Β§17.4 Basic Hypergeometric Functions

Β§17.4(i) ${{}_{r}\phi_{s}}$ Functions

 17.4.1 ${{}_{r+1}\phi_{s}}\left({a_{0},a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,% b_{s}};q,z\right)={{}_{r+1}\phi_{s}}\left(a_{0},a_{1},\dots,a_{r};b_{1},b_{2},% \dots,b_{s};q,z\right)=\sum_{n=0}^{\infty}\frac{\left(a_{0};q\right)_{n}\left(% a_{1};q\right)_{n}\cdots\left(a_{r};q\right)_{n}}{\left(q;q\right)_{n}\left(b_% {1};q\right)_{n}\cdots\left(b_{s};q\right)_{n}}\*\left((-1)^{n}q^{\genfrac{(}{% )}{0.0pt}{}{n}{2}}\right)^{s-r}z^{n}.$ β Defines: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Referenced by: Β§17.4(iv), Β§17.4(iv), Β§17.4(iv), Β§17.4(iv), Β§17.4(iv), Β§17.6, Β§18.27(i), Β§18.27(ii) Permalink: http://dlmf.nist.gov/17.4.E1 Encodings: TeX, pMML, png See also: Annotations for Β§17.4(i), Β§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 $|z|<1$ when $s=r$.

 17.4.2 $\lim_{q\to 1-}{{}_{r+1}\phi_{s}}\left({q^{a_{0}},q^{a_{1}},\dots,q^{a_{r}}% \atop q^{b_{1}},\dots,q^{b_{s}}};q,(q-1)^{s-r}z\right)={{}_{r+1}F_{s}}\left({a% _{0},a_{1},\dots,a_{r}\atop b_{1},\dots,b_{s}};z\right).$ β Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$: alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Source: Koekoek et al. (2010, p.Β 15) Referenced by: Erratum (V1.1.11) for Equation (17.4.2) Permalink: http://dlmf.nist.gov/17.4.E2 Encodings: TeX, pMML, png Generalization (effective with 1.1.11): This limit relation which was previously accurate for ${{}_{r+1}\phi_{r}}$ was updated to be accurate for the more general ${{}_{r+1}\phi_{s}}$. See also: Annotations for Β§17.4(i), Β§17.4 and Ch.17

For the function on the right-hand side see Β§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 $\left((-1)^{n}q^{\genfrac{(}{)}{0.0pt}{}{n}{2}}\right)^{s-r}$ is not included in the sum. In practice this discrepancy does not usually cause serious problems because the case most often considered is $r=s$.

Β§17.4(ii) ${{}_{r}\psi_{s}}$ Functions

 17.4.3 ${{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.$ β Defines: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable Referenced by: Β§17.8 Permalink: http://dlmf.nist.gov/17.4.E3 Encodings: TeX, pMML, png See also: Annotations for Β§17.4(ii), Β§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 $s\geq r$ 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 $\lim_{q\to 1-}{{}_{r}\psi_{r}}\left({q^{a_{1}},q^{a_{2}},\dots,q^{a_{r}}\atop q% ^{b_{1}},q^{b_{2}},\dots,q^{b_{r}}};q,z\right)={{}_{r}H_{r}}\left({a_{1},a_{2}% ,\dots,a_{r}\atop b_{1},b_{2},\dots,b_{r}};z\right).$

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

Β§17.4(iii) $q$-Appell Functions

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

 17.4.5 $\displaystyle\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)$ $\displaystyle=\sum_{m,n\geq 0}\frac{\left(a;q\right)_{m+n}\left(b;q\right)_{m}% \left(b^{\prime};q\right)_{n}x^{m}y^{n}}{\left(q;q\right)_{m}\left(q;q\right)_% {n}\left(c;q\right)_{m+n}},$ β Defines: $\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$: first $q$-Appell function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-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 $\Phi^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.4(iii), Β§17.4 and Ch.17 17.4.6 $\displaystyle\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$ $\displaystyle=\sum_{m,n\geq 0}\frac{\left(a;q\right)_{m+n}\left(b;q\right)_{m}% \left(b^{\prime};q\right)_{n}x^{m}y^{n}}{\left(q,c;q\right)_{m}\left(q,c^{% \prime};q\right)_{n}},$ β Defines: $\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$: second $q$-Appell function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: 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, Erratum (V1.1.3) for Equation (17.4.6) Permalink: http://dlmf.nist.gov/17.4.E6 Encodings: TeX, pMML, png Clarification (effective with 1.1.3): The explicit usage of $\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}$ in the denominator of the right-hand side was used. Correction (effective with 1.0.10): The notation for $\Phi^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.4(iii), Β§17.4 and Ch.17 17.4.7 $\displaystyle\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$ $\displaystyle=\sum_{m,n\geq 0}\frac{\left(a,b;q\right)_{m}\left(a^{\prime},b^{% \prime};q\right)_{n}x^{m}y^{n}}{\left(q;q\right)_{m}\left(q;q\right)_{n}\left(% c;q\right)_{m+n}},$ β Defines: $\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$: third $q$-Appell function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: 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 $\Phi^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.4(iii), Β§17.4 and Ch.17 17.4.8 $\displaystyle\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$ $\displaystyle=\sum_{m,n\geq 0}\frac{\left(a,b;q\right)_{m+n}x^{m}y^{n}}{\left(% q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}}.$ β Defines: $\Phi^{(4)}\left(\NVar{a},\NVar{b};\NVar{c},\NVar{c^{\prime}};\NVar{q};\NVar{x}% ,\NVar{y}\right)$: fourth $q$-Appell function Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: 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 $\Phi^{(4)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.4(iii), Β§17.4 and Ch.17

Β§17.4(iv) Classification

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

 17.4.9 $qa_{0}a_{1}\cdots a_{s}=b_{1}b_{2}\cdots b_{s}.$ β Symbols: $q$: complex base and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.4.E9 Encodings: TeX, pMML, png See also: Annotations for Β§17.4(iv), Β§17.4 and Ch.17

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

 17.4.10 $q^{k}a_{0}a_{1}\cdots a_{s}=b_{1}b_{2}\cdots b_{s}.$ β 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 Β§17.4(iv), Β§17.4 and Ch.17

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

 17.4.11 $a_{0}q=a_{1}b_{1}=a_{2}b_{2}=\dots=a_{s}b_{s}.$ β Symbols: $q$: complex base and $s$: nonnegative integer Referenced by: Β§17.4(iv) Permalink: http://dlmf.nist.gov/17.4.E11 Encodings: TeX, pMML, png See also: Annotations for Β§17.4(iv), Β§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 $b_{1}=-b_{2}=\sqrt{a_{0}}.$ β Permalink: http://dlmf.nist.gov/17.4.E12 Encodings: TeX, pMML, png See also: Annotations for Β§17.4(iv), Β§17.4 and Ch.17

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

 17.4.13 $a_{0}q=a_{1}b_{1}=a_{2}b_{2}=\dots=a_{s-1}b_{s-1}.$ β Symbols: $q$: complex base and $s$: nonnegative integer Permalink: http://dlmf.nist.gov/17.4.E13 Encodings: TeX, pMML, png See also: Annotations for Β§17.4(iv), Β§17.4 and Ch.17