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21: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

… ►

17.9.3

{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{% \infty}}{\left(bz/c;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq% /(bz)};q,q\right)+\frac{\left(a,bz,c/b;q\right)_{\infty}}{\left(c,z,c/(bz);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({z,abz/c,0\atop bz,bzq/c};q,q\right),

ⓘ

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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable
Referenced by:: Erratum (V1.0.23) for Equation (17.9.3)
Permalink:: http://dlmf.nist.gov/17.9.E3
Encodings:: pMML, png, TeX
Correction (effective with 1.0.23):: The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${{}_{2}\phi_{1}}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)).
Suggested 2019-04-26 by Roberto S. Costas-Santos
See also:: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

… ►provided that the series expansions of both

\phi

’s terminate. …

22: 5.11 Asymptotic Expansions

… ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at

k=n-1

(

k\geq 0

) and

z

is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …

23: 10.49 Explicit Formulas

…

24: 17.4 Basic Hypergeometric Functions

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

r=s

z=q

, and …

25: 13.4 Integral Representations

… ►The contour of integration starts and terminates at a point

\alpha

on the real axis between

0

and

1

. …

26: 15.6 Integral Representations

… ►In (15.6.5) the integration contour starts and terminates at a point

A

on the real axis between

0

and

1

. …

27: 29.3 Definitions and Basic Properties

… ►If

\nu

is a nonnegative integer and

2p\leq\nu

, then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with

2m\leq\nu

. …

28: 1.10 Functions of a Complex Variable

… ►(Or more generally, a simple contour that starts at the center and terminates on the boundary.) …

29: 2.7 Differential Equations

… ►Hence unless the series (2.7.8) terminate (in which case the corresponding

\Lambda_{j}

is zero) they diverge. …

30: 13.2 Definitions and Basic Properties

…