-Hypergeometric and Related Functions Properties 17.4 Basic Hypergeometric Functions 17.6 $\mathop{{{}_{{2}}\phi_{{1}}}\/}\nolimits$ Function

§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, -hypergeometric function

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

;

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

compare (17.3.2).

¶ -Binomial Series

Keywords:: -binomial series

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

;

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

compare (17.2.37).

¶ -Binomial Theorem

Keywords:: -binomial theorem

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

Symbols:: $\left(a;q\right)_{{n}}$ : -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, : complex base, : nonnegative integer and : 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, -hypergeometric function

17.5.4 $\mathop{{{}_{{1}}\phi_{{0}}}\/}\nolimits\!\left(0;-;q,z\right)=\sum_{{n=0}}^{% \infty}\frac{z^{n}}{\left(q;q\right)_{{n}}}=\frac{1}{\left(z;q\right)_{{\infty% }}},$

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