A new multivariable 6-psi-6 summation formula

By multidimensional matrix inversion, combined with an A_r extension of Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7 summation is derived. By a polynomial argument this 8-phi-7 summation is transformed to another multivariable 8-phi-7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6-phi-5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey's very-well-poised 6-psi-6 summation formula.Comment: 16 page

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_{\nu}(z)$, with respect to its degree $\nu$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, $P_{\nu}^{\pm m}(z)$, with respect to the degree $\nu$, for $m\in\mathbb{N}$. At first, we establish several contour-integral representations of $\partial P_{\nu}^{\pm m}(z)/\partial\nu$. They are then used to derive Rodrigues-type formulas for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ with $n\in\mathbb{N}$. Next, some closed-form expressions for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives $[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}$, $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ and $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}$, all with $m>n$, are evaluated in terms of $[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}$.Comment: LateX, 40 pages, 1 figure, extensive referencin

A roadmap for Antarctic and Southern Ocean science for the next two decades and beyond

Antarctic and Southern Ocean science is vital to understanding natural variability, the processes that govern global change and the role of humans in the Earth and climate system. The potential for new knowledge to be gained from future Antarctic science is substantial. Therefore, the international Antarctic community came together to ‘scan the horizon’ to identify the highest priority scientific questions that researchers should aspire to answer in the next two decades and beyond. Wide consultation was a fundamental principle for the development of a collective, international view of the most important future directions in Antarctic science. From the many possibilities, the horizon scan identified 80 key scientific questions through structured debate, discussion, revision and voting. Questions were clustered into seven topics: i)Antarctic atmosphere and global connections, ii) Southern Ocean and sea ice in a warming world, iii) ice sheet and sea level, iv) the dynamic Earth, v) life on the precipice, vi) near-Earth space and beyond, and vii) human presence in Antarctica. Answering the questions identified by the horizon scan will require innovative experimental designs, novel applications of technology, invention of next-generation field and laboratory approaches, and expanded observing systems and networks. Unbiased, non-contaminating procedures will be required to retrieve the requisite air, biota, sediment, rock, ice and water samples. Sustained year-round access toAntarctica and the Southern Ocean will be essential to increase winter-time measurements. Improved models are needed that represent Antarctica and the Southern Ocean in the Earth System, and provide predictions at spatial and temporal resolutions useful for decision making. A co-ordinated portfolio of cross-disciplinary science, based on new models of international collaboration, will be essential as no scientist, programme or nation can realize these aspirations alone.Tinker Foundation, Antarctica New Zealand, The New Zealand Antarctic Research Institute, the Scientific Committee on Antarctic Research (SCAR), the Council of Managers of National Antarctic Programs (COMNAP), the Alfred Wegner Institut, Helmholtz Zentrum für Polar und Meeresforschung (Germany), and the British Antarctic Survey (UK).http://journals.cambridge.org/action/displayJournal?jid=ANShb201