§17.5 $\mathop{{{}_{{0}}\phi _{{0}}}\/}\nolimits,\mathop{{{}_{{1}}\phi _{{0}}}\/}\nolimits,\mathop{{{}_{{1}}\phi _{{1}}}\/}\nolimits$ Functions

Notes:: See Andrews (1976, pp. 17, 19, 36) and Gasper and Rahman (2004, pp. 25–26).
Permalink:: http://dlmf.nist.gov/17.5

¶ Euler’s Second Sum

Keywords:: Euler’s sums, $q$ -hypergeometric function

17.5.1 $\mathop{{{}_{{0}}\phi _{{0}}}\/}\nolimits\!\left(-;-;q,z\right)=\sum _{{n=0}}^{\infty}\frac{(-1)^{n}q^{{\binom{n}{2}}}z^{n}}{\left(q;q\right)_{{n}}}=\left(z;q\right)_{{\infty}},$ $|z|<1$ ;

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii)
Permalink:: http://dlmf.nist.gov/17.5.E1
Encodings:: TeX, pMML, png

compare (17.3.2).

¶ $q$ -Binomial Series

Keywords:: $q$ -binomial series

17.5.2 $\mathop{{{}_{{1}}\phi _{{0}}}\/}\nolimits\!\left(a;-;q,z\right)=\frac{\left(az;q\right)_{{\infty}}}{\left(z;q\right)_{{\infty}}},$ $|z|<1$ ;

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E2
Encodings:: TeX, pMML, png

compare (17.2.37).

¶ $q$ -Binomial Theorem

Keywords:: $q$ -binomial theorem

17.5.3 $\mathop{{{}_{{1}}\phi _{{0}}}\/}\nolimits\!\left(q^{{-n}};-;q,z\right)=\left(zq^{{-n}};q\right)_{{n}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E3
Encodings:: TeX, pMML, png

This is (17.2.35) reformulated.

¶ Euler’s First Sum

Keywords:: Euler’s sums, $q$ -hypergeometric function

17.5.4 $\mathop{{{}_{{1}}\phi _{{0}}}\/}\nolimits\!\left(0;-;q,z\right)=\sum _{{n=0}}^{\infty}\frac{z^{n}}{\left(q;q\right)_{{n}}}=\frac{1}{\left(z;q\right)_{{\infty}}},$ $|z|<1$ ;

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii)
Permalink:: http://dlmf.nist.gov/17.5.E4
Encodings:: TeX, pMML, png

compare (17.3.1).

¶ Cauchy’s Sum

Keywords:: $q$ -hypergeometric function

17.5.5 $\mathop{{{}_{{1}}\phi _{{1}}}\/}\nolimits\!\left({a\atop c};q,c/a\right)=\frac{\left(c/a;q\right)_{{\infty}}}{\left(c;q\right)_{{\infty}}},$ $|c|<|a|$ .

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.5.E5
Encodings:: TeX, pMML, png

§17.5 Functions