General hypergeometric systems of equations and series of hypergeometric type

Abstract

Observations were made of the temporal variations of conduction currents through point-to-sphere and sphere-to-sphere gaps of insulating oil and of n-hexane under high direct voltages. Current bursts were distinguished from the normal fluctuations of the quasi-steady current component. Long-duration bursts were similar to some previous observations, but many short bursts had the polarity opposite to that of the quasi-steady component.

Essays on the theory of elliptic hypergeometric functions

“…The Pochhammer and Horn approach to the functions of hypergeometric type [1,2], which we used in the case of univariate functions, is not generalized yet to the level of elliptic hypergeometric functions of many variables. For that it is necessary to learn how to solve systems of difference equations of the first order for the kernels of multiple series or integrals with the coefficients which are elliptic functions of all summation or integration variables [10,11].…”

“…According to the general definition [10], formal series n∈Z c n are called the elliptic hypergeometric series, if the ratio of neighbouring coefficients c n+1 /c n is an elliptic function of n ∈ C. This definition lies in the stream of ideas of Pochhammer and Horn which are used for building the plain and q-hypergeometric series [2].…”

“…The Gauss hypergeomeric function 2 F 1 (a, b; c; x) is a canonical example of functions of such a type. According to the approach by Pochhammer and Horn [2], the generalized plain hypergeometric series can be defined as the sums n c n for which the ratio c n+1 /c n is a rational function of n. In 1847 Heine has introduced a q-analogue of the 2 F 1 -series 2 ϕ 1 (a, b; c; q; x) [3]. The general series of such type n c n are characterized by the property that for them the ratio c n+1 /c n is a rational function of q n , where q is some complex parameter.…”

Gevrey Expansions of Hypergeometric Integrals I

“…Hypergeometric systems also known as GKZ systems were introduced in [GGZ87] and [GZK89] as a far reaching generalisation of the Gauss hypergeometric differential equation. They appear as a special family of D-modules and they have been first studied in the regular case.…”

“…The toric ideal I A ⊂ C[∂] is the ideal generated by the family of binomials ∂ u − ∂ v where u, v ∈ N n and Au = Av. The ideal I A is a prime ideal and the Krull dimension of the quotient ring C[∂]/I A equals d. Following [GGZ87,GZK89], the hypergeometric ideal associated with the pair (A, β) is :…”