Belokolos et al. (1994, Chapter 5) and references therein. Here the
Riemann surface is represented by the action of a Schottkygroup on a region of
the complex plane. The same representation is used in Gianni et al. (1998).
►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices.
The elements of this group are of the form
…The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions.
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►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys.
►Quantum groups also apply -series extensively.
Quantum groups are really not groups at all but certain Hopf algebras.
They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics.
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►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform.
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§15.17(v) Monodromy Groups
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►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group.
These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic.
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►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC.
Books for which he has been editor or coeditor include Special Functions: Group Theoretical Aspects and Applications (with R.
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►Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials.
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►►►Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3).
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Magnify►Furthermore, the solutions of the KP equation solve the Schottky
problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)).
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► 1941 in Portland, Oregon) was the Group Leader of the Mathematical Software Group in the Applied and Computational Mathematics Division of NIST until his retirement in 2013.
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►He has also served several terms as an officer of the SIAM Activity Group on Orthogonal Polynomials and Special Functions.
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►Boisvert has served as Editor-in-Chief of the ACM Transactions on Mathematical Software 1992-2005, Co-chair of the Numerics Working Group of the Java Grande Forum 1998-2003, Co-chair of the Publications Board of the Association for Computing Machinery (ACM) 2005-2013, and Chair of the International Federation for Information Processing (IFIP) Working Group 2.
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►Brian Antonishek is on the staff of the Visualization and Usability Group in the Information Access Division of the Information Technology Laboratory at the National Institute of Standards and Technology.
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