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

ⓘ

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
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

ⓘ

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 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

ⓘ

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

ⓘ

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

ⓘ

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 Permalink: http://dlmf.nist.gov/17.8.E3 Encodings: TeX, pMML, png See also: Annotations for §17.8, §17.8 and Ch.17

Bailey’s Bilateral Summations

ⓘ

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)

ⓘ

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