Elliptic Faulhaber polynomials and lamé densities of states

International Mathematics Research Notices

Volumn 2006, Issue , 2006, Pages

  Grosset, M P ^a   Veselov, A P ^a,b  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

^b LANDAU INSTITUTE FOR THEORETICAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646459641     PISSN: 10737928     EISSN: 16870247     Source Type: Journal    
DOI: 10.1155/IMRN/2006/62120     Document Type: Article

References (33)

1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, DC, 1964, stated seventh printing May 1968.
2. E. D. Belokolos and V. Z. Enolskii, Reduction of Abelian functions and algebraically integrable systems, II. Complex analysis and representation theory, Journal of Mathematical Sciences (New York) 108 (2002), no. 3, 295-374.
3. E. D. Belokolos, V. Z. Enolskii, and M. Salerno, Wannier function for quasi-periodic finite-gap potential, Theoretical and Mathematical Physics 144 (2005), no. 2, 1081-1099.
4. P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1991.
5. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Russian Mathematical Surveys 31 (1976), no. 1, 55-136.
6. A. Erdelyi (ed.), Higher Transcendental Functions, vol. 1-3, McGraw-Hill, New York, 1953.
7. D. B. Fairlie and A. P. Veselov, Faulhaber and Bernoulli polynomials and solitons, Physica D 152-153 (2001), 47-50.
8. J. Faulhaber, Academia Algebra. Darinnen die Miraculosische inventiones zu den höchsten Gossen weiters continuirt und profitiert werden, Johann Ulrich Schönigs, Augsburg, 1631.
9. I. M. Gel'fand and L. A. Dikiǐ, Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations, Uspekhi Matematicheskikh Nauk 30 (1975), no. 5(185), 67-100.
10. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Advances in Mathematics 58 (1985), no. 3, 300-321.
11. F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach, Bulletin of the American Mathematical Society 35 (1998), no. 4, 271-317.
12. M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, Journal of Nonlinear Mathematical Physics 12 (2005), no. 4, 469-474.
13. _, Lamé equation, quantum top and elliptic Bernoulli polynomials, http://arxiv.org/abs/math-ph/0508066.
14. G. H. Halphen, Traité des Fonctions Elliptiques et de leurs Applications, Gauthier Villars et Fils Imprimeurs, Paris, 1888.
15. A. Hurwitz, Über die Entwicklungskoffizienten der lemniskatishen Functionen, Mathematische Annalen 51 (1898), no. 2, 196-226.
16. E. L. Ince, Further investigations into the periodic Lamé functions, Proceedings of the Royal Society of Edinburgh 60 (1940), 83-99.
17. C. G. J. Jacobi, De usu legitimo formulae summatoriae Maclaurinianae, Journal für die Reine und angewandte Mathematik 12 (1834), 263-272.
18. N. M. Katz, The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers, Mathematische Annalen 216 (1975), 1-4.
19. _, The Eisenstein measure and p-adic interpolation, American Journal of Mathematics 99 (1977), no. 2, 238-311.
20. D. E. Knuth, Johann Faulhaber and sums of powers, Mathematics of Computation 61 (1993), no. 203, 277-294.
21. I. M. Krichever, Elliptic solutions of the Kadomcev-Petviashvili equations, and integrable systems of particles, Function Analysis and Its Applications 14 (1980), no. 4, 45-54.
22. M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky, Korteweg-de Vries equation and generalizations. V, Journal of Mathematical Physics 11 (1970), no. 3, 952-953.
23. I. G. MacDonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1998.
24. R. M. Miura, C. S. Gardner, and M. D. Kruskal, Korteweg-de Vries equation and generalizations. II, Journal of Mathematical Physics 9 (1968), no. 8, 1204-1209.
25. S. P. Novikov, S. V. Manakov, L. P. Pitaevskiǐ, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Contemporary Soviet Mathematics, Plenum, New York, 1984.
26. Y. Onishi, Theory of the generalized Bernoulli-Hurwitz numbers for the algebraic functions of cyclotomic type and the universal Bernoulli numbers, http://arxiv.org/abs/math.NT/0406096.
27. D. Shenk and M. A. Shubin, Asymptotic expansion of state density and the spectral function of the Hill operator, Mathematics of the USSR Sbornik 56 (1987), 492-515.
28. K. Takemura, The Heun equation and the Calogero-Moser-Sutherland system. IV. The Hermite-Krichever ansatz, Communications in Mathematical Physics 258 (2005), no. 2, 367-403.
29. _, On eigenvalues of Lamé operator, 2004, http://arxiv.org/abs/math.CA/0409247.
30. J. Tannery and J. Molk, Éléments de la théorie des fonctions eliptiques. Tome III, Réimpression des éditions de 1898 et de 1902, Chelsea, New York, 1972.
31. A. Treibich and J.-L. Verdier, Revêtements tangentiels et sommes de 4 nombres triangulaires, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 311 (1990), no. 1, 51-54.
32. A. P. Veselov and J. P. Ward, On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials, Journal of Mathematical Analysis and Applications 305 (2005), no. 2, 712-721.
33. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1963.