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

Jacobiβs Triple Product

 17.8.1 $\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$

compare (20.5.9).

Analytic Continuation

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.

Ramanujanβs ${{}_{1}\psi_{1}}$ Summation

 17.8.2 ${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}},$ $\left|b/a\right|<\left|z\right|<1$. β 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 Β§17.8, Β§17.8 and Ch.17

Quintuple Product Identity

 17.8.3 $\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.$

Baileyβs Bilateral Summations

 17.8.4 $\displaystyle{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)$ $\displaystyle=\frac{\left(aq/(bc);q\right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c% ^{2},q^{2},aq,q/a;q^{2}\right)_{\infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q% \right)_{\infty}},$ $\left|qa\right|<\left|bc\right|$, β 17.8.5 $\displaystyle{{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)$ $\displaystyle=\frac{\left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,% q/c,q/d,q/(bcd);q\right)_{\infty}},$ $\left|q\right|<\left|bcd\right|$, β 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 Β§17.8, Β§17.8 and Ch.17
 17.8.6 ${{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},$ $\left|qa^{\frac{3}{2}}\right|<\left|bcd\right|$, β 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 Β§17.8, Β§17.8 and Ch.17
 17.8.7 ${{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}},$ $\left|qa^{2}\right|<\left|bcde\right|$. β 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 Β§17.8, Β§17.8 and Ch.17

Sum Related to (17.6.4)

 17.8.8 ${{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),$ $\left|cq^{2}\right|<\left|b^{2}\right|$. β 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: Β§17.6(i), Β§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 Β§17.8, Β§17.8 and Ch.17

For similar formulas see Verma and Jain (1983).