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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
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''11/2. 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. Gwynneth Coogan and Ken Ono give a related
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
: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


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. * . 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 Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
. *
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Leg ...
, ''Theorie der Kugelfunctionen'', (1878) ''1'', pp 97–125. * Eduard Heine, ''Handbuch die Kugelfunctionen. Theorie und Anwendung'' (1898) Springer, Berlin.


External links

*{{MathWorld, q-HypergeometricFunction, q-Hypergeometric Function Q-analogs Hypergeometric functions