Basic hypergeometric series

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Q-analog of hypergeometric series

title: "Basic hypergeometric series" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["q-analogs", "hypergeometric-functions"] description: "Q-analog of hypergeometric series" topic_path: "general/q-analogs" source: "https://en.wikipedia.org/wiki/Basic_hypergeometric_series" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Q-analog of hypergeometric series ::

In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series x**n is called hypergeometric if the ratio of successive terms x**n+1/x**n is a rational function of n. If the ratio of successive terms is a rational function of q**n, then the series is called a basic hypergeometric series. The number q is called the base.

The basic hypergeometric series {}_2\phi_1(q^{\alpha},q^{\beta};q^{\gamma};q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1.

Definition

There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as

:;{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a{j} \ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac {(a_1, a_2, \ldots, a_{j};q)n} {(b_1, b_2, \ldots, b_k,q;q)n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n where :(a_1,a_2,\ldots,a_m;q)n = (a_1;q)n (a_2;q)n \ldots (a_m;q)n and :(a;q)n = \prod{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}) is the q-shifted factorial. The most important special case is when j = k + 1, when it becomes :;{k+1}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a{k}&a{k+1} \ b_1 & b_2 & \ldots & b{k} \end{matrix} ; q,z \right] = \sum{n=0}^\infty \frac {(a_1, a_2, \ldots, a{k+1};q)n} {(b_1, b_2, \ldots, b_k,q;q)n} z^n. This series is called balanced if a1 ... a**k + 1 = b1 ...bkq. This series is called well poised if a1q = a2b1 = ... = a**k + 1b**k, and very well poised if in addition a2 = −a3 = qa11/2. The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since :\lim{q\to 1};{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right] holds ().

The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as

:;j\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_j;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} \left((-1)^nq^{n\choose 2}\right)^{k-j}z^n.

The most important special case is when j = k, when it becomes :;k\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_k \ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_k;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} z^n.

The unilateral series can be obtained as a special case of the bilateral one by setting one of the b variables equal to q, at least when none of the a variables is a power of q, as all the terms with n

Simple series

Some simple series expressions include

:\frac{z}{1-q} ;_{2}\phi_1 \left[\begin{matrix} q ; q \ q^2 \end{matrix}; ; q,z \right] = \frac{z}{1-q}

  • \frac{z^2}{1-q^2}
  • \frac{z^3}{1-q^3}
  • \ldots

and

:\frac{z}{1-q^{1/2}} ;_{2}\phi_1 \left[\begin{matrix} q ; q^{1/2} \ q^{3/2} \end{matrix}; ; q,z \right] = \frac{z}{1-q^{1/2}}

  • \frac{z^2}{1-q^{3/2}}
  • \frac{z^3}{1-q^{5/2}}
  • \ldots

and

:;_{2}\phi_1 \left[\begin{matrix} q ; -1 \ -q \end{matrix}; ; q,z \right] = 1+ \frac{2z}{1+q}

  • \frac{2z^2}{1+q^2}
  • \frac{2z^3}{1+q^3}
  • \ldots.

The ''q''-binomial theorem

The q-binomial theorem (first published in 1811 by Heinrich August Rothe){{citation | last = Bressoud | first = D. M. | authorlink=David Bressoud | doi = 10.1017/S0305004100058114 | issue = 2 | journal = Mathematical Proceedings of the Cambridge Philosophical Society | mr = 600238 | pages = 211–223 | title = Some identities for terminating q-series | volume = 89 | year = 1981| bibcode = 1981MPCPS..89..211B | last = Benaoum | first = H. B. | arxiv = math-ph/9812011 | doi = 10.1088/0305-4470/31/46/001 | issue = 46 | journal = Journal of Physics A: Mathematical and General | pages = L751–L754 | title = h-analogue of Newton's binomial formula | year = 1998 | volume = 31| bibcode = 1998JPhA...31L.751B | s2cid = 119697596 ;{1}\phi_0 (a;q,z) =\frac{(az;q)\infty}{(z;q)\infty}= \prod{n=0}^\infty \frac {1-aq^n z}{1-q^n z}. It can be proved by repeatedly applying the identity ;{1}\phi_0 (a;q,z) = \frac {1-az}{1-z} ;{1}\phi_0 (a;q,qz). When a= q^{-N} is a negative integer power of q, the hypergeometric sum is finite and one recovers the finite form \sum_{n=0}^{N}y^nq^{n(n+1)/2}\begin{bmatrix}N\n\end{bmatrix}q=\prod{k=1}^{N}\left(1+yq^k\right) of the q-binomial theorem (also sometimes known as the Cauchy binomial theorem). Here \begin{bmatrix}N\n\end{bmatrix}_q is a q-binomial coefficient.

The special case of a = 0 is closely related to the q-exponential.

Ramanujan's identity

Srinivasa Ramanujan gave the identity ;1\psi_1 \left[\begin{matrix} a \ b \end{matrix} ; q,z \right] = \sum{n=-\infty}^\infty \frac {(a;q)n} {(b;q)n} z^n = \frac {(b/a,q,q/az,az;q)\infty } {(b,b/az,q/a,z;q)\infty} valid for |q| ;_6\psi_6 have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as

:\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n = (q;q)\infty ; (-1/z;q)\infty ; (-zq;q)_\infty.

Gwynneth Coogan and Ken Ono give a related formal power series{{citation | last1=Coogan | first1=Gwynneth H. | last2=Ono | first2=Ken | authorlink2=Ken Ono | title=A q-series identity and the arithmetic of Hurwitz zeta functions | date=2003 | journal=Proceedings of the American Mathematical Society | volume=131 | issue=3 | pages=719–724 | doi=10.1090/S0002-9939-02-06649-2 | doi-access=free}}

:A(z;q) \stackrel{\rm{def}}{=} \frac{1}{1+z} \sum_{n=0}^\infty \frac{(z;q)_n}{(-zq;q)n}z^n = \sum{n=0}^\infty (-1)^n z^{2n} q^{n^2}.

Watson's contour integral

As an analogue of the Barnes integral for the hypergeometric series, Watson showed that : {}2\phi_1(a,b;c;q,z) = \frac{-1}{2\pi i}\frac{(a,b;q)\infty}{(q,c;q)\infty} \int{-i\infty}^{i\infty}\frac{(qq^s,cq^s;q)\infty}{(aq^s,bq^s;q)\infty}\frac{\pi(-z)^s}{\sin \pi s}ds where the poles of (aq^s,bq^s;q)_\infty lie to the left of the contour and the remaining poles lie to the right. There is a similar contour integral for r+1φr. This contour integral gives an analytic continuation of the basic hypergeometric function in z.

Matrix version

The basic hypergeometric matrix function can be defined as follows: : {}2\phi_1(A,B;C;q,z):= \sum{n=0}^\infty\frac{(A;q)_n(B;q)_n}{(C;q)_n(q;q)_n}z^n,\quad (A;q)_0:=1,\quad(A;q)n:=\prod{k=0}^{n-1}(1-Aq^k). The ratio test shows that this matrix function is absolutely convergent. Ahmed Salem (2014) The basic Gauss hypergeometric matrix function and its matrix q-difference equation, Linear and Multilinear Algebra, 62:3, 347-361, DOI: 10.1080/03081087.2013.777437

Notes

References

References

  1. [http://mathworld.wolfram.com/CauchyBinomialTheorem.html Wolfram Mathworld: Cauchy Binomial Theorem]

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