q-hypergeometric function

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1: 17.1 Special Notation

§17.1 Special Notation

$k, j, m, n, r, s$	nonnegative integers.
…

►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)

. ►Another function notation used is the “idem” function: …

2: 17.17 Physical Applications

§17.17 Physical Applications

… ►See Kassel (1995). …

3: 17.15 Generalizations

§17.15 Generalizations

…

4: 17.18 Methods of Computation

§17.18 Methods of Computation

…

5: 17.16 Mathematical Applications

§17.16 Mathematical Applications

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6: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

… ►

Euler’s Second Sum

… ►

17.5.2

{{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},

|z|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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 and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.3

{{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

Euler’s First Sum

… ►

Cauchy’s Sum

…

7: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$ -Hypergeometric and Related Functions

…

8: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

… ►

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

… ►The series (17.4.1) is said to be balanced or Saalschützian when it terminates,

r=s

z=q

, and … ►The series (17.4.1) is said to be k-balanced when

r=s

and … ►The series (17.4.1) is said to be well-poised when

r=s

and …

9: 17.12 Bailey Pairs

… ►

Bailey Transform

… ►

Bailey Pairs

… ►

Weak Bailey Lemma

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Strong Bailey Lemma

… ► …

10: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

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

… ►

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

… ►

Quintuple Product Identity

… ►

Bailey’s Bilateral Summations

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