q-Hypergeometric Function
The modern definition of the 
-hypergeometric
function is
![_rphi_s[alpha_1,alpha_2,...,alpha_r; beta_1,...,beta_s;q,z]
=sum_(n=0)^infty((alpha_1;q)_n(alpha_2;q)_n...(alpha_r;q)_n)/((beta_1;q)_n...(beta_s;q)_n)(z^n)/((q;q)_n)[(-1)^nq^((n; 2))]^(1+s-r),](/images/equations/q-HypergeometricFunction/NumberedEquation1.gif) |
(1)
|
where 
is a binomial
coefficient and 
is a q-Pochhammer symbol (Gasper and Rahman
1990; Bhatnagar 1995, p. 21; Koepf 1998, p. 25). This is the version of
the 
-hypergeometric function implemented in
the Wolfram Language as QHypergeometricPFQ[
a1, ..., ar
, 
b1, ...,
bs
, q, z].
An older form of definition omits the factor ![[(-1)^kq^((n; 2))]^(1+s-r)](/images/equations/q-HypergeometricFunction/Inline9.gif)
,
![_rphi_s^'[alpha_1,alpha_2,...,alpha_r; beta_1,...,beta_s;q,z]=sum_(n=0)^infty((alpha_1;q)_n(alpha_2;q)_n...(alpha_r;q)_n)/((beta_1;q)_n...(beta_s;q)_n)(z^n)/((q;q)_n),](/images/equations/q-HypergeometricFunction/NumberedEquation2.gif) |
(2)
|
This is the 
-hypergeometric function as defined by
Bailey (1935), Slater (1966), Andrews (1986), and Hardy (1999).
Note that the two definitions coincide when 
, including
the common case 
.
A particular case of 
is given
by
 |
(3)
|
(Andrews 1986, p. 10). A 
-analog of Gauss's
theorem (the q-Gauss identity) due to
Jacobi and Heine is given by
 |
(4)
|
for 
(Koepf 1998, p. 40).
Heine proved the transformation formula
 |
(5)
|
(Andrews 1986, pp. 10-11). Rogers (1893) obtained the formulas
 |
(6)
|
 |
(7)
|
(Andrews 1986, pp. 10-11).
The function 
has the simple confluent identity
![lim_(alpha_r->infty)_rphi_s[alpha_1,alpha_2,...,alpha_r; beta_1,...,beta_s;q,z/(alpha_r)]=_(r-1)phi_s[alpha_1,alpha_2,...,alpha_(r-1); beta_1,...,beta_s;q,z].](/images/equations/q-HypergeometricFunction/NumberedEquation8.gif) |
(8)
|
In the limit 
,
![lim_(q->1^-)_rphi_s[q^(alpha_1),q^(alpha_2),...,q^(alpha_r); q^(beta_1),...,q^(beta_s);q,(q-1)^(1+s-r)z]=_rF_s[alpha_1,alpha_2,...,alpha_r; beta_1,...,beta_s;z],](/images/equations/q-HypergeometricFunction/NumberedEquation9.gif) |
(9)
|
where 
is a generalized
hypergeometric function (Koepf 1998, p. 25).
Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 10, 1986.
Bailey, W. N. "Basic Hypergeometric Series." Ch. 8 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 65-72, 1935.
Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 21, 1995.
Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.
Gasper, G. "Elementary Derivations of Summation and Transformation Formulas for q-Series." In Fields Inst. Comm. 14 (Ed. M. E. H. Ismail et al. ), pp. 55-70, 1997.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 107-111, 1999.
Heine, E. "Über die Reihe 
."
J. reine angew. Math. 32, 210-212, 1846.
Heine, E. "Untersuchungen über die Reihe 
."
J. reine angew. Math. 34, 285-328, 1847.
Heine, E. Theorie der Kugelfunctionen und der verwandten Functionen, Bd. 1. Berlin: Reimer, pp. 97-125, 1878.
Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 25-26, 1998.
Krattenthaler, C. "HYP and HYPQ." J. Symb. Comput. 20, 737-744, 1995.
Rogers, L. J. "On a Three-Fold Symmetry in the Elements of Heine's Series." Proc. London Math. Soc. 24, 171-179, 1893.
Slater, L. J. Generalized Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1966.
Weisstein, Eric W. "q-Hypergeometric Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/q-HypergeometricFunction.html
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