17 q-Hypergeometric and Related Functions Notation 17 q-Hypergeometric and Related Functions 17.2 Calculus

§17.1 Special Notation

ⓘ

Defines:: $\operatorname{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\operatorname{idem}$ function
Keywords:: bilateral basic hypergeometric function, bilateral $q$ -hypergeometric function, idem function, notation, $q$ -hypergeometric function
Referenced by:: Erratum (V1.0.10) for Section 17.1
Permalink:: http://dlmf.nist.gov/17.1
Correction (effective with 1.0.10):: The notation for $q$ -Appell functions has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.
See also:: Annotations for Ch.17

(For other notation see Notation for the Special Functions.)

$k, j, m, n, r, s$	nonnegative integers.
$z$	complex variable.
$x$	real variable.
$q$ $(\in\mathbb{C})$	base: unless stated otherwise $\|q\|<1$ .
$\left(a;q\right)_{n}$	$q$ -shifted factorial: $(1-a)(1-aq)\cdots\left(1-aq^{n-1}\right)$ .

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:

f(\chi_{1};\chi_{2},\dots,\chi_{n})+\operatorname{idem}\left(\chi_{1};\chi_{2}% ,\dots,\chi_{n}\right)=\sum_{j=1}^{n}f(\chi_{j};\chi_{1},\chi_{2},\dots,\chi_{% j-1},\chi_{j+1},\dots,\chi_{n}).

These notations agree with Gasper and Rahman (2004). A slightly different notation is that in Bailey (1964) and Slater (1966); see §17.4(i). Fine (1988) uses $F(a,b;t:q)$ for a particular specialization of a ${{}_{2}\phi_{1}}$ function.

17 q-Hypergeometric and Related Functions 17.2 Calculus

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

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.

§17.1 Special Notation

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