Rogers-Ramanujan identities

The Rogers-Ramanujan identities are originally two identities between infinite series and products, which Leonard James Rogers first proved in 1894. S. Ramanujan found them independently before 1913 (without evidence). Ramanujan then came across the article by Rogers by chance, which had received little attention until then, and published a new proof with Rogers in 1919. Issai Schur independently found the identities and proof in 1917. There are also generalizations of identities.

Bulk

The identities are (with ): ${\displaystyle |q|<1}$

${\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots \,}$

and

${\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots \,}$

${\displaystyle G(q)}$and defined over the left part of the identities (as an infinite series) are called Rogers-Ramanujan functions. ${\displaystyle H(q)}$

Here, the q-Pochhammer symbols : ${\displaystyle (\cdot ;\cdot )_{n}}$

${\displaystyle (q;q)_{n}=\prod _{k=1}^{n}(1-q^{k})=(1-q)(1-q^{2})\cdots (1-q^{n})}$

${\displaystyle (q;q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+1})}$

${\displaystyle (q^{4};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+4})}$

${\displaystyle (q^{2};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+2})}$

${\displaystyle (q^{3};q^{5})_{\infty }=\prod _{k=0}^{\infty }(1-q^{5k+3})}$

So that the identities can also be written:

${\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}={\frac {1}{\prod _{k=0}^{\infty }(1-q^{5k+1})(1-q^{5k+4})}}={\frac {1}{\prod _{k=1}^{\infty }(1-q^{5k-1})(1-q^{5k-4})}}}$

and

${\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}={\frac {1}{\prod _{k=0}^{\infty }(1-q^{5k+2})(1-q^{5k+3})}}={\frac {1}{\prod _{k=1}^{\infty }(1-q^{5k-2})(1-q^{5k-3})}}}$

There are also generalized identities of the Rogers-Ramanujan type that have been established in the work of Wilfrid Norman Bailey , Freeman Dyson , Atle Selberg, and Lucy Joan Slater in particular (Slater lists 130 such identities in her 1952 essay). More found z. B. George E. Andrews (Andrews-Gordon identity, with Basil Gordon ), Heinz Göllnitz (Göllnitz-Gordon identities).

Ramanujan listed a total of 40 identities with the functions (in his notebooks). ${\displaystyle G(q),H(q)}$

Application to partitions

Since the terms occurring in the identity are generating functions of certain partitions , the identities make statements about partitions (decays) of natural numbers:

According to the first identity (for ), the number of decays of an integer n in which neighboring parts of the partition differ by at least 2 is equal to the number of decays where each part is equal to 1 or 4 mod 5. ${\displaystyle G(q)}$

The second identity can be formulated as follows: The number of disintegrations of an integer n in which adjacent parts of the partition differ by at least 2 and in which the smallest part is greater than or equal to 2 is equal to the number of disintegrations whose parts are the same 2 or 3 mod 5 are.

Others

If one sets (where the imaginary part of is positive), are ${\displaystyle q=e^{2\pi i\tau }}$ ${\displaystyle \tau \in \mathbb {C} }$

${\displaystyle q^{\frac {-1}{60}}G(q)}$

and

${\displaystyle q^{\frac {11}{60}}H(q)}$

Module functions .

The chain break

${\displaystyle R(q)=1+{\frac {q}{1+{\frac {q^{2}}{1+{\frac {q^{3}}{1+\cdots }}}}}}}$

is called the Rogers-Ramanujan continued fraction. Sometimes it is also defined with a factor (this gives you quotients of module functions). ${\displaystyle q^{\frac {1}{5}}}$

It applies

${\displaystyle R(q)=\prod _{k=0}^{\infty }{\frac {(1-q^{5k+1})(1-q^{5k+4})}{(1-q^{5k+2})(1-q^{5k+3})}}={\frac {H(q)}{G(q)}}}$

or with the Ramanujan's theta function

${\displaystyle f(a,b)=\sum _{k=-\infty }^{\infty }a^{\frac {k(k+1)}{2}}b^{\frac {k(k-1)}{2}}}$

is

${\displaystyle R(q)={\frac {f(-q,-q^{4})}{f(-q^{2},-q^{3})}}}$.

The continued fraction can also be expressed using the Dedekind η function . The connection of the continued fraction with the Rogers-Ramanujan functions was already found by Rogers in 1894 (and later by Ramanujan independently).

Application in statistical mechanics

The identities are used in statistical mechanics in the solution of the Hard Hexagon model by Rodney Baxter in 1980. The Hard Hexagon model is a gas of particles on a triangular grid, so that no two particles may be adjacent on the grid. They are also used in other precisely solvable models of statistical mechanics.

literature

• George E. Andrews : The theory of partitions, Addison-Wesley 1976, Cambridge University Press 1998
• David Bressoud , Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, American Mathematical Society 1980
• David Bressoud: An easy proof of the Rogers-Ramanujan identities, J. of Number Theory, Volume 16, 1983, pp. 235-241
• Godfrey Harold Hardy , EM Wright: Introduction to the theory of numbers, Oxford, Clarendon Press 1975 (p. 290ff, chapters 19-13)
• George E. Andrews , Rodney J. Baxter : A motivated proof of the Rogers-Ramanujan identities, American Mathematical Monthly, Volume 96, 1989, pp. 401-409

Web links

• Rogers-Ramanujan-Identities, Mathworld
• Anne Schilling, The Rogers-Ramanujan identities at Y2K, 2000, pdf

Individual evidence

1. ↑ Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc., Vol. 25, 1894, pp. 318-343
2. ↑ He reported it to Percy Alexander MacMahon , who published it in his book Combinatory Analysis , Cambridge University Press, Volume 2, 1916 (without evidence)
3. ↑ Rogers, Ramanujan, Proof of certain identities in combinatory analysis, Cambr. Phil. Soc. Proc., Vol. 19, 1919, pp. 211-216
4. ↑ Schur, A contribution to additive number theory and the theory of continued fractions, meeting reports of the Preuss. Academy of Sciences, Math.-Phys. Klasse, 1917, pp. 302–321, also in Schur, Gesammelte Abhandlungen, Volume 2, Springer, 1973
5. ^ Bailey, Generalized hypergeometric series, Cambridge University Press 1935
6. ^ Slater, Further identities of the Rogers-Ramanujan type, Proceedings of the London Mathematical Society. Second Series, Vol. 54, 1952, pp. 147-167
7. ^ Andrews-Gordon Identity, Mathworld
8. ↑ Bruce Berndt et al. a., Ramanujans forty identities for the Rogers- Ramanujan functions, pdf
9. ^ Rogers-Ramanujan Continued Fraction, Mathworld
10. ↑ Bruce Berndt et al. a., The Rogers-Ramanujan continued fraction, pdf
11. ↑ Baxter, Exactly solvable models in statistical mechanics, Academic Press 1982. First, Baxter, Journal of Physics, A, Volume 13, 1980, L61-L70. See also George E. Andrews , The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci., Vol. 78, 1981, pp. 5290-5292, pdf