q-Pochhammer Symbol

The q-analog of the Pochhammer symbol defined by

 (a;q)_k={product_(j=0)^(k-1)(1-aq^j)   if k>0; 1   if k=0; product_(j=1)^(|k|)(1-aq^(-j))^(-1)   if k<0; product_(j=0)^(infty)(1-aq^j)   if k=infty

(Koepf 1998, p. 25). q-Pochhammer symbols are frequently called q-series and, for brevity, (a;q)_n is often simply written (a)_n. Note that this contention has the slightly curious side-effect that the argument is not taken literally, so for example (-q)_n means (-q;q)_n, not (-q;-q)_n (cf. Andrews 1986b).

The q-Pochhammer symbol (a;q)_n is implemented in the Wolfram Language as QPochhammer[a, q, n], with the special cases (a;q)_infty and (q;q)_infty represented as QPochhammer[a, q] and QPochhammer[q], respectively.

Min Max
Powered by webMathematica
Min Max
Im Powered by webMathematica

Letting n->infty gives the special case (q)_infty, sometimes known as "the" Euler function phi(q) and defined by


This function is closely related to the pentagonal number theorem and other related and beautiful sum/product identities. As mentioned above, it is implemented in Mathematica as QPochhammer[q]. As can be seen in the plot above, along the real axis, (q)_infty reaches a maximum value (q^*)_infty=1.2283488670385... (OEIS A143440) at value q^*=-0.4112484... (OEIS A143441).

The general q-Pochhammer symbol is given by the sum

 sum_(k=0)^n(-a)^kq^((k; 2))[n; k]_q=(a;q)_n,

where [n; k]_q is a q-binomial coefficient (Koekoek and Swarttouw 1998, p. 11).

It is closely related to the Dedekind eta function,


where tau the half-period ratio and q^_=e^(2piitau) is the square of the nome (Berndt 1994, p. 139). Other representations in terms of special functions include


where theta_n(z,q) is a Jacobi theta function (and in the latter case, care must be taken with the definition of the principal value the cube root).

Asymptotic results for q-Pochhammer symbols include


for q=e^(-t) (Watson 1936, Gordon and McIntosh 2000).

For q->1^-,


gives the normal Pochhammer symbol (alpha)_n (Koekoek and Swarttouw 1998, p. 7). The q-Pochhammer symbols are also called q-shifted factorials (Koekoek and Swarttouw 1998, pp. 8-9).

The q-Pochhammer symbol satisfies

 (a;q)_n=(q^(1-n)/a;q)_n(-a)^nq^((n; 2))
 (a;q^(-1))_n=(a^(-1);q)_n(-a)^nq^(-(n; 2))
 (a;q)_(-n)=1/((aq^(-n);q)_n)=((-q/a)^n)/((q/a;q)_n)q^((n; 2)),

(here, (n; k) is a binomial coefficient so (n; 2)=n(n-1)/2), as well as many other identities, some of which are given by Koekoek and Swarttouw (1998, p. 9).

A generalized q-Pochhammer symbol can be defined using the concise notation


(Gordon and McIntosh 2000).

The q-bracket

 [n]_q=[n; 1]_q

and q-binomial


symbols are sometimes also used when discussing q-series, where [n; 1]_q is a q-binomial coefficient.

See also

Borwein Conjectures, Dedekind Eta Function, Fine's Equation, Jackson's Identity, Jacobi Identities, Mock Theta Function, Pochhammer Symbol, q-Analog, q-Binomial Coefficient, q-Binomial Theorem, q-Cosine, q-Factorial, Q-Function, q-Gamma Function, q-Hypergeometric Function, q-Multinomial Coefficient, q-Series, q-Series Identities, q-Sine, Ramanujan Psi Sum, Ramanujan Theta Functions, Rogers-Ramanujan Identities

Explore with Wolfram|Alpha


Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986a.Andrews, G. E. "The Fifth and Seventh Order Mock Theta Functions." Trans. Amer. Soc. 293, 113-134, 1986b.Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, 1998.Andrews, G. E.; Askey, R.; and Roy, R. Special Functions. Cambridge, England: Cambridge University Press, 1999.Berndt, B. C. "q-Series." Ch. 27 in Ramanujan's Notebooks, Part IV. New York:Springer-Verlag, pp. 261-286, 1994.Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan's Lost Notebook." Trans. Amer. Math. Soc. 352, 2157-2177, 2000.Bhatnagar, G. "A Multivariable View of One-Variable q-Series." In Special Functions and Differential Equations. Proceedings of the Workshop (WSSF97) held in Madras, January 13-24, 1997) (Ed. K. S. Rao, R. Jagannathan, G. van den Berghe, and J. Van der Jeugt). New Delhi, India: Allied Pub., pp. 60-72, 1998.Gasper, G. "Lecture Notes for an Introductory Minicourse on q-Series." 25 Sep 1995., 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.Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.Gosper, R. W. "Experiments and Discoveries in q-Trigonometry." In Symbolic Computation, Number Theory,Special Functions, Physics and Combinatorics. Proceedings of the Conference Held at the University of Florida, Gainesville, FL, November 11-13, 1999 (Ed. F. G. Garvan and M. E. H. Ismail). Dordrecht, Netherlands: Kluwer, pp. 79-105, 2001.Gordon, B. and McIntosh, R. J. "Some Eighth Order Mock Theta Functions." J. London Math. Soc. 62, 321-335, 2000.Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, p. 7, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 25 and 30, 1998.Sloane, N. J. A. Sequences A143440 and A143441 in "The On-Line Encyclopedia of Integer Sequences."Watson, G. N. "The Final Problem: An Account of the Mock Theta Functions." J. London Math. Soc. 11, 55-80, 1936.

Referenced on Wolfram|Alpha

q-Pochhammer Symbol

Cite this as:

Weisstein, Eric W. "q-Pochhammer Symbol." From MathWorld--A Wolfram Web Resource.

Subject classifications