Basic hypergeometric series
Encyclopedia
In mathematics
, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog
generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series
.
A series xn is called hypergeometric if the ratio of successive terms xn+1/xn 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 2φ1(qα,qβ;qγ;q,x) was first considered by . It becomes the hypergeometric series F(α,β;γ;x) in the limit when the base q is 1.
The unilateral basic hypergeometric series is defined as
where
and where
is the q-shifted factorial.
The most important special case is when j = k+1, when it becomes
This series is called balanced if a1...ak+1 = b1...bkq.
This series is called well poised if a1q = a2b1 = ... = ak+1bk, and very well poised if in addition a2 = −a3 = qa11/2.
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series
, is defined as
The most important special case is when j = k, when it becomes
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.
and
and
which follows by repeatedly applying the identity
The special case of
is closely related to the q-exponential.
valid for
and 
. Similar identities for 
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
Ken Ono
gives a related formal power series
for the hypergeometric series, Watson showed that
where the poles of
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog
Q-analog
Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1...
generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series
Elliptic hypergeometric series
In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratiocn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the...
.
A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
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 2φ1(qα,qβ;qγ;q,x) was first considered by . It becomes the hypergeometric series F(α,β;γ;x) in the limit when the base q is 1.
Definition
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic geometric series ψ.The unilateral basic hypergeometric series is defined as
where
and where
is the q-shifted factorial.
The most important special case is when j = k+1, when it becomes
This series is called balanced if a1...ak+1 = b1...bkq.
This series is called well poised if a1q = a2b1 = ... = ak+1bk, and very well poised if in addition a2 = −a3 = qa11/2.
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series
Bilateral hypergeometric series
In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratioof two terms is a rational function of n...
, is defined as
The most important special case is when j = k, when it becomes
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 includeand
and
The q-binomial theorem
The q-binomial theorem states thatwhich follows by repeatedly applying the identity
The special case of
is closely related to the q-exponential.Ramanujan's identity
Ramanujan gave the identityvalid for
and . Similar identities for 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 asKen Ono
Ken Ono
Ken Ono is an American mathematician who specializes in number theory, especially in integer partitions, modular forms, and the fields of interest to Srinivasa Ramanujan...
gives a related formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
Watson's contour integral
As an analogue of the Barnes integralBarnes integral
In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by...
for the hypergeometric series, Watson showed that
where the poles of
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.