-Hypergeometric and Related Functions Properties 17.6 $\mathop{{{}_{{2}}\phi_{{1}}}\/}\nolimits$ Function 17.8 Special Cases of $\mathop{{{}_{{r}}\psi_{{r}}}\/}\nolimits$ Functions

§17.7 Special Cases of Higher $\mathop{{{}_{{r}}\phi_{{s}}}\/}\nolimits$ Functions

Keywords:: -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.7

§17.7(i) $\mathop{{{}_{{2}}\phi_{{2}}}\/}\nolimits$ Functions
§17.7(ii) $\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits$ Functions
§17.7(iii) Other $\mathop{{{}_{{r}}\phi_{{s}}}\/}\nolimits$ Functions

§17.7(i) $\mathop{{{}_{{2}}\phi_{{2}}}\/}\nolimits$ Functions

Notes:: See Gasper and Rahman (2004, pp. 28–29). For (17.7.3) combine Gasper and Rahman (2004, p. 359, Equation (III.4)) with (17.6.4).
Permalink:: http://dlmf.nist.gov/17.7.i

¶ -Analog of Bailey’s $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(-1\right)$ Sum

Keywords:: Bailey’s $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits(-1)$ sum, -hypergeometric function

17.7.1 $\mathop{{{}_{{2}}\phi_{{2}}}\/}\nolimits\!\left({a,q/a\atop-q,b};q,-b\right)=% \frac{\left(ab,bq/a;q^{2}\right)_{{\infty}}}{\left(b;q\right)_{{\infty}}},$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E1
Encodings:: TeX, pMML, png

¶ -Analog of Gauss’s $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(-1\right)$ Sum

Keywords:: Gauss’s $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits(-1)$ sum, -hypergeometric function

17.7.2 $\mathop{{{}_{{2}}\phi_{{2}}}\/}\nolimits\!\left({a^{2},b^{2}\atop abq^{{\frac{% 1}{2}}},-abq^{{\frac{1}{2}}}};q,-q\right)=\frac{\left(a^{2}q,b^{2}q;q^{2}% \right)_{{\infty}}}{\left(q,a^{2}b^{2}q;q^{2}\right)_{{\infty}}}.$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E2
Encodings:: TeX, pMML, png

¶ Sum Related to (17.6.4)

Keywords:: -hypergeometric function

17.7.3 $\mathop{{{}_{{2}}\phi_{{2}}}\/}\nolimits\!\left({\ifrac{c^{2}}{b^{2}},b^{2}% \atop c,cq};q^{2},q\right)=\frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{{% \infty}}}{\left(c,cq;q^{2}\right)_{{\infty}}}{\left(\frac{\left(c/b;q\right)_{% {\infty}}}{\left(b;q\right)_{{\infty}}}+\frac{\left(-c/b;q\right)_{{\infty}}}{% \left(-b;q\right)_{{\infty}}}\right)}.$

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 and : complex base
Referenced by:: §17.7(i)
Permalink:: http://dlmf.nist.gov/17.7.E3
Encodings:: TeX, pMML, png

§17.7(ii) $\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits$ Functions

Notes:: See Gasper and Rahman (2004, pp. 17, 58, 356).
Permalink:: http://dlmf.nist.gov/17.7.ii

¶ -Pfaff–Saalschütz Sum

Keywords:: -hypergeometric function

17.7.4 $\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits\!\left({a,b,q^{{-n}}\atop c,abq^{{1-n% }}/c};q,q\right)=\frac{\left(c/a,c/b;q\right)_{{n}}}{\left(c,c/(ab);q\right)_{% {n}}}.$

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 and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E4
Encodings:: TeX, pMML, png

¶ Nonterminating Form of the -Saalschütz Sum

Keywords:: -hypergeometric function

17.7.5 $\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits\!\left({a,b,c\atop e,f};q,q\right)+% \frac{\left(q/e,a,b,c,qf/e;q\right)_{{\infty}}}{\left(e/q,aq/e,bq/e,cq/e,f;q% \right)_{{\infty}}}\*\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits\!\left({aq/e,bq/% e,cq/e\atop q^{2}/e,qf/e};q,q\right)=\frac{\left(q/e,f/a,f/b,f/c;q\right)_{{% \infty}}}{\left(aq/e,bq/e,cq/e,f;q\right)_{{\infty}}},$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E5
Encodings:: TeX, pMML, png

where .

