-Hypergeometric and Related Functions Properties 17.9 Transformations of Higher $\mathop{{{}_{{r}}\phi_{{r}}}\/}\nolimits$ Functions 17.11 Transformations of

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

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

¶ Bailey’s $\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits$ Transformations

17.10.1 $\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits\!\left({a,b\atop c,d};q,z\right)=% \frac{\left(az,d/a,c/b,dq/(abz);q\right)_{{\infty}}}{\left(z,d,q/b,cd/(abz);q% \right)_{{\infty}}}\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits\!\left({a,abz/d% \atop az,c};q,\frac{d}{a}\right),$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r}}\psi_{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b% _{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : complex base and : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E1
Encodings:: TeX, pMML, png

17.10.2 $\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits\!\left({a,b\atop c,d};q,z\right)=% \frac{\left(az,bz,cq/(abz),dq/(abz);q\right)_{{\infty}}}{\left(q/a,q/b,c,d;q% \right)_{{\infty}}}\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits\!\left({abz/c,abz/% d\atop az,bz};q,\frac{cd}{abz}\right).$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r}}\psi_{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b% _{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : complex base and : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E2
Encodings:: TeX, pMML, png

¶ Other Transformations

Keywords:: -hypergeometric function

17.10.3 $\mathop{{{}_{{8}}\psi_{{8}}}\/}\nolimits\!\left({qa^{{\frac{1}{2}}},-qa^{{% \frac{1}{2}}},c,d,e,f,aq^{{-n}},q^{{-n}}\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{% 2}}},aq/c,aq/d,aq/e,aq/f,q^{{n+1}},aq^{{n+1}}};q,\frac{a^{2}q^{{2n+2}}}{cdef}% \right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{{n}}}{\left(q/c,q/d,aq/e,% aq/f;q\right)_{{n}}}\*\mathop{{{}_{{4}}\psi_{{4}}}\/}\nolimits\!\left({e,f,aq^% {{n+1}}/(cd),q^{{-n}}\atop aq/c,aq/d,q^{{n+1}},ef/(aq^{n})};q,q\right),$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r}}\psi_{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b% _{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E3
Encodings:: TeX, pMML, png

17.10.4 $\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits\!\left({e,f\atop aq/c,aq/d};q,\frac{% aq}{ef}\right)=\frac{\left(q/c,q/d,aq/e,aq/f;q\right)_{{\infty}}}{\left(aq,q/a% ,aq/(cd),aq/(ef);q\right)_{{\infty}}}\*\sum_{{n=-\infty}}^{{\infty}}\frac{(1-% aq^{{2n}})\left(c,d,e,f;q\right)_{{n}}}{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right% )_{{n}}}\left(\frac{qa^{3}}{cdef}\right)^{n}q^{{n^{2}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r}}\psi_{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b% _{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E4
Encodings:: TeX, pMML, png

17.10.5 $\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{{\infty}% }}{\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{{\infty}}}\*\mathop{{{}_{{8}}% \psi_{{8}}}\/}\nolimits\!\left({qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g};q,\frac{q^{2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf)% ,q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q\right)_{{\infty}}}{\left(fa,q/(fa),aq/f,f% /a,g/f,fg,qf^{2};q\right)_{{\infty}}}\*\mathop{{{}_{{8}}\phi_{{7}}}\/}% \nolimits\!\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d,fq/e,fq% /g};q,\frac{q^{2}}{bcdefg}\right)+\mathop{\mathrm{idem}\/}\nolimits\!\left(f;g% \right).$

Symbols:: $\mathop{\mathrm{idem}\/}\nolimits\!\left(\chi_{1};\chi_{2},\dots,\chi_{n}\right)$ : $\mathop{\mathrm{idem}\/}\nolimits$ function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r+1}}\phi_{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}% \atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or -hypergeometric) function, $\mathop{{{}_{{r}}\psi_{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b% _{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral -hypergeometric) function and : complex base
Permalink:: http://dlmf.nist.gov/17.10.E5
Encodings:: TeX, pMML, png

17.10.6 $\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{{\infty}}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{{% \infty}}}\*\mathop{{{}_{{10}}\psi_{{10}}}\/}\nolimits\!\left({qa,-qa,ba,ca,da,% ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{2% }}{bcdefghk}\right)=\frac{\left(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c% ,qg/d,qg/e,qg/f;q\right)_{{\infty}}}{\left(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg^{2}% ;q\right)_{{\infty}}}\*\mathop{{{}_{{10}}\phi_{{9}}}\/}\nolimits\!\left({g^{2}% ,qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k};q,% \frac{q^{2}}{bcdefghk}\right)+\mathop{\mathrm{idem}\/}\nolimits\!\left(g;h,k% \right).$

Symbols:: $\mathop{\mathrm{idem}\/}\nolimits\!\left(\chi_{1};\chi_{2},\dots,\chi_{n}\right)$ : $\mathop{\mathrm{idem}\/}\nolimits$ function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r+1}}\phi_{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}% \atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or -hypergeometric) function, $\mathop{{{}_{{r}}\psi_{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b% _{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral -hypergeometric) function, : nonnegative integer and : complex base
Permalink:: http://dlmf.nist.gov/17.10.E6
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 17.9 Transformations of Higher $\mathop{{{}_{{r}}\phi_{{r}}}\/}\nolimits$ Functions 17.11 Transformations of

-Appell Functions

§17.10 Transformations of Functions

¶ Bailey’s Transformations

¶ Other Transformations

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

¶ Bailey’s $\mathop{{{}_{{2}}\psi_{{2}}}\/}\nolimits$ Transformations