Göllnitz-Gordon identities and Ramanujan-Göllnitz-Gordon continued fractions

introduction

• 틀:수학노트

q-hypergeometric series

identities

• Slater 34

\[\sum_{n=0}^{\infty}\frac{q^{n^2+2n}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=1/(q^{3};q^{8})_{\infty}(q^{4};q^{8})_{\infty}(q^{5};q^{8})_{\infty}\]

• Slater 36

\[\sum_{n=0}^{\infty}\frac{q^{n^2}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=1/(q^{1};q^{8})_{\infty}(q^{4};q^{8})_{\infty}(q^{7};q^{8})_{\infty}\]

• G. E. Andrews, The Theory of Partitions, 1976, Corollary 2.7., page 21,

\[\sum_{n=0}^{\infty}\frac{q^{n^2+n}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=1/(q^{1};q^{8})_{\infty}(q^{5};q^{8})_{\infty}(q^{6};q^{8})_{\infty}\label{GA}\]

• is \ref{GA} modular?

lifting to rank 2 case

Using the Gauss formula (in useful techniques in q-series) \[\prod_{r=0}^{n-1}(1+zq^r)=(1+z)(1+zq)\cdots(1+zq^{n-1})= \sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r(r-1)/2}z^r\]

we can rewrite \((-q;q^{2})_{n}\) as \[(-q;q^{2})_{n}=\sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r^{2}}\]

Therefore, \[\sum_{n=0}^{\infty}\frac{q^{n^2}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^2}\sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r^{2}}}{ (q^{2};q^{2})_{n}}\] \[=\sum_{i,j\geq 0}\frac{q^{(i+j)^2}q^{j^2}}{(-q)_{i+j}(q)_{j}(q)_{i}}=\sum_{i,j\geq 0}\frac{q^{i^2+2ij+2j^2}}{(-q)_{i+j}(q)_{j}(q)_{i}}\]

and \[\sum_{n=0}^{\infty}\frac{q^{n^2+2n}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=\sum_{i,j\geq 0}\frac{q^{i^2+2ij+2j^2}q^{2(i+j)}}{(-q)_{i+j}(q)_{j}(q)_{i}}\]

Note that we have used the identity $$ (-q)_{n}=\frac{(q^2;q^2)_{n}}{(q;q)_{n}} $$

• one can obtain the following matrix from the above hypergeometric series

\[ \begin{bmatrix} 4 & 2 \\ 2 & 2 \end{bmatrix}\]

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• Andrews-Gordon identity
• rank 2 case
• rank 2 continued fraction

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxeDBITGE5WEJ5dHM/edit

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/

books

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976
  • Sec 7.4

expositions

• http://dspace.thapar.edu:8080/dspace/bitstream/10266/1935/1/joined_document.pdf

articles

• Explicit evaluations of Ramanujan-Göllnitz-Gordon continued fraction
  • Nayandeep Deka Baruah and Nipen Saikia, 2008
• Certain identities for Ramanujan–Göllnitz–Gordon continued fraction
  • K.R. Vasuki, and B.R. Srivatsa Kumar, 2005
• On the Expansion of a Continued Fraction of Gordon
  • Michael D. Hirschhorn, 2001
• On the Ramanujan-G¨ollnitz-Gordon Continued Fraction
  • Heng Huat Chan and Sen-Shan Huang, 1997
• Partition identities and a continued fraction of Ramanujan
  • Krishnaswami Alladi and Basil Gordon, 1993
• Partitionen mit Differenzenbedingungen
  • H. Göllnitz, J. Reine Angew. Math. 225 (1967), 154–190.
• Gordon, B. Some continued fractions of the Rogers-Ramanujan type. Duke Math. J. 32 1965 741–748.
• A Combinatorial Generalization of the Rogers-Ramanujan Identities
  • Gordon, B. Amer. J. Math. 83, 393-399, 1961