We derive double product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We also present some generating functions that arise from it in the $q$ and $q$-inverse Askey schemes. Using this $q$-Chaundy theorem which expresses a product of two nonterminating basic hypergeometric series as a sum over a terminating basic hypergeometric series, we study generating functions for the symmetric families of orthogonal polynomials in the $q$ and $q$-inverse Askey scheme. By applying the $q$-Chaundy theorem to $q$-exponential generating functions due to Ismail, we are able to derive alternative expansions of these generating functions and from these, new representations for the continuous $q$-Hermite and $q$-inverse Hermite polynomials which are connected by a quadratic transformation for the terminating basic hypergeometric series representations. less

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of $C^*$-algebras with fibre $D \otimes \mathbb{K}$ equipped with a fibrewise $\mathbb{T}$-action, where $\mathbb{T}$ denotes the circle group and $D = \operatorname{End}\left(V\right)^{\otimes \infty}$ for a $\mathbb{T}$-representation $V$. In particular, we show that the group of $\mathbb{T}$-equivariant $*$-automorphisms $\operatorname{Aut}_{\mathbb{T}}(D \otimes \mathbb{K})$ is an infinite loop space giving rise to a cohomology theory $E^*_{D,\mathbb{T}}(X)$. Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to $E^1_{D,\mathbb{T}}(X) \cong [X, B\operatorname{Aut}_{\mathbb{T}}(D \otimes \mathbb{K})]$. We compute this group for tori and compare the case $D = \mathbb{C}$ to the equivariant Brauer group for trivial actions on the base space. less

The four-dimensional variational data assimilation methodology for assimilating noisy observations into a deterministic model has been the workhorse of forecasting centers for over three decades. While this method provides a computationally efficient framework for dynamic data assimilation, it is largely silent on the important question concerning the minimum number and placement of observations. To answer this question, we demonstrate the dual advantage of placing the observations where the square of the sensitivity of the model solution with respect to the unknown control variables, called forward sensitivities, attains its maximum. Therefore, we can force the observability Gramian to be of full rank, which in turn guarantees efficient recovery of the optimal values of the control variables, which is the first of the two advantages of this strategy. We further show that the proposed strategy of placing observations has another inherent optimality: the square of the sensitivity of the optimal estimates of the control with respect to the observations (used to obtain these estimates) attains its minimum value, a second advantage that is a direct consequence of the above strategy for placing observations. Our analytical framework and numerical experiments on linear and nonlinear systems confirm the effectiveness of our proposed strategy. less

We study special values for the continuous $q$-Jacobi polynomials and present applications of these special values which arise from bilinear generating functions, and in particular the Poisson kernel for these polynomials. less