§17.4 Basic Hypergeometric Functions

Keywords:: basic hypergeometric functions, $q$ -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.4

§17.4(i) $\mathop{{{}_{{r}}\phi _{{s}}}\/}\nolimits$ Functions
§17.4(ii) $\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits$ Functions
§17.4(iii) Appell Functions
§17.4(iv) Classification

§17.4(i) $\mathop{{{}_{{r}}\phi _{{s}}}\/}\nolimits$ Functions

Referenced by:: §17.1, §17.18, §18.27(i), §18.28(i), ¶ ‣ §18.33(iv)
Permalink:: http://dlmf.nist.gov/17.4.i

17.4.1 $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)=\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left(a_{0},a_{1},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)=\sum _{{n=0}}^{\infty}\frac{\left(a_{0};q\right)_{{n}}\left(a_{1};q\right)_{{n}}\cdots\left(a_{r};q\right)_{{n}}}{\left(q;q\right)_{{n}}\left(b_{1};q\right)_{{n}}\cdots\left(b_{s};q\right)_{{n}}}\*\left((-1)^{n}q^{{\binom{n}{2}}}\right)^{{s-r}}z^{n}.$

Defines:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function
Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv), §17.4(iv)
Permalink:: http://dlmf.nist.gov/17.4.E1
Encodings:: TeX, pMML, png

Here and elsewhere it is assumed that the $b_{j}$ do not take any of the values $q^{{-n}}$ . The infinite series converges for all $z$ when $s>r$ , and for $|z|<1$ when $s=r$ .

17.4.2 $\lim _{{q\to 1-}}\mathop{{{}_{{r+1}}\phi _{{r}}}\/}\nolimits\!\left({q^{{a_{0}}},q^{{a_{1}}},\dots,q^{{a_{r}}}\atop q^{{b_{1}}},\dots,q^{{b_{r}}}};q,z\right)=\mathop{{{}_{{r+1}}F_{{r}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},\dots,b_{r}};z\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $r$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.4.E2
Encodings:: TeX, pMML, png

For the function on the right-hand side see §16.2(i).

This notation is from Gasper and Rahman (2004). It is slightly at variance with the notation in Bailey (1935) and Slater (1966). In these references the factor $\left((-1)^{n}q^{{\binom{n}{2}}}\right)^{{s-r}}$ is not included in the sum. In practice this discrepancy does not usually cause serious problems because the case most often considered is $r=s$ .

§17.4(ii) $\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits$ Functions

Keywords:: bilateral $q$ -hypergeometric function, $q$ -hypergeometric function
Referenced by:: §17.18
Permalink:: http://dlmf.nist.gov/17.4.ii

17.4.3 $\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)=\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits\!\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)=\sum _{{n=-\infty}}^{{\infty}}\frac{\left(a_{1},a_{2},\dots,a_{r};q\right)_{{n}}(-1)^{{(s-r)n}}q^{{(s-r)\binom{n}{2}}}z^{n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{{n}}}=\sum _{{n=0}}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q\right)_{{n}}(-1)^{{(s-r)n}}q^{{(s-r)\binom{n}{2}}}z^{n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{{n}}}+\sum _{{n=1}}^{\infty}\frac{\left(q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{{n}}}{\left(q/a_{1},q/a_{2},\dots,q/a_{r};q\right)_{{n}}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}\right)^{n}.$

Defines:: $\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function
Symbols:: $\binom{m}{n}$ : binomial coefficient, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.4.E3
Encodings:: TeX, pMML, png

Here and elsewhere the $b_{j}$ must not take any of the values $q^{{-n}}$ , and the $a_{j}$ must not take any of the values $q^{{n+1}}$ . The infinite series converge when $s\geq r$ provided that $|(b_{1}\cdots b_{s})/(a_{1}\cdots a_{r}z)|<1$ and also, in the case $s=r$ , $|z|<1$ .

17.4.4 $\lim _{{q\to 1-}}\mathop{{{}_{{r}}\psi _{{r}}}\/}\nolimits\!\left({q^{{a_{1}}},q^{{a_{2}}},\dots,q^{{a_{r}}}\atop q^{{b_{1}}},q^{{b_{2}}},\dots,q^{{b_{r}}}};q,z\right)=\mathop{{{}_{{r}}H_{{r}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{r}};z\right).$

Symbols:: $\mathop{{{}_{{p}}H_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : bilateral 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 $q$ -hypergeometric) function, $q$ : complex base, $r$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.4.E4
Encodings:: TeX, pMML, png

For the function $\mathop{{{}_{{r}}H_{{r}}}\/}\nolimits$ see §16.4(v).

§17.4(iii) Appell Functions

Keywords:: $q$ -Appell functions
Permalink:: http://dlmf.nist.gov/17.4.iii

The following definitions apply when $|x|<1$ and $|y|<1$ :

17.4.5 $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)=\sum _{{m,n\geq 0}}\frac{\left(a;q\right)_{{m+n}}\left(b;q\right)_{{m}}\left(b^{{\prime}};q\right)_{{n}}x^{m}y^{n}}{\left(q;q\right)_{{m}}\left(q;q\right)_{{n}}\left(c;q\right)_{{m+n}}},$

Defines:: $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ : first $q$ -Appell function
Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/17.4.E5
Encodings:: TeX, pMML, png

17.4.6 $\mathop{\Phi^{{(2)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c,c^{{\prime}};x,y\right)=\sum _{{m,n\geq 0}}\frac{\left(a;q\right)_{{m+n}}\left(b;q\right)_{{m}}\left(b^{{\prime}};q\right)_{{n}}x^{m}y^{n}}{\left(q;q\right)_{{m}}\left(q;q\right)_{{n}}\left(c;q\right)_{{m}}\left(c^{{\prime}};q\right)_{{n}}},$

Symbols:: $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ : first $q$ -Appell function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/17.4.E6
Encodings:: TeX, pMML, png

17.4.7 $\mathop{\Phi^{{(3)}}\/}\nolimits\!\left(a,a^{{\prime}};b,b^{{\prime}};c;x,y\right)=\sum _{{m,n\geq 0}}\frac{\left(a,b;q\right)_{{m}}\left(a^{{\prime}},b^{{\prime}};q\right)_{{n}}x^{m}y^{n}}{\left(q;q\right)_{{m}}\left(q;q\right)_{{n}}\left(c;q\right)_{{m+n}}},$

Symbols:: $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ : first $q$ -Appell function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/17.4.E7
Encodings:: TeX, pMML, png

17.4.8 $\mathop{\Phi^{{(4)}}\/}\nolimits\!\left(a;b;c,c^{{\prime}};x,y\right)=\sum _{{m,n\geq 0}}\frac{\left(a,b;q\right)_{{m+n}}x^{m}y^{n}}{\left(q,c;q\right)_{{m}}\left(q,c^{{\prime}};q\right)_{{n}}}.$