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We produce two-dimensional contiguous relations for generalized hypergeometric functions by starting with linearization coefficients for some continuous generalized hypergeometric orthogonal polynomials in the Askey-scheme.

On even-dimensional Euclidean space for integer powers of the Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on spheres in arbitrary even dimensions.

In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlev\'e equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra.

Searches for radio signatures of ultra-high energy neutrinos and cosmic rays could benefit from improved efficiency by using real-time beamforming or correlation triggering. For missions with power limitations, such as the ANITA-3 Antarctic balloon experiment, full speed high resolution digitization of incoming signals is not practical. To this end, the University of Hawaii has developed the Realtime Independent Three-bit Converter (RITC), a 3-channel, 3-bit, streaming analog-to-digital converter implemented in the IBM-8RF 0.13 um process. RITC is primarily designed to digitize broadband radio signals produced by the Askaryan effect, and thus targets an analog bandwidth of >1 GHz, with a sample-and-hold architecture capable of storing up to 2.6 gigasamples-per-second. An array of flash analog-to-digital converters perform 3-bit conversion of sets of stored samples while acquisition continues elsewhere in the sampling array. A serial interface is provided to access an array of on chip digital-to-analog converters that control the digitization thresholds for each channel as well as the overall sampling rate. Demultiplexed conversion outputs are read out simultaneously for each channel via a set of 36 LVDS links, each running at 650 Mb/s. We briefly describe the design architecture of RITC. Evaluation of the RITC is currently under way, and we will report testing updates as they become available, including prospects for the use of this architecture as the analog half of a novel triggering system for the ANITA-3 ultra-high energy neutrino experiment., Comment: 3 pages, 5 figures