17 q-Hypergeometric and Related FunctionsProperties17.4 Basic Hypergeometric Functions17.6 ${}_2{}^{}\varphi _1^{}$ Function

§17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

ⓘ

Notes:

See Andrews (1976, pp. 17, 19, 36) and Gasper and Rahman (2004, pp. 25–26).

Referenced by:

Erratum (V1.2.1) for §17.5

Permalink:

http://dlmf.nist.gov/17.5

Clarification (effective with 1.2.1):

Below (17.5.2) and (17.5.4), a sentence was added indicating that these equations can be used as the analytic continuation for the ${}_1{}^{}\varphi _0^{}$ contained in those equations.

See also:

Annotations for Ch.17

Euler’s Second Sum

ⓘ

Keywords:

Euler’s sums, Euler’s sums (first, second, third), $q$-hypergeometric function

See also:

Annotations for §17.5 and Ch.17

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

compare (17.3.2).

$q$-Binomial Series

ⓘ

Keywords:

$q$-binomial series

See also:

Annotations for §17.5 and Ch.17

17.5.2		$${}_1{}^{}\varphi _0^{}(a;-;q,z)=\frac{{(az;q)}_{\mathrm{\infty }}}{{(z;q)}_{\mathrm{\infty }}},$$
;
ⓘ 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 and $z$: complex variable Referenced by: §17.5, Erratum (V1.2.1) for §17.5 Permalink: http://dlmf.nist.gov/17.5.E2 Encodings: TeX, pMML, png See also: Annotations for §17.5, §17.5 and Ch.17

compare (17.2.37). This equation can be used as the analytic continuation for this ${}_1{}^{}\varphi _0^{}$.

$q$-Binomial Theorem

ⓘ

Keywords:

$q$-binomial theorem

See also:

Annotations for §17.5 and Ch.17

17.5.3		$${}_1{}^{}\varphi _0^{}(q^{-n};-;q,z)={(z{q}^{-n};q)}_n.$$
ⓘ 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: TeX, pMML, png See also: Annotations for §17.5, §17.5 and Ch.17

This is (17.2.35) reformulated.

Euler’s First Sum

ⓘ

Keywords:

Euler’s sums, Euler’s sums (first, second, third), $q$-hypergeometric function

See also:

Annotations for §17.5 and Ch.17

17.5.4		$${}_1{}^{}\varphi _0^{}(0;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(q;q)}_n}=\frac{1}{{(z;q)}_{\mathrm{\infty }}},$$
;
ⓘ 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 Referenced by: §17.2(iii), §17.5, Erratum (V1.2.1) for §17.5 Permalink: http://dlmf.nist.gov/17.5.E4 Encodings: TeX, pMML, png See also: Annotations for §17.5, §17.5 and Ch.17

compare (17.3.1). This equation can be used as the analytic continuation for this ${}_1{}^{}\varphi _0^{}$.

Cauchy’s Sum

ⓘ

Keywords:

Cauchy’s sum, $q$-hypergeometric function

See also:

Annotations for §17.5 and Ch.17

17.5.5		$${}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a}{c};q,c/a)=\frac{{(c/a;q)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}}.$$
ⓘ 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 and $q$: complex base Referenced by: Erratum (V1.1.10) for Equation (17.5.5) Permalink: http://dlmf.nist.gov/17.5.E5 Encodings: TeX, pMML, png Correction (effective with 1.1.10): The constraint is not necessary and was removed. See also: Annotations for §17.5, §17.5 and Ch.17

17.4 Basic Hypergeometric Functions17.6 ${}_2{}^{}\varphi _1^{}$ Function

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy