q-hypergeometric function

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1: 17.1 Special Notation

§17.1 Special Notation

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$k,j,m,n,r,s$	nonnegative integers.
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►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. ►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

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4: 17.18 Methods of Computation

§17.18 Methods of Computation

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5: 17.16 Mathematical Applications

§17.16 Mathematical Applications

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6: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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Euler’s Second Sum

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17.5.1 $${}_0{}^{}\varphi _0^{}(-;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(q;q)}_n}={(z;q)}_{\mathrm{\infty }};$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)

Permalink:

http://dlmf.nist.gov/17.5.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.11):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

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17.5.3 $${}_1{}^{}\varphi _0^{}(q^{-n};-;q,z)={(z{q}^{-n};q)}_n.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $n$: 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

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Euler’s First Sum

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Cauchy’s Sum

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7: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$-Hypergeometric and Related Functions

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8: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

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§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

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Bailey Transform

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Bailey Pairs

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Weak Bailey Lemma

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Strong Bailey Lemma

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10: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

§17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

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Quintuple Product Identity

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Bailey’s Bilateral Summations

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