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Melham's Conjecture on Odd Power Sums of Fibonacci Numbers
Ozeki and Prodinger showed that the odd power sum of the first several
consecutive Fibonacci numbers of even order is equal to a polynomial evaluated
at certain Fibonacci number of odd order. We prove that this polynomial and its
derivative both vanish at , and will be an integer polynomial after
multiplying it by a product of the first consecutive Lucas numbers of odd
order. This presents an affirmative answer to a conjecture of Melham.Comment: 15page
Quantum tunneling through planar p-n junctions in HgTe quantum wells
We demonstrate that a p-n junction created electrically in HgTe quantum wells
with inverted band-structure exhibits interesting intraband and interband
tunneling processes. We find a perfect intraband transmission for electrons
injected perpendicularly to the interface of the p-n junction. The opacity and
transparency of electrons through the p-n junction can be tuned by changing the
incidence angle, the Fermi energy and the strength of the Rashba spin-orbit
interaction. The occurrence of a conductance plateau due to the formation of
topological edge states in a quasi-one-dimensional p-n junction can be switched
on and off by tuning the gate voltage. The spin orientation can be
substantially rotated when the samples exhibit a moderately strong Rashba
spin-orbit interaction.Comment: 4 pages, 4 figure
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