Bailey pair and lemma

introduction

q-Pfaff-Sallschutz sum

Bailey lemma

If the sequence \(\{\alpha_r\}, \{\beta_r\}\), \(\{\delta_r\}, \{\gamma_r\}\) satisfy the following

\(\beta_L=\sum_{r=0}^{L}{\alpha_r}{u_{L-r}v_{L+r}}\), \(\gamma_L=\sum_{r=L}^{\infty}{\delta_r}{u_{r-L}v_{r+L}}\)

then,

\(\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\sum_{n=0}^{\infty}\beta_n\delta_{n}\)

(corollay)

Choose the following

\(u_{n}=\frac{1}{(q)_n}\) ,\(v_{n}=\frac{1}{(x)_n}\),\(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\)

Then by the basic analogue of Gauss's theorem (q-Gauss sum)

(Recall \(\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\) )

\(\gamma_n=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\)

Hence by Bailey's lemma,

\(\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}\)

Bailey pair

the sequence \(\{\alpha_r\}, \{\beta_r\}\) satisfying the following is called a Bailey pair
\(\beta_L=\sum_{r=0}^{L}\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}\)
conjugate Bailey pair \(\{\delta_r\}, \{\gamma_r\}\)
\(\gamma_L=\sum_{r=L}^{\infty}\frac{\delta_r}{(q)_{r-L}(aq)_{r+L}}\)

Bailey chain

we derive a new Bailey chain from a known Bailey pair
\(\alpha^\prime_n= \frac{(\rho_1;q)_n(\rho_2;q)_n(aq/\rho_1\rho_2)^n}{(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\alpha_n\)
\(\beta^\prime_n = \sum_{j\ge0}\frac{(\rho_1;q)_j(\rho_2;q)_j(aq/\rho_1\rho_2;q)_{n-j}(aq/\rho_1\rho_2)^j}{(q;q)_{n-j}(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\beta_j\)
corollary. by taking \(\rho_1,\rho_2\to \infty\) , we get
\(\alpha^\prime_n= a^nq^{n^2}\alpha_n\)
\(\beta^\prime_n = \sum_{r=0}^{L}\frac{a^rq^{r^2}}{(q)_{L-r}}\beta_j\)

history

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related items

encyclopedia

http://en.wikipedia.org/wiki/Bailey_pair
http://en.wikipedia.org/wiki/Wilfrid_Norman_Bailey
http://en.wikipedia.org/wiki/
http://www.scholarpedia.org/
Princeton companion to mathematics(Companion_to_Mathematics.pdf)

books

[[4909919|]]

articles

50 Years of Bailey's lemma
- S. Ole Warnaar, 2009
A generalization of the q-Saalschutz sum and the Burge transform
- A. Schilling, S.O. Warnaa, 2009
Rogers-Ramanujan-Slater Type identities
- Mc Laughlin, 2008
Andrews–Gordon type identities from combinations of Virasoro characters
- Boris Feigin, Omar Foda, Trevor Welsh, 2007
Finite Rogers-Ramanujan Type Identities
- Andrew V. Sills, 2003
Virasoro character identities from the Andrews–Bailey construction
- Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)

Multiple series Rogers-Ramanujan type identities.
- George E. Andrews, Pacific J. Math. Volume 114, Number 2 (1984), 267-283.
Special values of the dilogarithm function
- J. H. Loxton, 1984
Wilfrid Norman Bailey
- Slater, L. J. (1962), Journal of the London Mathematical Society. Second Series 37: 504–512
Further identities of the Rogers-Ramanujan type
- Slater, L. J. (1952), Proceedings of the London Mathematical Society. Second Series 54: 147–167
A New Proof of Rogers's Transformations of Infinite Series
- Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475
On two theorems of combinatory analysis and some allied identities
http://www.ams.org/mathscinet

question and answers(Math Overflow)

blogs

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experts on the field

http://arxiv.org/