¶ F. H. Jackson’s Terminating -Analog of Dixon’s Sum

Keywords:: Dixon’s sum, -hypergeometric function

17.7.6 $\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits\!\left({q^{{-2n}},b,c\atop q^{{1-2n}}% /b,q^{{1-2n}}/c};q,\frac{q^{{2-n}}}{bc}\right)=\frac{\left(b,c;q\right)_{{n}}% \left(q,bc;q\right)_{{2n}}}{\left(q,bc;q\right)_{{n}}\left(b,c;q\right)_{{2n}}}.$

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 : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E6
Encodings:: TeX, pMML, png

¶ Continued Fractions

Keywords:: continued fractions, -hypergeometric function

For continued-fraction representations of a ratio of $\mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits$ functions, see Cuyt et al. (2008, pp. 399–400).

§17.7(iii) Other $\mathop{{{}_{{r}}\phi_{{s}}}\/}\nolimits$ Functions

Notes:: See Gasper and Rahman (2004, pp. 19, 43–44, 61–62, 81–83, 355–356) and Andrews (1996).
Permalink:: http://dlmf.nist.gov/17.7.iii

¶ -Analog of Dixon’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left(1\right)$ Sum

Keywords:: Dixon’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits(1)$ sum, -hypergeometric function

17.7.7 $\mathop{{{}_{{4}}\phi_{{3}}}\/}\nolimits\!\left({a,-qa^{{\frac{1}{2}}},b,c% \atop-a^{{\frac{1}{2}}},aq/b,aq/c};q,\frac{qa^{{\frac{1}{2}}}}{bc}\right)=% \frac{\left(aq,qa^{{\frac{1}{2}}}/b,qa^{{\frac{1}{2}}}/c,aq/(bc);q\right)_{{% \infty}}}{\left(aq/b,aq/c,qa^{{\frac{1}{2}}},qa^{{\frac{1}{2}}}/(bc);q\right)_% {{\infty}}}.$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E7
Encodings:: TeX, pMML, png

¶ Gasper–Rahman -Analog of Watson’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ Sum

Keywords:: Watson’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ sum

17.7.8 $\mathop{{{}_{{8}}\phi_{{7}}}\/}\nolimits\!\left({\lambda,q\lambda^{{\frac{1}{2% }}},-q\lambda^{{\frac{1}{2}}},a,b,c,-c,\lambda q/c^{2}\atop\lambda^{{\frac{1}{% 2}}},-\lambda^{{\frac{1}{2}}},\lambda q/a,\lambda q/b,\lambda q/c,-\lambda q/c% ,c^{2}};q,-\frac{\lambda q}{ab}\right)=\frac{\left(\lambda q,c^{2}/\lambda;q% \right)_{{\infty}}\left(aq,bq,c^{2}q/a,c^{2}q/b;q^{2}\right)_{{\infty}}}{\left% (\lambda q/a,\lambda q/b;q\right)_{{\infty}}\left(q,abq,c^{2}q,c^{2}q/(ab);q^{% 2}\right)_{{\infty}}},$

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 and : complex base
Referenced by:: ¶ ‣ §17.7(iii)
Permalink:: http://dlmf.nist.gov/17.7.E8
Encodings:: TeX, pMML, png

where $\lambda=-c(ab/q)^{{\frac{1}{2}}}$ .

