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q-Hypergeometric Function

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The modern definition of the q-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),
(1)

where (n; 2)=1/2n(n-1) is a binomial coefficient and (a;q)_n is a q-Pochhammer symbol (Gasper and Rahman 1990; Bhatnagar 1995, p. 21; Koepf 1998, p. 25). This is the version of the q-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),

_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),
(2)

This is the q-hypergeometric function as defined by Bailey (1935), Slater (1966), Andrews (1986), and Hardy (1999).

Note that the two definitions coincide when r=1+s, including the common case _2phi_1(a,b;c;q).

A particular case of _rphi_s is given by

_2psi_1(a,b;c;q,z)=sum_(n=0)^infty((a;q)_n(b;q)_nz^n)/((q;q)_n(c;q)_n)
(3)

(Andrews 1986, p. 10). A q-analog of Gauss's theorem (the q-Gauss identity) due to Jacobi and Heine is given by

_2phi_1(a,b;c;q,c/(ab))=((c/a;q)_infty(c/b;q)_infty)/((c;q)_infty(c/(ab);q)_infty)
(4)

for |c/(ab)|<1 (Koepf 1998, p. 40). Heine proved the transformation formula

_2phi_1(a,b;c;q,z)=((b;q)_infty(az;q)_infty)/((c;q)_infty(z;q)_infty)_2phi_1(c/b,z;az;q,b),
(5)

(Andrews 1986, pp. 10-11). Rogers (1893) obtained the formulas

_2phi_1(a,b;c;q,z)=((c/b;q)_infty(bz;q)_infty)/((z;q)_infty(c;q)_infty)_2phi_1(b,abz/c;bz;q,c/b)
(6)
_2phi_1(a,b,c;q,z)=((abz/c;q)_infty)/((z;q)_infty)_2phi_1(c/a,c/b;c;q,abz/c)
(7)

(Andrews 1986, pp. 10-11).

The function _rphi_s 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].
(8)

In the limit q->1^-,

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],
(9)

where _rF_s is a generalized hypergeometric function (Koepf 1998, p. 25).

See also

Generalized Hypergeometric Function, q-Pochhammer Symbol, q-Saalschütz Sum, q-Series

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References

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 1+((q^alpha-1)(q^beta-1))/((q-1)(q^gamma-1))x +((q^alpha-1)(q^(alpha+1)-1)(q^beta-1)(q^(beta+1)-1))/((q-1)(q^2-1)(q^gamma-1)(q^(gamma+1)-1))x^2+...." J. reine angew. Math. 32, 210-212, 1846.Heine, E. "Untersuchungen über die Reihe 1+((1-q^alpha)(1-q^beta))/((1-q)(1-q^gamma))·x+((1-q^alpha)(1-q^(alpha+1))(1-q^beta)(1-q^(beta+1)))/((1-q)(1-q^2)(1-q^gamma)(1-q^(gamma+1)))·x^2+...." 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.

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q-Hypergeometric Function

Cite this as:

Weisstein, Eric W. "q-Hypergeometric Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-HypergeometricFunction.html

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