Pochhammer symbol

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11—20 of 125 matching pages

11: 18.27 $q$ -Hahn Class

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18.27.9

v_{x}=\frac{\left(a^{-1}x,c^{-1}x;q\right)_{\infty}}{\left(x,bc^{-1}x;q\right)% _{\infty}},

0<a<q^{-1}

0<b<q^{-1}

c<0

ⓘ

Symbols:: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $x$ : real variable and $v_{x}$
Permalink:: http://dlmf.nist.gov/18.27.E9
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: $q$ -Pochhammersymbols now link to their definition.
See also:: Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.9_5

h_{n}=\frac{\left(-c\right)^{n}a^{n+1}}{1-abq^{2n+1}}\frac{\left(q;q\right)_{n% }q^{\genfrac{(}{)}{0.0pt}{}{n+2}{2}}}{\left(aq,cq;q\right)_{n}}\*\frac{\left(q% ,c^{-1}aq,a^{-1}c,abq^{n+1};q\right)_{\infty}}{\left(aq,cq,bq^{n+1},c^{-1}abq^% {n+1};q\right)_{\infty}},

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $n$ : nonnegative integer and $h_{n}$
Referenced by:: §18.27(iii), Erratum (V1.2.0) Section 18.27
Permalink:: http://dlmf.nist.gov/18.27.E9_5
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.12

v_{x}=\frac{\left(qx/c,-qx/d;q\right)_{\infty}}{\left(q^{\alpha+1}x/c,-q^{% \beta+1}x/d;q\right)_{\infty}},

\alpha,\beta>-1

c,d>0

ⓘ

Symbols:: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $x$ : real variable and $v_{x}$
Permalink:: http://dlmf.nist.gov/18.27.E12
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: $q$ -Pochhammersymbols now link to their definition.
See also:: Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.14_1

h_{n}=\frac{\left(aq\right)^{n}}{1-abq^{2n+1}}\frac{\left(q,bq;q\right)_{n}}{% \left(aq;q\right)_{n}}\*\frac{\left(abq^{n+1};q\right)_{\infty}}{\left(aq;q% \right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $n$ : nonnegative integer and $h_{n}$
Source:: Koekoek et al. (2010, (14.12.2))
Referenced by:: §18.27(iv), (18.28.27), Erratum (V1.2.0) Section 18.27
Permalink:: http://dlmf.nist.gov/18.27.E14_1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.27(iv), §18.27 and Ch.18

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18.27.19

\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\,\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : Stieltjes–Wigert polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.27.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(vi), §18.27 and Ch.18

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12: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

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18.29.1

\left(bc,bd,cd;q\right)_{n}\*\left(Q_{n}({\mathrm{e}}^{\mathrm{i}\theta};a,b,c% ,d\mid q)+Q_{n}({\mathrm{e}}^{-\mathrm{i}\theta};a,b,c,d\mid q)\right),

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.29.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.29 and Ch.18

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18.29.2

Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},

n\to\infty

;

z, a, b, c, d, q

fixed.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $z$ : complex variable, $q$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.29.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.29 and Ch.18

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13: 17.14 Constant Term Identities

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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}}.

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base, $j$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §17.14
Permalink:: http://dlmf.nist.gov/17.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.14, §17.14 and Ch.17

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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

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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

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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

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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

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14: 5.18 $q$ -Gamma and $q$ -Beta Functions

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5.18.1

\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),

n=0,1,2,\dots

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable, $n$ : nonnegative integer, $k$ : nonnegative integer and $a$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(i), §5.18 and Ch.5

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5.18.2

n!_{q}=1(1+q)\cdots(1+q+\dots+q^{n-1})=\left(q;q\right)_{n}(1-q)^{-n}.

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Defines:: $!_{\NVar{q}}$ : $q$ -factorial (as in $n!_{q}$ )
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(i), §5.18 and Ch.5

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5.18.3

\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}).

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable, $k$ : nonnegative integer and $a$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(i), §5.18 and Ch.5

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5.18.4

\Gamma_{q}\left(z\right)=\left(q;q\right)_{\infty}(1-q)^{1-z}/\left(q^{z};q% \right)_{\infty},

ⓘ

Defines:: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real or complex variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.18.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(ii), §5.18 and Ch.5

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5.18.12

\mathrm{B}_{q}\left(a,b\right)=\int_{0}^{1}\frac{t^{a-1}\left(tq;q\right)_{% \infty}}{\left(tq^{b};q\right)_{\infty}}\,{\mathrm{d}}_{q}t,

