Basic hypergeometric series explained

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 qⁿ, then the series is called a basic hypergeometric series. The number q is called the base.

The basic hypergeometric series

{}_2\phi

	\alpha

	1(q

,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}a₁&a₂&\ldots&a_j\ b₁&b₂&\ldots&b_k\end{matrix};q,z\right]=

	infty
\sum
	n=0

	(a_1,a_2,\ldots,a_j;q)_n
	(b_1,b_2,\ldots,b_k,q;q)_n

\left((-1)^nq^n\choose\right)^1+k-jzⁿ

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=

	n-1
\prod
	k=0

(1-aq^{k)=(1-a)(1-aq)(1-aq}^{2) … (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}a₁&a₂&\ldots&a_k&a_k+1\ b₁&b₂&\ldots&b_k\end{matrix};q,z\right]=

	infty
\sum
	n=0

	(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 ...b_kq.This series is called well poised if a₁q = a₂b₁ = ... = a_{k + 1}b_k, and very well poised if in addition a₂ = −a₃ = qa₁^1/2. The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since

\lim_q\to _j\phi_k\left[\begin{matrix}

	a₁
q

	a₂
q

&\ldots&

	a_j
q

	b₁
\ q

	b₂
q

&\ldots&

	b_k
q

\end{matrix};q,(q-1)^1+k-jz\right]= _jF_k\left[\begin{matrix}a₁&a₂&\ldots&a_j\ b₁&b₂&\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}a₁&a₂&\ldots&a_j\ b₁&b₂&\ldots&b_k\end{matrix};q,z\right]=

	infty
\sum
	n=-infty

	(a_1,a_2,\ldots,a_j;q)_n
	(b_1,b_2,\ldots,b_k;q)_n

\left((-1)^nq^n\choose\right)^k-jz^n.

The most important special case is when j = k, when it becomes

_k\psi_k\left[\begin{matrix}a₁&a₂&\ldots&a_k\ b₁&b₂&\ldots&b_k\end{matrix};q,z\right]=

	infty
\sum
	n=-infty

	(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 < 0 then vanish.

Simple series

Some simple series expressions include

	z
	1-q

₂\phi₁\left[\begin{matrix}q q\ q²\end{matrix} ;q,z\right]=

	z
	1-q

	z²
	1-q²

	z³
	1-q³

+\ldots

and

	z
	1-q^1/2

₂\phi₁\left[\begin{matrix}q q^1/2\ q^3/2\end{matrix} ;q,z\right]=

	z
	1-q^1/2

	z²
	1-q^3/2

	z³
	1-q^5/2

+\ldots

and

₂\phi₁\left[\begin{matrix}q -1\ -q\end{matrix} ;q,z\right]=1+

	2z
	1+q

	2z²
	1+q²

	2z³
	1+q³

+\ldots.

The q-binomial theorem

The q-binomial theorem (first published in 1811 by Heinrich August Rothe)^[1] ^[2] states that $\;_\phi_0 (a;q,z) =\frac= \prod_^\infty \frac .$ It can be proved by repeatedly applying the identity $\;_\phi_0 (a;q,z) = \frac \;_\phi_0 (a;q,qz).$ When $a= q^$ is a negative integer power of q, the hypergeometric sum is finite and one recovers the finite form $\sum_^y^nq^\beginN\\n\end_q=\prod_^\left(1+yq^k\right)$ of the q-binomial theorem (also sometimes known as the Cauchy binomial theorem).^[3] Here $\beginN\\n\end_q$ 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 $\;_1\psi_1 \left[\begin{matrix} a \\ b \end{matrix} ; q,z \right] = \sum_^\infty \frac z^n= \frac$ valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for

_6\psi₆

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

	infty
\sum
	n=-infty

q^n(n+1)/2zⁿ=(q;q)_infty (-1/z;q)_infty (-zq;q)_infty.

Gwynneth Coogan and Ken Ono give a related formal power series

A(z;q)\stackrel{\rm{def}}{=}

	1
	1+z

	infty
\sum
	n=0

	(z;q)_n
	(-zq;q)_n

zⁿ=

	infty
\sum
	n=0

(-1)ⁿz²ⁿ

	n²
q

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)=

	-1
	2\pii

	(a,b;q)_infty
	(q,c;q)_infty

	iinfty
\int
	-iinfty

s,cq

(qq

	s;q)

	infty

s,bq

(aq

	s;q)

	infty

	\pi(-z)^s
	\sin\pis

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+1φ_r. 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):=}

infty	(A;q)_n(B;q)_n
	(C;q)_n(q;q)_n

\sum

n=0

z^n,(A;q)_{0:=1, (A;q)}_n:=\prod

	n-1

	k=0

(1-Aq^k).

The ratio test shows that this matrix function is absolutely convergent.^[4]

References

- W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
William Y. C. Chen and Amy Fu, Semi-Finite Forms of Bilateral Basic Hypergeometric Series (2004)
Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood,,,
Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujan's
_1\psi₁

Summation
- - - Victor Kac, Pokman Cheung, Quantum calculus, Universitext, Springer-Verlag, 2002.
Roelof . Koekoek . Rene F. . Swarttouw . 1996 . The Askey scheme of orthogonal polynomials and its q-analogues . Technical University Delft . no. 98-17. . Section 0.2
Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions, Encyclopedia of Mathematics and its Applications, volume 71, Cambridge University Press.
Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.
Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.

Notes and References

.
.
http://mathworld.wolfram.com/CauchyBinomialTheorem.html Wolfram Mathworld: Cauchy Binomial Theorem
Ahmed Salem (2014) The basic Gauss hypergeometric matrix functionand its matrix q-difference equation, Linear and Multilinear Algebra, 62:3, 347-361, DOI:10.1080/03081087.2013.777437