zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A proof of Andrews’ $q$-Dyson conjecture. (English) Zbl 0565.33001
The $q$-shifted factorial is defined by $(x;q)\sb n=(1-x)(1-xq)\cdots(1- xq\sp{n-1})$. Andrews conjectured that the constant term in the Laurent polynomial $$ \prod\sb{1\le i\le j<n}(x\sb i/x\sb j;q)\sb{a\sb i}(qx\sb j/x\sb i;q)\sb{a\sb j}$$ is $$(q;q)\sb{a\sb 1+\dots+a\sb n}/(q;q)\sb{a\sb 1}\dots(q;q)\sb{a\sb n}, $$ as an extension of an earlier conjecture of {\it F. J. Dyson} [J. Math. Phys. 3, 140--156 (1962; Zbl 0105.41604)] that was proved by{\it J. Gunson} [ibid. 3, 752--753 (1962; Zbl 0111.43903)] and {\it K. G. Wilson} [ibid. 3, 1040--1043 (1962; Zbl 0113.21403)] (the case $q=1)$. A combinatorial proof is given in the present paper. This is a major advance, and is one more indication that enumerative combinatorics has come of age, and should be learned by many people who could use it, as well as those who are developing it.
Reviewer: R.Askey

MSC:
33D15Basic hypergeometric functions of one variable, ${}_r\phi_s$
05A15Exact enumeration problems, generating functions
WorldCat.org
Full Text: DOI
References:
[1] Andrews, G. E.: Problems and prospects for basic hypergeometric functions. Theory and applications of special functions, 191-224 (1975)
[2] Andrews, G. E.: 2nd ed. The theory of partitions, encyclopedia of math. And its appl.. The theory of partitions, encyclopedia of math. And its appl. 2 (1976)
[3] Dyson, F. J.: Statistical theory of the energy levels of complex systems I. J. math. Phys. 3, 140-156 (1962) · Zbl 0105.41604
[4] Foata, D.; Cartier, P.: Problèmes combinatoires de commutation et réarrangements. Lecture notes in mathematics 85 (1969) · Zbl 0186.30101
[5] Foata, D.: On the netto inversion number of a sequence. Proc. amer. Math. soc. 19, 236-240 (1968) · Zbl 0157.03403
[6] Gessel, I.: Tournaments and Vandermonde’s determinant. J. graph theory 3, 305-307 (1979) · Zbl 0433.05008
[7] Good, I. J.: Short proof of a conjecture of Dyson. J. math. Phys. 11, 1984 (1970) · Zbl 0194.05903
[8] Gunson, J.: Proof of a conjecture of Dyson in the statistical theory of energy levels. J. math. Phys. 3, 752-753 (1962) · Zbl 0111.43903
[9] K. Kadell, Andrews q-Dyson Conjecture: n=4, Trans. Amer. Math. Soc. (to appear).
[10] Knuth, D. E.: 2nd ed. Fundamental algorithms, the art of computer programming. Fundamental algorithms, the art of computer programming 1 (1973) · Zbl 0191.17903
[11] Knuth, D. E.: Sorting and searching, the art of computer programming. 3 (1973) · Zbl 0302.68010
[12] Melzak, Z. A.: Mathematical ideas, modelling and applications. (1976) · Zbl 0331.00002
[13] Morris, W.: Constant term identities for finite and affine root systems, conjectures and theorems. Ph.d. thesis, university of wisconsin, Madison, WI (1982)
[14] Wilson, K.: Proof of a conjecture of Dyson. J. math. Phys. 3, 1040-1043 (1962) · Zbl 0113.21403
[15] Zeilberger, D.: Sister celine’s technique and its generalizations. J. math. Anal. appl. 85, 114-145 (1980) · Zbl 0485.05003
[16] Zeilberger, D.: A combinatorial proof of Dyson’s conjecture. Discrete math. 41, 317-321 (1982) · Zbl 0492.05007