0<q<1

\Re a>0

\Re b>0

ⓘ

Symbols:: $\int$ : integral, $\mathrm{B}_{\NVar{q}}\left(\NVar{a},\NVar{b}\right)$ : $q$ -beta function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $\Re$ : real part, $q$ : real or complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Permalink:: http://dlmf.nist.gov/5.18.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §5.18(iii), §5.18 and Ch.5

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15: 18.28 Askey–Wilson Class

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18.28.1

p_{n}(x)=p_{n}\left(x;a,b,c,d\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}% \left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},% abcdq^{n-1};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2% aq^{j}x+a^{2}q^{2j})},

ⓘ

Defines:: $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,|\,\NVar{q}\right)$ : Askey–Wilson polynomial
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $p_{n}(x)$ : polynomial of degree $n$ , $q$ : real variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: (18.28.29), §18.28(ii), §18.38(iii), Erratum (V1.2.0) for Equation (18.28.1)
Permalink:: http://dlmf.nist.gov/18.28.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.28(ii), §18.28 and Ch.18

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18.28.3

2\pi\sin\theta\,w(\cos\theta)={\left|\frac{\left({\mathrm{e}}^{2i\theta};q% \right)_{\infty}}{\left(a{\mathrm{e}}^{\mathrm{i}\theta},b{\mathrm{e}}^{% \mathrm{i}\theta},c{\mathrm{e}}^{\mathrm{i}\theta},d{\mathrm{e}}^{\mathrm{i}% \theta};q\right)_{\infty}}\right|}^{2},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $\sin\NVar{z}$ : sine function, $w(x)$ : weight function and $q$ : real variable
Permalink:: http://dlmf.nist.gov/18.28.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.4

h_{0}=\frac{\left(abcd;q\right)_{\infty}}{\left(q,ab,ac,ad,bc,bd,cd;q\right)_{% \infty}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.28.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.5

h_{n}=h_{0}\frac{(1-abcdq^{n-1})\left(q,ab,ac,ad,bc,bd,cd;q\right)_{n}}{(1-% abcdq^{2n-1})\left(abcd;q\right)_{n}},

n=1,2,\dotsc

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $n$ : nonnegative integer and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.28.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.22

h_{n}=\frac{\left(\alpha\beta\right)^{n+1}q^{\left(n+1\right)^{2}}}{\alpha% \beta q^{2n+1}-1}\frac{\left(q;q\right)_{n}}{\left(\alpha q,\beta\delta q,% \gamma q;q\right)_{n}}\*\frac{\left(\frac{\gamma}{\alpha\beta}q^{-n},\frac{% \delta}{\alpha}q^{-n},\frac{1}{\beta}q^{-n},\gamma\delta q;q\right)_{\infty}}{% \left(\frac{1}{\alpha\beta}q^{-n},\frac{\gamma\delta}{\alpha}q,\frac{\gamma}{% \beta}q,\delta q;q\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $\delta$ : arbitrary small positive constant, $q$ : real variable, $n$ : nonnegative integer and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.28.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §18.28(viii), §18.28 and Ch.18

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16: 17.6 ${{}_{2}\phi_{1}}$ Function

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17.6.3

{{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,q\right)=\frac{a^{n}\left(c/a;q% \right)_{n}}{\left(c;q\right)_{n}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

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17.6.5

{{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{% \infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q% \right)_{\infty}},

|b|>|q|

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

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17.6.10

(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{% \left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left(bq,zq;q\right)_{n}},

|z|<1

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

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17.6.14

\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}\left(b;q^{2}\right)_{n}z^{n}}{% \left(q;q\right)_{n}\left(azb;q^{2}\right)_{n}}=\frac{\left(az,bz;q^{2}\right)% _{\infty}}{\left(z,azb;q^{2}\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop bz% };q^{2},zq\right).

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.6(ii)
Permalink:: http://dlmf.nist.gov/17.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

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17.6.25

\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(% a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq% ^{n},bq^{n}\atop cq^{n}};q,dz\right),

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator
Permalink:: http://dlmf.nist.gov/17.6.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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17: 17.4 Basic Hypergeometric Functions

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17.4.1

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

ⓘ

Defines:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §18.27(i), §18.27(ii)
Permalink:: http://dlmf.nist.gov/17.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(i), §17.4 and Ch.17

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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

►

17.4.6

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};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,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}},

ⓘ

Defines:: $\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : second $q$ -Appell function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : 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, Erratum (V1.1.3) for Equation (17.4.6)
Permalink:: http://dlmf.nist.gov/17.4.E6
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.3):: The explicit usage of $\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}$ in the denominator of the right-hand side was used.
Correction (effective with 1.0.10):: The notation for $\Phi^{(2)}$ 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

►

17.4.7

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a,b;q\right)_{m}\left(a^{\prime},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:: $\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : third $q$ -Appell function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : 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.E7
Encodings:: pMML, png, TeX
Correction (effective with 1.0.10):: The notation for $\Phi^{(3)}$ 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

►

17.4.8

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)=\sum_{m,n\geq 0}\frac{\left(a,b;% q\right)_{m+n}x^{m}y^{n}}{\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n% }}.

