Bailey pair

Bailey pair

In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by harvs|txt|first=W. N. |last=Bailey|authorlink=Wilfrid Norman Bailey|year1=1947|year2=1948 while studyingthe second proof harvtxt|Rogers|1917 of the Rogers-Ramanujan identities, and Bailey chains were introduced by harvtxt|Andrews|1984.

Definition

The q-Pochhammer symbols (a;q)_n are defined as:

:(a;q)_n = prod_{0le j

A pair of sequences (α"n",β"n") is called a Bailey pair if they are related by:eta_n=sum_{r=0}^nfrac{alpha_r}{(q;q)_{n-r}(aq;q)_{n+ror equivalently:alpha_n = (1-aq^{2n})sum_{j=0}^nfrac{(aq;q)_{n+j-1}(-1)^{n-j}q^{n-jchoose 2}eta_j}{(q;q)_{n-j.

Bailey's lemma

Bailey's lemma states that if (α"n",β"n") is a Bailey pair, then so is(α'"n",β'"n") where:alpha^prime_n= frac{( ho_1;q)_n( ho_2;q)_n(aq/ ho_1 ho_2)^nalpha_n}{(aq/ ho_1;q)_n(aq/ ho_2;q)_n}:eta^prime_n = sum_{jge0}frac{( ho_1;q)_j( ho_2;q)_j(aq/ ho_1 ho_2;q)_{n-j}(aq/ ho_1 ho_2)^jeta_j}{(q;q)_{n-j}(aq/ ho_1;q)_n(aq/ ho_2;q)_n}.In other words, given one Bailey pair, one can construct a second using the formulas above. This process can be iterated to produce an infinite sequence of Bailey pairs, called a Bailey chain.

Examples

An example of a Bailey pair is given by harv|Andrews|Askey|Roy|1999|loc=p. 590:alpha_n = q^{n^2+n}sum_{j=-n}^n(-1)^jq^{-j^2}, quad eta_n = frac{(-q)^n}{(q^2;q^2)_n}.

harvs|txt|first=L. J. |last=Slater|authorlink=L. J. Slater|year=1952 gave a list of 130 examples related to Bailey pairs.

References

*Citation | last1=Andrews | first1=George E. | title=Multiple series Rogers-Ramanujan type identities | url=http://projecteuclid.org/euclid.pjm/1102708707 | id=MathSciNet | id = 757501 | year=1984 | journal=Pacific Journal of Mathematics | issn=0030-8730 | volume=114 | issue=2 | pages=267–283
*Citation | author1-link=George Andrews (mathematician) | last1=Andrews | first1=George E. | author2-link=Richard Askey |last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=Cambridge University Press | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6; 978-0-521-78988-2 | id=MathSciNet | id = 1688958 | year=1999 | volume=71
*Citation | last1=Bailey | first1=W. N. | title=Some identities in combinatory analysis | doi=10.1112/plms/s2-49.6.421 | id=MathSciNet | id = 0022816 | year=1947 | journal=Proceedings of the London Mathematical Society. Second Series | issn=0024-6115 | volume=49 | pages=421–425
*Citation | last1=Bailey | first1=W. N. | title=Identities of the Rogers-Ramanujan Type | doi=10.1112/plms/s2-50.1.1 | year=1948 | journal=Proc. London Math. Soc. | volume=s2-50 | pages=1–10
*citation|url=http://www.emis.de/journals/SLC/opapers/s18paule.pdf|title=The Concept of Bailey Chains|first= Peter |last=Paule
*Citation | last1=Slater | first1=L. J. | title=Further identies of the Rogers-Ramanujan type | doi=10.1112/plms/s2-54.2.147 | id=MathSciNet | id = 0049225 | year=1952 | journal=Proceedings of the London Mathematical Society. Second Series | issn=0024-6115 | volume=54 | pages=147–167
*Citation | last1=Warnaar | first1=S. Ole | title=Algebraic combinatorics and applications (Gössweinstein, 1999) | url=http://www.maths.uq.edu.au/~uqowarna/pubs/Bailey50.pdf | publisher=Springer-Verlag | location=Berlin, New York | id=MathSciNet | id = 1851961 | year=2001 | chapter=50 years of Bailey's lemma | pages=333–347


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