-Hypergeometric and Related Functions Properties 17.13 Integrals 17.15 Generalizations

§17.14 Constant Term Identities

Notes:: See Zeilberger and Bressoud (1985), Andrews (1986, p. 34).
Keywords:: -hypergeometric function
Referenced by:: §17.2(vi), §27.14(vi)
Permalink:: http://dlmf.nist.gov/17.14

¶ Zeilberger–Bressoud Theorem (Andrews’ -Dyson Conjecture)

Keywords:: -Dyson conjecture, -hypergeometric function

17.14.1 $\frac{\left(q;q\right)_{{a_{1}+a_{2}+\cdots+a_{n}}}}{\left(q;q\right)_{{a_{1}}% }\left(q;q\right)_{{a_{2}}}\cdots\left(q;q\right)_{{a_{n}}}}=\mbox{ coeff. of % }x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}\mbox{ in }\prod_{{1\leq j<k\leq n}}\left(% \frac{x_{j}}{x_{k}};q\right)_{{a_{j}}}\left(\frac{qx_{k}}{x_{j}};q\right)_{{a_% {k}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base, : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: ¶ ‣ §17.14
Permalink:: http://dlmf.nist.gov/17.14.E1
Encodings:: TeX, pMML, png

¶ Rogers–Ramanujan Constant Term Identities

Keywords:: Rogers–Ramanujan identities

In the following, and denote the left-hand sides of (17.2.49) and (17.2.50), respectively.

17.14.2 $\sum_{{n=0}}^{{\infty}}\frac{q^{{n(n+1)}}}{\left(q^{2};q^{2}\right)_{{n}}\left% (-q;q^{2}\right)_{{n+1}}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^% {2}\right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};q^{2}% \right)_{{\infty}}}{\left(z^{{-1}}q^{2};q^{2}\right)_{{\infty}}\left(-q;q^{2}% \right)_{{\infty}}\left(z^{{-1}}q;q^{2}\right)_{{\infty}}}=\frac{1}{\left(-q;q% ^{2}\right)_{{\infty}}}\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}% \right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};q^{2}% \right)_{{\infty}}}{\left(z^{{-1}}q;q\right)_{{\infty}}}=\frac{H(q)}{\left(-q;% q^{2}\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.50)
Referenced by:: ¶ ‣ §17.14
Permalink:: http://dlmf.nist.gov/17.14.E2
Encodings:: TeX, pMML, png

17.14.3 $\sum_{{n=0}}^{{\infty}}\frac{q^{{n(n+1)}}}{\left(q^{2};q^{2}\right)_{{n}}\left% (-q;q^{2}\right)_{{n+1}}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^% {2}\right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};q^{2}% \right)_{{\infty}}}{\left(z^{{-1}};q^{2}\right)_{{\infty}}\left(-q;q^{2}\right% )_{{\infty}}\left(z^{{-1}}q;q^{2}\right)_{{\infty}}}=\frac{1}{\left(-q;q^{2}% \right)_{{\infty}}}\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}% \right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};q^{2}% \right)_{{\infty}}}{\left(z^{{-1}};q\right)_{{\infty}}}=\frac{G(q)}{\left(-q;q% ^{2}\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E3
Encodings:: TeX, pMML, png

17.14.4 $\sum_{{n=0}}^{{\infty}}\frac{q^{{n^{2}}}}{\left(q^{2};q^{2}\right)_{{n}}\left(% q;q^{2}\right)_{{n}}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}% \right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};q^{2}% \right)_{{\infty}}}{\left(-z^{{-1}};q^{2}\right)_{{\infty}}\left(q;q^{2}\right% )_{{\infty}}\left(z^{{-1}};q^{2}\right)_{{\infty}}}=\frac{1}{\left(q;q^{2}% \right)_{{\infty}}}\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}% \right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};q^{2}% \right)_{{\infty}}}{\left(z^{{-2}};q^{4}\right)_{{\infty}}}=\frac{G(q^{4})}{% \left(q;q^{2}\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E4
Encodings:: TeX, pMML, png

17.14.5 $\sum_{{n=0}}^{{\infty}}\frac{q^{{n^{2}+2n}}}{\left(q^{2};q^{2}\right)_{{n}}% \left(q;q^{2}\right)_{{n+1}}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-% zq;q^{2}\right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2};% q^{2}\right)_{{\infty}}}{\left(-q^{2}z^{{-1}};q^{2}\right)_{{\infty}}\left(q;q% ^{2}\right)_{{\infty}}\left(z^{{-1}}q^{2};q^{2}\right)_{{\infty}}}=\frac{1}{% \left(q;q^{2}\right)_{{\infty}}}\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(% -zq;q^{2}\right)_{{\infty}}\left(-z^{{-1}}q;q^{2}\right)_{{\infty}}\left(q^{2}% ;q^{2}\right)_{{\infty}}}{\left(q^{4}z^{{-2}};q^{4}\right)_{{\infty}}}=\frac{H% (q^{4})}{\left(q;q^{2}\right)_{{\infty}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : complex variable and : LHS of (17.2.50)
Referenced by:: ¶ ‣ §17.14
Permalink:: http://dlmf.nist.gov/17.14.E5
Encodings:: TeX, pMML, png

Macdonald (1982) includes extensive conjectures on generalizations of (17.14.1) to root systems. These conjectures were proved in Cherednik (1995), Habsieger (1986), and Kadell (1994); see also Macdonald (1998). For additional results of the type (17.14.2)–(17.14.5) see Andrews (1986, Chapter 4).

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