Basic Bilateral 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 $_2\phi_1(q^,q^;q^;q,x)$ 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

:\;_\phi_k \left begin
a_1 & a_2 & \ldots & a_ \\
b_1 & b_2 & \ldots & b_k \end
; q,z \right= \sum_^\infty
\frac \left((-1)^nq^\right)^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
:$(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^)$
is the ''q''-shifted factorial.
The most important special case is when ''j'' = ''k'' + 1, when it becomes
:\;_\phi_k \left begin
a_1 & a_2 & \ldots & a_&a_ \\
b_1 & b_2 & \ldots & b_ \end
; q,z \right= \sum_^\infty
\frac z^n.
This series is called ''balanced'' if ''a''₁ ... ''a''_{''k'' + 1} = ''b''₁ ...''b''_''k''''q''.
This series is called ''well poised'' if ''a''₁''q'' = ''a''₂''b''₁ = ... = ''a''_{''k'' + 1}''b''_''k'', and ''very well poised'' if in addition ''a''₂ = −''a''₃ = ''qa''₁^1/2.
The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since
:\lim_\;_\phi_k \left begin
q^ & q^ & \ldots & q^ \\
q^ & q^ & \ldots & q^ \end
; q,(q-1)^ z \right\;_F_k \left begin
a_1 & a_2 & \ldots & a_j \\
b_1 & b_2 & \ldots & b_k \end
;z \right/math>
holds ().

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

:\;_j\psi_k \left begin
a_1 & a_2 & \ldots & a_j \\
b_1 & b_2 & \ldots & b_k \end
; q,z \right= \sum_^\infty
\frac \left((-1)^nq^\right)^z^n.

The most important special case is when ''j'' = ''k'', when it becomes
:\;_k\psi_k \left begin
a_1 & a_2 & \ldots & a_k \\
b_1 & b_2 & \ldots & b_k \end
; q,z \right= \sum_^\infty
\frac 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

:\frac \;_\phi_1 \left begin
q \; q \\
q^2 \end\; ; q,z \right=
\frac
+ \frac
+ \frac
+ \ldots

and

:\frac \;_\phi_1 \left begin
q \; q^ \\
q^ \end\; ; q,z \right=
\frac
+ \frac
+ \frac
+ \ldots

and

:\;_\phi_1 \left begin
q \; -1 \\
-q \end\; ; q,z \right= 1+
\frac
+ \frac
+ \frac
+ \ldots.

The ''q''-binomial theorem

The ''q''-binomial theorem (first published in 1811 by Heinrich August Rothe) states that

:$\;_\phi_0 (a;q,z) =\frac= \prod_^\infty \frac$

which follows by repeatedly applying the identity
:$\;_\phi_0 (a;q,z) = \frac \;_\phi_0 (a;q,qz).$

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

Cauchy binomial theorem

Cauchy binomial theorem is a special case of the q-binomial theorem.
: $\sum_^y^nq^\beginN\\n\end_q=\prod_^\left(1+yq^k\right)\qquad(, q, <1)$

Ramanujan's identity

Srinivasa Ramanujan gave the identity

:\;_1\psi_1 \left begin a \\ b \end ; q,z \right
= \sum_^\infty \frac z^n
= \frac

valid for , ''q'',  < 1 and , ''b''/''a'',  < , ''z'',  < 1. Similar identities for $\;_6\psi_6$ 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

:$\sum_^\infty q^z^n = (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.$

Ken Ono gives a related formal power series

:$A(z;q) \stackrel \frac \sum_^\infty \fracz^n = \sum_^\infty (-1)^n z^ q^.$

Watson's contour integral

As an analogue of the Barnes integral for the hypergeometric series, Watson showed that
:$_2\phi_1(a,b;c;q,z) = \frac\frac \int_^\frac\fracds$
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:
:$_2\phi_1(A,B;C;q,z):= \sum_^\infty\fracz^n,\quad (A;q)_0:=1,\quad(A;q)_n:=\prod_^(1-Aq^k).$
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

See also

*Dixon's identity
*Rogers–Ramanujan identities

Notes

External links

Wolfram Mathworld – q-Hypergeometric Functions

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 $\,_1\psi_1$ Summation
'
*
*
*
*Victor Kac, Pokman Cheung, Quantum calculus'', Universitext, Springer-Verlag, 2002.
* {{cite report , first1=Roelof , last1=Koekoek , first2=Rene F. , last2=Swarttouw , date=1996 , title=The Askey scheme of orthogonal polynomials and its q-analogues , url=http://fa.its.tudelft.nl/~koekoek/askey/ , publisher=Technical University Delft , id=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.

Q-analogs
Hypergeometric functions