Rogers–Dougall very well-poised sum

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1: 16.4 Argument Unity

… ►It is very well-poised if it is well-poised and

a_{1}=b_{1}+1

. … ►

Dixon’s Well-Poised Sum

… ►

Rogers–Dougall Very Well-Poised Sum

… ►

Dougall’s Very Well-Poised Sum

…

2: 17.4 Basic Hypergeometric Functions

… ►In these references the factor

\left((-1)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}\right)^{s-r}

is not included in the sum. … ►

17.4.3

{{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.

ⓘ

Defines:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(ii), §17.4 and Ch.17

… ►

17.4.5

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac{\left(a;q% \right)_{m+n}\left(b;q\right)_{m}\left(b^{\prime};q\right)_{n}x^{m}y^{n}}{% \left(q;q\right)_{m}\left(q;q\right)_{n}\left(c;q\right)_{m+n}},

ⓘ

Defines:: $\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$ : first $q$ -Appell function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Referenced by:: Erratum (V1.0.10) for Section 17.1
Permalink:: http://dlmf.nist.gov/17.4.E5
Encodings:: pMML, png, TeX
Correction (effective with 1.0.10):: The notation for $\Phi^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.
See also:: Annotations for §17.4(iii), §17.4 and Ch.17

… ►The series (17.4.1) is said to be well-poised when

r=s

and … ►The series (17.4.1) is said to be very-well-poised when

r=s

, (17.4.11) is satisfied, and …

3: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

Bailey’s Transformation of Very-Well-Poised ${{}_{8}\phi_{7}}$

… ►

§17.9(iv) Bibasic Series

… ►

17.9.19

\sum_{n=0}^{\infty}\frac{\left(a;q^{2}\right)_{n}\left(b;q\right)_{n}}{\left(q% ^{2};q^{2}\right)_{n}\left(c;q\right)_{n}}z^{n}=\frac{\left(b;q\right)_{\infty% }\left(az;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{2}\right)% _{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n}\left(z;q^{2}\right)% _{n}b^{2n}}{\left(q;q\right)_{2n}\left(az;q^{2}\right)_{n}}+\frac{\left(b;q% \right)_{\infty}\left(azq;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}% \left(zq;q^{2}\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n% +1}\left(zq;q^{2}\right)_{n}b^{2n+1}}{\left(q;q\right)_{2n+1}\left(azq;q^{2}% \right)_{n}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

►

17.9.20

\sum_{n=0}^{\infty}\frac{\left(a;q^{k}\right)_{n}\left(b;q\right)_{kn}z^{n}}{% \left(q^{k};q^{k}\right)_{n}\left(c;q\right)_{kn}}=\frac{\left(b;q\right)_{% \infty}\left(az;q^{k}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{k}% \right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{n}\left(z;q^{k}% \right)_{n}b^{n}}{\left(q;q\right)_{n}\left(az;q^{k}\right)_{n}},

k=1,2,3,\dots

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

4: Bibliography M

… ►

S. C. Milne (1985a) A

q

-analog of the

{}_{5}F_{4}(1)

summation theorem for hypergeometric series well-poised in

\mathit{SU}(n)

. Adv. in Math. 57 (1), pp. 14–33.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (1985d) A

q

-analog of hypergeometric series well-poised in

\mathit{SU}(n)

and invariant

G

-functions. Adv. in Math. 58 (1), pp. 1–60.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

ⓘ

External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

S. Moch, P. Uwer, and S. Weinzierl (2002) Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Driss Essouabri), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

…

5: 17.18 Methods of Computation

… ►Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …

6: 17.12 Bailey Pairs

… ►

17.12.1

\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},

ⓘ

Symbols:: $n$ : nonnegative integer, $\alpha_{n}$ : part of Bailey pair and $\beta_{n}$ : part of Bailey pair
Permalink:: http://dlmf.nist.gov/17.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.12, §17.12 and Ch.17

… ►

\beta_{n}=\sum_{j=0}^{n}\alpha_{j}u_{n-j}v_{n+j},

►

\gamma_{n}=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.

… ► ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …

7: 17.14 Constant Term Identities

… ►

Rogers–Ramanujan Constant Term Identities

… ►

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:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $H(q)$ : LHS of (17.2.50)
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

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:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $G(q)$ : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

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:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $G(q)$ : LHS of (17.2.49)
Permalink:: http://dlmf.nist.gov/17.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

►

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:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $H(q)$ : LHS of (17.2.50)
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

…

8: David M. Bressoud

… ►His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …

9: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over

m\geq 0

of the generating functions for partitions into distinct parts with largest part equal to

m

. … ►where the inner sum is the sum of all positive odd divisors of

t

. … ►where the inner sum is the sum of all positive divisors of

t

that are in

S

. ►

§26.10(iv) Identities

►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …

10: 18.33 Polynomials Orthogonal on the Unit Circle

… ►

18.33.2

\phi_{n}(z)=\kappa_{n}z^{n}+\sum_{\ell=1}^{n}\kappa_{n,n-\ell}z^{n-\ell},

ⓘ

Symbols:: $z$ : complex variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $\phi_{n}(z)$ : polynomials
Permalink:: http://dlmf.nist.gov/18.33.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.33(ii), §18.33 and Ch.18

… ►

18.33.3

\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline{z}}^{-1})}={\kappa_{n}}+% \sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell}z^{\ell},

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $z$ : complex variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $\phi_{n}(z)$ : polynomials
Referenced by:: Erratum (V1.0.10) for Equation (18.33.3)
Permalink:: http://dlmf.nist.gov/18.33.E3
Encodings:: pMML, png, TeX
Errata (effective with 1.0.10):: Originally this equation was written incorrectly as $\phi_{n}^{*}(z)={\kappa_{n}}z^{n}+\sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell% }z^{n-\ell}$ . Also, the equality $\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline{z}}^{-1})}$ has been added.
Reported 2014-11-10 by Roderick Wong
See also:: Annotations for §18.33(ii), §18.33 and Ch.18

… ►

18.33.13

\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{\ell}}{\left(% \lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=\frac{{\left(\lambda% \right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-\lambda-n+1};z\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $\phi_{n}(z)$ : polynomials
Referenced by:: §18.33(v)
Permalink:: http://dlmf.nist.gov/18.33.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

… ►When

a=0

the Askey case is also known as the Rogers–Szegő case. …