§17.4 Basic Hypergeometric Functions

17.4.1		$$\begin{array}{l}{}_{r+1}\varphi _s(\genfrac{}{}{0pt}{}{a_0,a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_s};q,z)={}_{r+1}\varphi _s(a_0,a_1,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)\\ =\sum _{n=0}^{\mathrm{\infty }}\frac{{(a_0;q)}_n{(a_1;q)}_n\mathrm{\cdots }{(a_r;q)}_n}{{(q;q)}_n{(b_1;q)}_n\mathrm{\cdots }{(b_s;q)}_n}{\left({(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\right)}^{s-r}{z}^n.\end{array}$$
ⓘ Defines: ${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}\varphi _s(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (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 See also: Annotations for §17.4(i), §17.4 and Ch.17

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 when $s=r$.

17.4.2		$$\underset{q\to 1-}{lim}{}_{r+1}\varphi _r(\genfrac{}{}{0pt}{}{q^{a_0},q^{a_1},\mathrm{\dots },q^{a_r}}{q^{b_1},\mathrm{\dots },q^{b_r}};q,z)={}_{r+1}F_r(\genfrac{}{}{0pt}{}{a_0,a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_r};z).$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, ${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}\varphi _s(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: 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 See also: Annotations for §17.4(i), §17.4 and Ch.17

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 (1964) and Slater (1966). In these references the factor ${\left({(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\right)}^{s-r}$ is not included in the sum. In practice this discrepancy does not usually cause serious problems because the case most often considered is $r=s$.

17.4.3		$$\begin{array}{l}{}_r\psi _s(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_s};q,z)={}_r\psi _s(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)\\ =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}\\ =\begin{array}{l}\sum _{n=0}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}\\ +\sum _{n=1}^{\mathrm{\infty }}\frac{{(q/b_1,q/b_2,\mathrm{\dots },q/b_s;q)}_n}{{(q/a_1,q/a_2,\mathrm{\dots },q/a_r;q)}_n}{\left(\frac{b_1{b}_2\mathrm{\cdots }{b}_s}{a_1{a}_2\mathrm{\cdots }{a}_{r}z}\right)}^n.\end{array}\end{array}$$
ⓘ Defines: ${}_r\psi _s(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r\psi _s(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: 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 See also: Annotations for §17.4(ii), §17.4 and Ch.17

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 $s\ge r$ provided that and also, in the case $s=r$, .

17.4.4		$$\underset{q\to 1-}{lim}{}_r\psi _r(\genfrac{}{}{0pt}{}{q^{a_1},q^{a_2},\mathrm{\dots },q^{a_r}}{q^{b_1},q^{b_2},\mathrm{\dots },q^{b_r}};q,z)={}_{r}H_r(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_r};z).$$
ⓘ Symbols: ${}_{p}H_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: bilateral hypergeometric function, ${}_r\psi _s(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r\psi _s(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: 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 See also: Annotations for §17.4(ii), §17.4 and Ch.17

For the function ${}_{r}H_r$ see §16.4(v).

The following definitions apply when and :

17.4.5		${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$	$={\displaystyle \sum _{m,n\ge 0}}{\displaystyle \frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}}},$
ⓘ Defines: ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable Referenced by: Erratum (V1.0.10) for Section 17.1 Permalink: http://dlmf.nist.gov/17.4.E5 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for §17.4(iii), §17.4 and Ch.17
17.4.6		${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$	$={\displaystyle \sum _{m,n\ge 0}}{\displaystyle \frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_m{(c^{\prime };q)}_n}},$
ⓘ Defines: ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable Referenced by: Erratum (V1.0.10) for Section 17.1 Permalink: http://dlmf.nist.gov/17.4.E6 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for §17.4(iii), §17.4 and Ch.17
17.4.7		${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$	$={\displaystyle \sum _{m,n\ge 0}}{\displaystyle \frac{{(a,b;q)}_m{(a^{\prime },b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}}},$
ⓘ Defines: ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$: third $q$-Appell function Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable Referenced by: Erratum (V1.0.10) for Section 17.1 Permalink: http://dlmf.nist.gov/17.4.E7 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for ${\mathrm{\Phi }}^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for §17.4(iii), §17.4 and Ch.17
17.4.8		${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$	$={\displaystyle \sum _{m,n\ge 0}}{\displaystyle \frac{{(a,b;q)}_{m+n}{x}^m{y}^n}{{(q,c;q)}_m{(q,c^{\prime };q)}_n}}.$
ⓘ Defines: ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$: fourth $q$-Appell function Symbols: ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable Referenced by: Erratum (V1.0.10) for Section 17.1 Permalink: http://dlmf.nist.gov/17.4.E8 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for ${\mathrm{\Phi }}^{(4)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for §17.4(iii), §17.4 and Ch.17

The series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and

17.4.9		$$q{a}_0{a}_1\mathrm{\cdots }{a}_s=b_1{b}_2\mathrm{\cdots }{b}_s.$$
ⓘ Symbols: $q$: complex base and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.4.E9 Encodings: TeX, pMML, png See also: Annotations for §17.4(iv), §17.4 and Ch.17

The series (17.4.1) is said to be k-balanced when $r=s$ and

17.4.10		$$q^k{a}_0{a}_1\mathrm{\cdots }{a}_s=b_1{b}_2\mathrm{\cdots }{b}_s.$$
ⓘ Symbols: $k$: nonnegative integer, $q$: complex base and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.4.E10 Encodings: TeX, pMML, png See also: Annotations for §17.4(iv), §17.4 and Ch.17

The series (17.4.1) is said to be well-poised when $r=s$ and

17.4.11		$$a_{0}q=a_1{b}_1=a_2{b}_2=\mathrm{\cdots }=a_s{b}_s.$$
ⓘ Symbols: $q$: complex base and $s$: nonnegative integer Referenced by: §17.4(iv) Permalink: http://dlmf.nist.gov/17.4.E11 Encodings: TeX, pMML, png See also: Annotations for §17.4(iv), §17.4 and Ch.17

The series (17.4.1) is said to be very-well-poised when $r=s$, (17.4.11) is satisfied, and

17.4.12		$$b_1=-b_2=\sqrt{a_0}.$$
ⓘ Permalink: http://dlmf.nist.gov/17.4.E12 Encodings: TeX, pMML, png See also: Annotations for §17.4(iv), §17.4 and Ch.17

The series (17.4.1) is said to be nearly-poised when $r=s$ and

17.4.13		$$a_{0}q=a_1{b}_1=a_2{b}_2=\mathrm{\cdots }=a_{s-1}{b}_{s-1}.$$
ⓘ Symbols: $q$: complex base and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.4.E13 Encodings: TeX, pMML, png See also: Annotations for §17.4(iv), §17.4 and Ch.17