¶ Andrews’ Terminating -Analog of (17.7.8)

Keywords:: Watson’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ sum

17.7.9 $\mathop{{{}_{{4}}\phi_{{3}}}\/}\nolimits\!\left({q^{{-n}},aq^{n},c,-c\atop(aq)% ^{{\frac{1}{2}}},-(aq)^{{\frac{1}{2}}},c^{2}};q,q\right)=\begin{cases}0,&\mbox% {$n$ odd,}\\ \dfrac{c^{n}\left(q,aq/c^{2};q^{2}\right)_{{n/2}}}{\left(aq,c^{2}q;q^{2}\right% )_{{n/2}}},&\mbox{$n$ even.}\end{cases}$

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 : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E9
Encodings:: TeX, pMML, png

¶ Gasper–Rahman -Analog of Whipple’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ Sum

Keywords:: Whipple’s $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ sum

17.7.10 $\mathop{{{}_{{8}}\phi_{{7}}}\/}\nolimits\!\left({-c,q(-c)^{{\frac{1}{2}}},-q(-% c)^{{\frac{1}{2}}},a,q/a,c,-d,-q/d\atop(-c)^{{\frac{1}{2}}},-(-c)^{{\frac{1}{2% }}},-cq/a,-ac,-q,cq/d,cd};q,c\right)=\frac{\left(-c,-cq;q\right)_{{\infty}}% \left(acd,acq/d,cdq/a,cq^{2}/(ad);q^{2}\right)_{{\infty}}}{\left(cd,cq/d,-ac,-% cq/a;q\right)_{{\infty}}}.$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E10
Encodings:: TeX, pMML, png

¶ Andrews’ Terminating -Analog

17.7.11 $\mathop{{{}_{{4}}\phi_{{3}}}\/}\nolimits\!\left({q^{{-n}},q^{{n+1}},c,-c\atop e% ,c^{2}q/e,-q};q,q\right)=\frac{\left(eq^{{-n}},eq^{{n+1}},c^{2}q^{{1-n}}/e,c^{% 2}q^{{n+2}}/e;q^{2}\right)_{{\infty}}}{\left(e,c^{2}q/e;q\right)_{{\infty}}}.$

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 : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E11
Encodings:: TeX, pMML, png

¶ First -Analog of Bailey’s $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left(1\right)$ Sum

Keywords:: Bailey’s $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits(1)$ sum, -hypergeometric function

17.7.12 $\mathop{{{}_{{4}}\phi_{{3}}}\/}\nolimits\!\left({a,aq,b^{2}q^{{2n}},q^{{-2n}}% \atop b,bq,a^{2}q^{2}};q^{2},q^{2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{{n% }}}{\left(-aq,b;q\right)_{{n}}}.$

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 and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E12
Encodings:: TeX, pMML, png

¶ Second -Analog of Bailey’s $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left(1\right)$ Sum

17.7.13 $\mathop{{{}_{{4}}\phi_{{3}}}\/}\nolimits\!\left({a,aq,b^{2}q^{{2n-2}},q^{{-2n}% }\atop b,bq,a^{2}};q^{2},q^{2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{{n}}(1% -bq^{{n-1}})}{\left(-a,b;q\right)_{{n}}(1-bq^{{2n-1}})}.$

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 : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E13
Encodings:: TeX, pMML, png

¶ F. H. Jackson’s -Analog of Dougall’s $\mathop{{{}_{{7}}F_{{6}}}\/}\nolimits\!\left(1\right)$ Sum

Keywords:: Dougall’s $\mathop{{{}_{{7}}F_{{6}}}\/}\nolimits(1)$ sum, -hypergeometric function

17.7.14 $\mathop{{{}_{{8}}\phi_{{7}}}\/}\nolimits\!\left({a,qa^{{\frac{1}{2}}},-qa^{{% \frac{1}{2}}},b,c,d,e,q^{{-n}}\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,% aq/c,aq/d,aq/e,aq^{{n+1}}};q,q\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd);q% \right)_{{n}}}{\left(aq/b,aq/c,aq/d,aq/(bcd);q\right)_{{n}}},$

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 and : nonnegative integer
Referenced by:: ¶ ‣ §17.7(iii)
Permalink:: http://dlmf.nist.gov/17.7.E14
Encodings:: TeX, pMML, png

where $a^{2}q=bcdeq^{{-n}}$ .

