17 q-Hypergeometric and Related Functions Properties 17.9 Further Transformations of

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

q

-Appell Functions

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

ⓘ

Keywords:: Bailey’s ${{}_{2}\psi_{2}}$ transformations, bilateral $q$ -hypergeometric function
Notes:: See Gasper and Rahman (2004, pp. 147–150, 364–366).
Permalink:: http://dlmf.nist.gov/17.10
See also:: Annotations for Ch.17

Bailey’s ${{}_{2}\psi_{2}}$ Transformations

ⓘ

See also:: Annotations for §17.10 and Ch.17

17.10.1		${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),$
ⓘ 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, $q$ : complex base and $z$ : complex variable Permalink: http://dlmf.nist.gov/17.10.E1 Encodings: TeX, pMML, png See also: Annotations for §17.10, §17.10 and Ch.17

17.10.2		${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).$
ⓘ 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, $q$ : complex base and $z$ : complex variable Permalink: http://dlmf.nist.gov/17.10.E2 Encodings: TeX, pMML, png See also: Annotations for §17.10, §17.10 and Ch.17

Other Transformations

ⓘ

Keywords:: idem function, $q$ -hypergeometric function
See also:: Annotations for §17.10 and Ch.17

17.10.3		${{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),$
ⓘ 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, $q$ : complex base and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.10.E3 Encodings: TeX, pMML, png See also: Annotations for §17.10, §17.10 and Ch.17

17.10.4		${{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.$
ⓘ 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 $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.10.E4 Encodings: TeX, pMML, png See also: Annotations for §17.10, §17.10 and Ch.17

17.10.5		$\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).$
ⓘ Symbols: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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 Permalink: http://dlmf.nist.gov/17.10.E5 Encodings: TeX, pMML, png See also: Annotations for §17.10, §17.10 and Ch.17

17.10.6		$\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{2}}{bcdefghk}\right)=\frac{\left(q,q% /(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q\right)_{\infty}}{% \left(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{\infty}}\{{}_{10}\phi_{9}% }\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,% qg/h,qg/k};q,\frac{q^{2}}{bcdefghk}\right)+\operatorname{idem}\left(g;h,k% \right).$
ⓘ Symbols: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $k$ : nonnegative integer and $q$ : complex base Permalink: http://dlmf.nist.gov/17.10.E6 Encodings: TeX, pMML, png See also: Annotations for §17.10, §17.10 and Ch.17

17.9 Further Transformations of

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

Functions 17.11 Transformations of

q

-Appell Functions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§17.10 Transformations of ψrr Functions

Bailey’s ψ22 Transformations

Other Transformations

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

Bailey’s ${{}_{2}\psi_{2}}$ Transformations