Dixon's identity

Dixon's identity sums a well-poised bilateral hypergeometric series as a quotient of gamma functions, under conditions that ensure convergence.

Connect

Bilateral hypergeometric series: A hypergeometric series whose index runs over all integers, including negative values. Dixon's identity evaluates a particular bilateral series rather than a one-sided expansion.
Generalized hypergeometric function: A function defined by a power series whose successive-term ratio is a rational function of the index. Dixon's bilateral sum belongs to the broader hypergeometric framework.
Alfred Cardew Dixon: A British mathematician associated with classical results on algebraic equations and hypergeometric series. Dixon's name is attached to the bilateral summation identity.
Gamma function identities: Equations relating gamma-function values at different arguments. Applying Dixon's formula produces identities among gamma values after suitable parameter choices.
Well-poised series: A hypergeometric series whose parameters satisfy specified pairwise balancing relations. Its parameter relations define the specialized series summed by Dixon's identity.
Hypergeometric series: A series in which the ratio of successive terms is a rational function of the index. This general series structure supplies the term ratios behind Dixon's identity.
Dixon's theorem: A classical finite sum identity involving binomial coefficients and factorials. It shares Dixon's name but is a distinct result from this bilateral hypergeometric summation.
Hypergeometric transformations: Identities that rewrite a hypergeometric function using different parameters or arguments. Summation formulas can serve as evaluations at special parameter configurations.
Very-well-poised series: A well-poised series with additional parameter structure that often simplifies its coefficients. The extra structure clarifies the symmetry underlying Dixon-type summations.
Bilateral series: A series indexed by all integers, with separate positive- and negative-index tails. Understanding both tails is essential to interpreting the identity's infinite sum.