17 q-Hypergeometric and Related Functions Properties 17.7 Special Cases of Higher

{{}_{r}\phi_{s}}

Functions 17.9 Further Transformations of

{{}_{r+1}\phi_{r}}

Functions

§17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

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Keywords:: bilateral $q$ -hypergeometric function, special cases
Notes:: See Gasper and Rahman (2004, pp. 15–16, 52–53, 140–141, 146–147, 149–150, 153).
Referenced by:: §17.3(iv), Erratum (V1.2.1) for §17.8
Permalink:: http://dlmf.nist.gov/17.8
Correction (effective with 1.2.3):: Text was added below (17.8.3) discussing higher-order tuple identities.
Addition (effective with 1.2.1):: Just above the paragraph Ramanujan’s ${{}_{1}\psi_{1}}$ Summation, a paragraph entitled “Analytic continuation” was inserted describing the analytic continuation of formulas which follow.
Addition (effective with 1.1.0):: Equation (17.8.8) was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for Ch.17

Jacobi’s Triple Product

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Keywords:: Jacobi’s triple product, $q$ -version
See also:: Annotations for §17.8 and Ch.17

17.8.1		$\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$
ⓘ Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §17.8 Permalink: http://dlmf.nist.gov/17.8.E1 Encodings: TeX, pMML, png See also: Annotations for §17.8, §17.8 and Ch.17

compare (20.5.9).

Analytic Continuation

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Referenced by:: Erratum (V1.2.1) for §17.8
See also:: Annotations for §17.8 and Ch.17

Note that for the equations below, the constraints are included to guarantee that the infinite series representation (17.4.3) of the ${{}_{r}\psi_{r}}$ functions converges. These equations can also be used as analytic continuation of these ${{}_{r}\psi_{r}}$ functions.

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

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Keywords:: Ramanujan’s ${{}_{1}\psi_{1}}$ summation, bilateral $q$ -hypergeometric function
Referenced by:: Erratum (V1.2.1) for §17.8
See also:: Annotations for §17.8 and Ch.17

17.8.2		${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}},$
$\left\|b/a\right\|<\left\|z\right\|<1$ .
ⓘ Symbols: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable Referenced by: Erratum (V1.2.1) for Equation (17.8.2) Permalink: http://dlmf.nist.gov/17.8.E2 Encodings: TeX, pMML, png Correction (effective with 1.2.1): The constraint $\left\|b/a\right\|<\left\|z\right\|<1$ was added. See also: Annotations for §17.8, §17.8 and Ch.17

Quintuple Product Identity

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Keywords:: $q$ -hypergeometric function, quintuple product identity
See also:: Annotations for §17.8 and Ch.17

17.8.3		$\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.$
ⓘ Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §17.8, §17.8, Erratum (V1.2.3) for Additions Permalink: http://dlmf.nist.gov/17.8.E3 Encodings: TeX, pMML, png See also: Annotations for §17.8, §17.8 and Ch.17

Apart from Jacobi’s triple product identity (17.8.1) and the quintuple product identity (17.8.3) (see Cooper (2006) for a review), there also exist higher-order tuple product identities. One may see Pascadi (2021) for discussions and derivations of sextuple, septuple, octuple, nonuple and undecuple product identities. These identities are all given in terms of sums and products of basic bilateral hypergeometric series.

Bailey’s Bilateral Summations

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Keywords:: Bailey’s bilateral summations, bilateral $q$ -hypergeometric function
See also:: Annotations for §17.8 and Ch.17

17.8.4	$\displaystyle{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)$	$\displaystyle=\frac{\left(aq/(bc);q\right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c% ^{2},q^{2},aq,q/a;q^{2}\right)_{\infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q% \right)_{\infty}},$
$\left\|qa\right\|<\left\|bc\right\|$ ,
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Referenced by: Erratum (V1.2.1) for Equation (17.8.4) Permalink: http://dlmf.nist.gov/17.8.E4 Encodings: TeX, pMML, png Correction (effective with 1.2.1): The constraint $\left\|qa\right\|<\left\|bc\right\|$ was added. See also: Annotations for §17.8, §17.8 and Ch.17
17.8.5	$\displaystyle{{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)$	$\displaystyle=\frac{\left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,% q/c,q/d,q/(bcd);q\right)_{\infty}},$
$\left\|q\right\|<\left\|bcd\right\|$ ,
ⓘ Symbols: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Referenced by: Erratum (V1.2.1) for Equation (17.8.5) Permalink: http://dlmf.nist.gov/17.8.E5 Encodings: TeX, pMML, png Correction (effective with 1.2.1): The constraint $\left\|q\right\|<\left\|bcd\right\|$ was added. See also: Annotations for §17.8, §17.8 and Ch.17

17.8.6		${{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},$
$\left\|qa^{\frac{3}{2}}\right\|<\left\|bcd\right\|$ ,
ⓘ Symbols: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Referenced by: Erratum (V1.2.1) for Equation (17.8.6) Permalink: http://dlmf.nist.gov/17.8.E6 Encodings: TeX, pMML, png Correction (effective with 1.2.1): The constraint $\left\|qa^{\frac{3}{2}}\right\|<\left\|bcd\right\|$ was added. See also: Annotations for §17.8, §17.8 and Ch.17

17.8.7		${{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}},$
$\left\|qa^{2}\right\|<\left\|bcde\right\|$ .
ⓘ Symbols: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Referenced by: Erratum (V1.2.1) for Equation (17.8.7) Permalink: http://dlmf.nist.gov/17.8.E7 Encodings: TeX, pMML, png Correction (effective with 1.2.1): The constraint $\left\|qa^{2}\right\|<\left\|bcde\right\|$ was added. See also: Annotations for §17.8, §17.8 and Ch.17

Sum Related to (17.6.4)

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Keywords:: bilateral $q$ -hypergeometric function, special cases
See also:: Annotations for §17.8 and Ch.17

17.8.8		${{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),$
$\left\|cq^{2}\right\|<\left\|b^{2}\right\|$ .
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Source: Verma and Jain (1983, (3.9)) Referenced by: §17.6(i), §17.8, Erratum (V1.1.0) for Additions, Erratum (V1.2.1) for Equation (17.8.8) Permalink: http://dlmf.nist.gov/17.8.E8 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. Suggested 2019-10-19 by Slobodan Damjanovic See also: Annotations for §17.8, §17.8 and Ch.17

For similar formulas see Verma and Jain (1983).

17.7 Special Cases of Higher

{{}_{r}\phi_{s}}

Functions 17.9 Further Transformations of

{{}_{r+1}\phi_{r}}

Functions

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§17.8 Special Cases of ψrr Functions