§17.8 Special Cases of $\mathop{{{}_{{r}}\psi _{{r}}}\/}\nolimits$ Functions

Notes:: See Gasper and Rahman (2004, pp. 15–16, 52–53, 140–141, 146–147, 149–150, 153).
Keywords:: bilateral $q$ -hypergeometric function
Referenced by:: §17.3(iv)
Permalink:: http://dlmf.nist.gov/17.8

¶ Jacobi’s Triple Product

Keywords:: Jacobi’s triple product

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E1
Encodings:: TeX, pMML, png

compare (20.5.9).

¶ Ramanujan’s $\mathop{{{}_{{1}}\psi _{{1}}}\/}\nolimits$ Summation

Keywords:: Ramanujan’s $\mathop{{{}_{{1}}\psi _{{1}}}\/}\nolimits$ summation, bilateral $q$ -hypergeometric function

17.8.2 $\mathop{{{}_{{1}}\psi _{{1}}}\/}\nolimits\!\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q\right)_{{\infty}}}{\left(b,q/a,z,b/(az);q\right)_{{\infty}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E2
Encodings:: TeX, pMML, png

¶ Quintuple Product Identity

Keywords:: $q$ -hypergeometric function

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:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E3
Encodings:: TeX, pMML, png

¶ Bailey’s Bilateral Summations

Keywords:: Bailey’s bilateral summations, bilateral $q$ -hypergeometric function

17.8.4 $\mathop{{{}_{{2}}\psi _{{2}}}\/}\nolimits\!\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q\right)_{{\infty}}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{{\infty}}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E4
Encodings:: TeX, pMML, png

17.8.5 $\mathop{{{}_{{3}}\psi _{{3}}}\/}\nolimits\!\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{\left(q,q/(bc),q/(bd),q/(cd);q\right)_{{\infty}}}{\left(q/b,q/c,q/d,q/(bcd);q\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E5
Encodings:: TeX, pMML, png

17.8.6 $\mathop{{{}_{{4}}\psi _{{4}}}\/}\nolimits\!\left({-qa^{{\frac{1}{2}}},b,c,d\atop-a^{{\frac{1}{2}}},aq/b,aq/c,aq/d};q,\frac{qa^{{\frac{3}{2}}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd),qa^{{\frac{1}{2}}}/b,qa^{{\frac{1}{2}}}/c,qa^{{\frac{1}{2}}}/d,q,q/a;q\right)_{{\infty}}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{{\frac{1}{2}}},qa^{{-\frac{1}{2}}},qa^{{\frac{3}{2}}}/(bcd);q\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E6
Encodings:: TeX, pMML, png

17.8.7 $\mathop{{{}_{{6}}\psi _{{6}}}\/}\nolimits\!\left({qa^{{\frac{1}{2}}},-qa^{{\frac{1}{2}}},b,c,d,e\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right)_{{\infty}}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{{\infty}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E7
Encodings:: TeX, pMML, png

§17.8 Special Cases of Functions

¶ Jacobi’s Triple Product

¶ Ramanujan’s Summation

¶ Quintuple Product Identity

¶ Bailey’s Bilateral Summations

§17.8 Special Cases of $\mathop{{{}_{{r}}\psi _{{r}}}\/}\nolimits$ Functions

¶ Ramanujan’s $\mathop{{{}_{{1}}\psi _{{1}}}\/}\nolimits$ Summation