¶ Limiting Cases of (17.7.14)

17.7.15 $\mathop{{{}_{{6}}\phi_{{5}}}\/}\nolimits\!\left({a,qa^{{\frac{1}{2}}},-qa^{{% \frac{1}{2}}},b,c,d\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,aq/c,aq/d};% q,\frac{aq}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{{% \infty}}}{\left(aq/b,aq/c,aq/d,aq/(bcd);q\right)_{{\infty}}},$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E15
Encodings:: TeX, pMML, png

and when $d=q^{{-n}}$ ,

17.7.16 $\mathop{{{}_{{6}}\phi_{{5}}}\/}\nolimits\!\left({a,qa^{{\frac{1}{2}}},-qa^{{% \frac{1}{2}}},b,c,q^{{-n}}\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,aq/c% ,aq^{{n+1}}};q,\frac{aq^{{n+1}}}{bc}\right)=\frac{\left(aq,aq/(bc);q\right)_{{% n}}}{\left(aq/b,aq/c;q\right)_{{n}}}.$

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 and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E16
Encodings:: TeX, pMML, png

¶ Bailey’s Nonterminating Extension of Jackson’s $\mathop{{{}_{{8}}\phi_{{7}}}\/}\nolimits$ Sum

17.7.17 $\mathop{{{}_{{8}}\phi_{{7}}}\/}\nolimits\!\left({a,qa^{{\frac{1}{2}}},-qa^{{% \frac{1}{2}}},b,c,d,e,f\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,aq/c,aq% /d,aq/e,aq/f};q,q\right)-\frac{b}{a}\frac{\left(aq,c,d,e,f,bq/a,bq/c,bq/d,bq/e% ,bq/f;q\right)_{{\infty}}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,% b^{2}q/a;q\right)_{{\infty}}}\*\mathop{{{}_{{8}}\phi_{{7}}}\/}\nolimits\!\left% ({b^{2}/a,qba^{{-\frac{1}{2}}},-qba^{{-\frac{1}{2}}},b,bc/a,bd/a,be/a,bf/a% \atop ba^{{-\frac{1}{2}}},-ba^{{-\frac{1}{2}}},bq/a,bq/c,bq/d,bq/e,bq/f};q,q% \right)=\frac{\left(aq,b/a,aq/(cd),aq/(ce),aq/(cf),aq/(de),aq/(df),aq/(ef);q% \right)_{{\infty}}}{\left(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q\right)_{{% \infty}}},$

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 and : complex base
Permalink:: http://dlmf.nist.gov/17.7.E17
Encodings:: TeX, pMML, png

where $qa^{2}=bcdef$ .

¶ Gasper–Rahman -Analogs of the Karlsson–Minton Sums

17.7.18 $\mathop{{{}_{{r+2}}\phi_{{r+1}}}\/}\nolimits\!\left({a,b,b_{1}q^{{m_{1}}},% \dots,b_{r}q^{{m_{r}}}\atop bq,b_{1},\dots,b_{r}};q,a^{{-1}}q^{{1-(m_{1}+% \cdots+m_{r})}}\right)=\frac{\left(q,bq/a;q\right)_{{\infty}}\left(b_{1}/b;q% \right)_{{m_{1}}}\cdots\left(b_{r}/b;q\right)_{{m_{r}}}}{\left(bq,q/a;q\right)% _{{\infty}}\left(b_{1};q\right)_{{m_{1}}}\cdots\left(b_{r};q\right)_{{m_{r}}}}% b^{{m_{1}+\cdots+m_{r}}},$

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 : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E18
Encodings:: TeX, pMML, png

and

17.7.19 $\mathop{{{}_{{r+1}}\phi_{{r}}}\/}\nolimits\!\left({a,b_{1}q^{{m_{1}}},\dots,b_% {r}q^{{m_{r}}}\atop b_{1},\dots,b_{r}};q,a^{{-1}}q^{{1-(m_{1}+\cdots+m_{r})}}% \right)=0,$

Symbols:: $\mathop{{{}_{{r+1}}\phi_{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}% \atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E19
Encodings:: TeX, pMML, png

where $m_{1},m_{2},\dots,m_{r}$ are arbitrary nonnegative integers.

