- Basic hypergeometric series
In
mathematics , the basic hypergeometric series, also sometimes called the hypergeometric q-series, areq-analog generalizations of ordinaryhypergeometric series . Two basic series are commonly defined, the unilateral basic hypergeometric series, and the bilateral basic geometric series.The naming is in analogy to an ordinary hypergeometric series. An ordinary series x_n} is termed an ordinary hypergeometric series if the ratio of successive terms x_{n+1}/x_n is a
rational function of "n". But if the ratio of successive terms is a rational function of q^n, then the series is called a basic hypergeometric series.The basic hypergeometric series was first considered by
Eduard Heine in the 19th century, as a way of capturing the common features of the Jacobitheta function s andelliptic function s.Definition
The unilateral basic hypergeometric series is defined as
:j}phi_k left [egin{matrix} a_1 & a_2 & ldots & a_{j} \ b_1 & b_2 & ldots & b_k end{matrix} ; q,z ight] = 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^{nchoose 2} ight)^{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
is the
q-shifted factorial .The most important special case is when "j" = "k"+1, when it becomes:k+1}phi_k left [egin{matrix} a_1 & a_2 & ldots & a_{k+1} \ b_1 & b_2 & ldots & b_k end{matrix} ; q,z ight] = 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.The bilateral basic hypergeometric series, corresponding to the
bilateral hypergeometric series , is defined as:jpsi_k left [egin{matrix} a_1 & a_2 & ldots & a_j \ b_1 & b_2 & ldots & b_k end{matrix} ; q,z ight] = 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^{nchoose 2} ight)^{k-j}z^n.
The most important special case is when "j" = "k", when it becomes:kpsi_k left [egin{matrix} a_1 & a_2 & ldots & a_k \ b_1 & b_2 & ldots & b_k end{matrix} ; q,z ight] = 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"<0 then vanish.
imple series
Some simple series expressions include
:frac{z}{1-q} ;_{2}phi_1 left [egin{matrix} q ; q \ q^2 end{matrix}; ; q,z ight] = 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 [egin{matrix} q ; q^{1/2} \ q^{3/2} end{matrix}; ; q,z ight] = 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 [egin{matrix} q ; -1 \ -q end{matrix}; ; q,z ight] = 1+frac{2z}{1+q}+ frac{2z^2}{1+q^2}+ frac{2z^3}{1+q^3}+ ldots.
imple identities
Some simple identities include
:1}phi_0 (a;q,z) = prod_{n=0}^infty frac {1-aq^n z}{1-q^n z}
and :1}phi_0 (a;q,z) = frac {1-az}{1-z} ;_{1}phi_0 (a;q,qz).
The special case of a=0 is closely related to the
q-exponential .Ramanujan's identity
Ramanujan gave the identity:1psi_1 left [egin{matrix} a \ b end{matrix} ; q,z ight] = sum_{n=-infty}^infty frac {(a;q)_n} {(b;q)_n} = frac {(b/a;q)_infty; (q;q)_infty; (q/az;q)_infty; (az;q)_infty }{(b;q)_infty; (b/az;q)_infty; (q/a;q)_infty; (z;q)_infty}
valid for q|<1 and b/a| < |z| < 1. Similar identities for 6psi_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.
Ken Ono gives a relatedformal power series :A(z;q) = 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}.
References
*
Eduard Heine , "Theorie der Kugelfunctionen", (1878) "1", pp 97-125.
* Eduard Heine, "Handbuch die Kugelfunctionen. Theorie und Anwendung" (1898) Springer, Berlin.
* W.N. Bailey, "Generalized Hypergeometric Series", (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
*Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=Cambridge University Press | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | id=MathSciNet | id = 2128719 | year=2004 | volume=96
* William Y. C. Chen and Amy Fu, " [http://cfc.nankai.edu.cn/publications/04-accepted/Chen-Fu-04A/semi.pdf Semi-Finite Forms of Bilateral Basic Hypergeometric Series] " (2004)
* Sylvie Corteel and Jeremy Lovejoy, " [http://www.labri.fr/Perso/~lovejoy/1psi1.pdf Frobenius Partitions and the Combinatorics of Ramanujan's 1psi_1 Summation] ", (undated)
* Gwynneth H. Coogan andKen Ono , " [http://www.math.wisc.edu/~ono/reprints/067.pdf A q-series identity and the Arithmetic of Hurwitz Zeta Functions] ", (2003) Proceedings of theAmerican Mathematical Society 131, pp. 719-724
Wikimedia Foundation. 2010.