The connection formula for the Jackson integral of type $BC_n$ is obtained in the form of a Sears--Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific bilateral multiple series. The co
efficients of this expansion are expressed by certain elliptic Lagrange interpolation functions. Analyzing basic properties of the elliptic Lagrange interpolation functions, an explicit determinant formula is provided for a fundamental solution matrix of the associated system of $q$-difference equations.
We give an alternative proof of an elliptic summation formula of type $BC_n$ by applying the fundamental $BC_n$ invariants to the study of Jackson integrals associated with the summation formula.
The Ramanujan $_1psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Baileys very-well-poised $_6psi_6$ summ
ation. Bilateral Jackson integral generalizations of the Dixon--Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_1psi_1$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulas for Jackson integrals relating to other roots systems obtained from the same perspective.
The Dixon--Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The $_1psi_1$ summation is a bilateral generalization of the $q$-binomial theorem. It is shown that a $q$-generalization of the
Dixon--Anderson integral, due to Evans, and multi-dimensional generalizations of the $_1psi_1$ summation, due to Milne and Gustafson, can be viewed as having a common origin in the theory of $q$-difference equations as expounded by Aomoto. Each is shown to be determined by a $q$-difference equation of rank one, and a certain asymptotic behavior. In calculating the latter, essential use is made of the concepts of truncation, regularization and connection formulae.
A hexagonal oxynitride (Li0.88_0.12)Nb3.0(O0.13N0.87)4 was synthesized through ammonia nitridation of LiNb3O8. The structural analysis revealed that this oxynitride consists of alternate stacking of octahedral and prismatic layers with different Li/N
b ratios: significant amounts of Li and Nb atoms (Li/Nb = 43/57) coexist in the octahedral layer, while the prismatic site is preferentially occupied by Nb (Li/Nb = 3/97). A metallic behavior was accompanied by an abrupt drop of electrical resistivity at about 3 K. Furthermore, large diamagnetism and specific-heat anomaly were observed below this temperature, suggesting the appearance of superconductivity in the Li-Nb oxynitride.
A compound determinant identity for minors of rectangular matrices is established. As an application, we derive Vandermonde type determinant formulae for classical group characters.