ⓘ

Defines:: $\Phi^{(4)}\left(\NVar{a},\NVar{b};\NVar{c},\NVar{c^{\prime}};\NVar{q};\NVar{x}% ,\NVar{y}\right)$ : fourth $q$ -Appell function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : 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.E8
Encodings:: pMML, png, TeX
Correction (effective with 1.0.10):: The notation for $\Phi^{(4)}$ 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

…

18: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►

17.7.1

{{}_{2}\phi_{2}}\left({a,q/a\atop-q,b};q,-b\right)=\frac{\left(ab,bq/a;q^{2}% \right)_{\infty}}{\left(b;q\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: Erratum (V1.1.10) for Equation (17.7.1)
Permalink:: http://dlmf.nist.gov/17.7.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.10):: The constraint $|b|<1$ is not necessary and was removed.
See also:: Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

… ►

17.7.3

{{}_{2}\phi_{2}}\left({\ifrac{c^{2}}{b^{2}},b^{2}\atop c,cq};q^{2},q\right)=% \frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(c,cq;q^{2}\right)_% {\infty}}{\left(\frac{\left(c/b;q\right)_{\infty}}{\left(b;q\right)_{\infty}}+% \frac{\left(-c/b;q\right)_{\infty}}{\left(-b;q\right)_{\infty}}\right)}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base
Referenced by:: §17.6(i), §17.7(i)
Permalink:: http://dlmf.nist.gov/17.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

… ►

17.7.18

{{}_{r+2}\phi_{r+1}}\left({a,b,b_{1}q^{m_{1}},\dots,b_{r}q^{m_{r}}\atop bq,b_{% 1},\dots,b_{r}};q,a^{-1}q^{1-(m_{1}+\cdots+m_{r})}\right)=\frac{\left(q,bq/a;q% \right)_{\infty}\left(b_{1}/b;q\right)_{m_{1}}\cdots\left(b_{r}/b;q\right)_{m_% {r}}}{\left(bq,q/a;q\right)_{\infty}\left(b_{1};q\right)_{m_{1}}\cdots\left(b_% {r};q\right)_{m_{r}}}b^{m_{1}+\cdots+m_{r}},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $m$ : nonnegative integer and $r$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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17.7.20

\sum_{k=0}^{n}\frac{1-ap^{k}q^{k}}{1-a}\frac{\left(a;p\right)_{k}\left(c;q% \right)_{k}}{\left(q;q\right)_{k}\left(ap/c;p\right)_{k}}c^{-k}=\frac{\left(ap% ;p\right)_{n}\left(cq;q\right)_{n}}{\left(q;q\right)_{n}\left(ap/c;p\right)_{n% }}c^{-n}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

… ►

17.7.23

\left(1-\frac{a}{q}\right)\left(1-\frac{b}{q}\right)\sum_{k=0}^{n}\frac{\left(% ap^{k},bp^{-k};q\right)_{n-1}(1-(ap^{2k}/b))}{\left(p;p\right)_{n}\left(p;p% \right)_{n-k}\left(ap^{k}/b;q\right)_{n+1}}(-1)^{k}p^{\genfrac{(}{)}{0.0pt}{}{% k}{2}}=\delta_{n,0}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $k$ : nonnegative integer, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

19: 10.31 Power Series

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10.31.3

I_{\nu}\left(z\right)I_{\mu}\left(z\right)=(\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}% ^{\infty}\frac{{\left(\nu+\mu+k+1\right)_{k}}(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma% \left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.31
Permalink:: http://dlmf.nist.gov/10.31.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammersymbol now links to its definition.
See also:: Annotations for §10.31 and Ch.10

20: 18.1 Notation

… ►

$q$ -Pochhammer Symbol

… ►

18.1.1

C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{\left% (\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right),

n=1,2,3,\dots

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.1.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.1(iii), §18.1 and Ch.18

… ►

18.1.2

G_{n}\left(p,q,x\right)=\frac{n!}{{\left(n+p\right)_{n}}}P^{(p-q,q-1)}_{n}% \left(2x-1\right),

ⓘ

Defines:: $G_{\NVar{n}}\left(\NVar{p},\NVar{q},\NVar{x}\right)$ : shifted Jacobi polynomial
Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $q$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.1.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.1(iii), §18.1 and Ch.18

…

Pochhammer symbol

11—20 of 125 matching pages

11: 18.27 q -Hahn Class

12: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes