty }\;\left(-{\tfrac {1}{z}};q\right)_{\infty }\;(-zq;q)_{\infty },

{\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} q ^ {\ frac {n (n + 1)} {2}} z ^ {n} = (q; q) _ {\ infty } \; \ left (- {\ tfrac {1} {z}}; q \ right) _ {\ infty} \; (- zq; q) _ {\ infty},}

Where $(a;q)_{\infty }$ ${\ displaystyle (a; q) _ {\ infty}}$ Is the infinite q- symbol of Pohhammer.

The formula takes on a particularly elegant appearance when expressed in terms of the theta function of Ramanujan . For $|ab|<1$ ${\ displaystyle | ab | <1}$ it can be rewritten as

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

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

Proof

For an analytical case, see the book of the Apostle ^[2] , the first edition of which was published in 1976. See also the link below for evidence stimulated by physicists.

Notes

↑ Jacobi, 1829 .
↑ Apostol, 1976 , p. theorem 14.6.

Literature

Andrews GE A simple proof of Jacobi's triple product identity // Proc. Amer. Math. Soc .. - American Mathematical Society , 1965. - T. 16 . - ISSN 0002-9939 .
Tom M. Apostol. chapter 14, theorem 14.6 of // Introduction to analytic number theory. - New York-Heidelberg: Springer-Verlag, 1976. - (Undergraduate Texts in Mathematics). - ISBN 978-0-387-90163-3 .
Peter J. Cameron. Combinatorics: Topics, Techniques, Algorithms . - Cambridge University Press , 1994. - ISBN 0-521-45761-0 .
Jacobi CGJ Fundamenta nova theoriae functionum ellipticarum . - Reprinted by Cambridge University Press 2012, language: Latin. - Königsberg: Borntraeger, 1829. - ISBN 978-1-108-05200-9 .
Carlitz L. A note on the Jacobi theta formula // Bull. Amer. Math. Soc .. - American Mathematical Society , 1962. - T. 68 , No. 6 . - S. 591-592. .
Wright EM An Enumerative Proof of An Identity of Jacobi // Journl of the London Mathematical Society. - London Mathematical Society , 1965. - T. s1-40 , no. 1 . - S. 55-57 .

Links

A brief combinatorial proof of identity, stimulated by physicists.

[_4943eec2bbc2deab-1] Jacobi, 1829 .

[_970d86fc73220593-2] Apostol, 1976 , p. theorem 14.6.