DLMF: Untitled Document

§17.18 Methods of Computation

…

§17.1 Special Notation

… ►The main functions treated in this chapter are the basic hypergeometric (or

q

-hypergeometric) function

{{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, the bilateral basic hypergeometric (or bilateral

q

-hypergeometric) function

{{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, and the

q

-analogs of the Appell functions

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)

,

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)

,

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)

, and

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)

. …

§17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

… ►These identities are all given in terms of sums and products of basic bilateral hypergeometric series. ►

Bailey’s Bilateral Summations

… ►For similar formulas see Verma and Jain (1983).

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

… ►

17.10.1

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.2

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

… ►

17.10.3

{{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

►

17.10.4

{{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.

ⓘ

Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

…

§17.4 Basic Hypergeometric Functions

… ►

§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}.

ⓘ

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:: §17.8
Permalink:: http://dlmf.nist.gov/17.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(ii), §17.4 and Ch.17

… ►

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).

ⓘ

Symbols:: ${{}_{\NVar{p}}H_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : bilateral hypergeometric function, ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $q$ : complex base, $r$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(ii), §17.4 and Ch.17

…

… ►

§16.4(v) Bilateral Series

►Denote, formally, the bilateral hypergeometric function ►

16.4.16

{{}_{p}H_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum_{k% =-\infty}^{\infty}\frac{{\left(a_{1}\right)_{k}}\dots{\left(a_{p}\right)_{k}}}% {{\left(b_{1}\right)_{k}}\dots{\left(b_{q}\right)_{k}}}z^{k}.

ⓘ

Defines:: ${{}_{\NVar{p}}H_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : bilateral hypergeometric function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.4.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §16.4(v), §16.4 and Ch.16

… ►This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).

… ►

Dougall’s Bilateral Sum

…

… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

… ►

Equation (17.10.6)

17.10.6

\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \*{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,ka\atop a,-a,aq/b,aq/c,% aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{3}}{bcdefghk}\right)=\frac{\left(q,q% /(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q\right)_{\infty}}{% \left(gh,gk,h/g,k/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{\infty}}\*{{}_{10}\phi% _{9}}\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg% /f,qg/h,qg/k};q,\frac{q^{3}}{bcdefghk}\right)+\operatorname{idem}\left(g;h,k\right)

In the numerator of the argument of the basic bilateral hypergeometric function ${{}_{10}\psi_{10}}$ and in the numerator of the arguments of the basic hypergeometric ${{}_{10}\phi_{9}}$ functions, we replaced $q^{2}$ by $q^{3}$ . We also added a missing factor $1/\left(k/g;q\right)_{\infty}$ in the first term on the right-hand side.

…

bilateral

9 matching pages

1: 17.18 Methods of Computation

§17.18 Methods of Computation

2: 17.1 Special Notation

§17.1 Special Notation

3: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

§17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

Bailey’s Bilateral Summations

4: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

5: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

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

6: 16.4 Argument Unity

§16.4(v) Bilateral Series

7: 15.4 Special Cases

Dougall’s Bilateral Sum

8: 18.12 Generating Functions

9: Errata

bilateral

9 matching pages

1: 17.18 Methods of Computation

§17.18 Methods of Computation

2: 17.1 Special Notation

§17.1 Special Notation

3: 17.8 Special Cases of ψ r r Functions

§17.8 Special Cases of ψ r r Functions

Ramanujan’s ψ 1 1 Summation

Bailey’s Bilateral Summations

4: 17.10 Transformations of ψ r r Functions

§17.10 Transformations of ψ r r Functions

5: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

§17.4(ii) ψ s r Functions

6: 16.4 Argument Unity

§16.4(v) Bilateral Series

7: 15.4 Special Cases

Dougall’s Bilateral Sum

8: 18.12 Generating Functions

9: Errata

3: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

§17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

4: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

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