¶ Gosper’s Bibasic Sum

Keywords:: -hypergeometric function

17.7.20 $\sum_{{k=0}}^{n}\frac{1-ap^{k}q^{k}}{1-a}\frac{\left(a;p\right)_{{k}}\left(c;q% \right)_{{k}}}{\left(q;q\right)_{{k}}\left(ap/c;p\right)_{{k}}}c^{{-k}}=\frac{% \left(ap;p\right)_{{n}}\left(cq;q\right)_{{n}}}{\left(q;q\right)_{{n}}\left(ap% /c;p\right)_{{n}}}c^{{-n}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E20
Encodings:: TeX, pMML, png

¶ Gasper’s Extensions of Gosper’s Bibasic Sum

17.7.21 $\sum_{{k=0}}^{n}\frac{(1-ap^{k}q^{k})(1-bp^{k}q^{{-k}})}{(1-a)(1-b)}\frac{% \left(a,b;p\right)_{{k}}\left(c,a/(bc);q\right)_{{k}}}{\left(q,aq/b;q\right)_{% {k}}\left(ap/c,bcp;p\right)_{{k}}}q^{k}=\frac{\left(ap,bp;p\right)_{{n}}\left(% cq,aq/(bc);q\right)_{{n}}}{\left(q,aq/b;q\right)_{{n}}\left(ap/c,bcp;p\right)_% {{n}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E21
Encodings:: TeX, pMML, png

17.7.22 $\sum_{{k=-m}}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq^{k}))}{(1-ad)(1-(b/d))}% \frac{\left(a,b;p\right)_{{k}}\left(c,ad^{2}/(bc);q\right)_{{k}}}{\left(dq,adq% /b;q\right)_{{k}}\left(adp/c,bcp/d;p\right)_{{k}}}q^{k}=\frac{(1-a)(1-b)(1-c)(% 1-(ad^{2}/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))}\left(\frac{\left(ap,% bp;p\right)_{{n}}\left(cq,ad^{2}q/(bc);q\right)_{{n}}}{\left(dq,adq/b;q\right)% _{{n}}\left(adp/c,bcp/d;p\right)_{{n}}}-\frac{\left(c/(ad),d/(bc);p\right)_{{m% +1}}\left(1/d,b/(ad);q\right)_{{m+1}}}{\left(1/c,bc/(ad^{2});q\right)_{{m+1}}% \left(1/a,1/b;p\right)_{{m+1}}}\right),$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E22
Encodings:: TeX, pMML, png

and -th difference generalization:

17.7.23 $\left(1-\frac{a}{q}\right)\left(1-\frac{b}{q}\right)\sum_{{k=0}}^{n}\frac{% \left(ap^{k},bp^{{-k}};q\right)_{{n-1}}(1-(ap^{{2k}}/b))}{\left(p;p\right)_{{n% }}\left(p;p\right)_{{n-k}}\left(ap^{k}/b;q\right)_{{n+1}}}(-1)^{k}p^{{\binom{k% }{2}}}=\delta_{{n,0}}.$

Symbols:: $\delta_{{j,k}}$ : Kronecker delta, $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : nonnegative integer, : complex base and : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.7.E23
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. 17.6 $\mathop{{{}_{{2}}\phi_{{1}}}\/}\nolimits$ Function 17.8 Special Cases of $\mathop{{{}_{{r}}\psi_{{r}}}\/}\nolimits$ Functions

§17.7 Special Cases of Higher Functions

Contents

§17.7(i) Functions

¶ -Analog of Bailey’s Sum

¶ -Analog of Gauss’s Sum