Uniform treatment of Darboux’s method and the Heisenberg polynomials

Liu, Sai-Yu; Wong, R.; Zhao, Yu-Qiu

doi:10.1090/S0002-9939-2013-11587-X

Uniform treatment of Darboux’s method and the Heisenberg polynomials
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by Sai-Yu Liu, R. Wong and Yu-Qiu Zhao PDF

Proc. Amer. Math. Soc. 141 (2013), 2683-2691 Request permission

Abstract:

We show that the set of Heisenberg polynomials furnishes a simple non-trivial example in the uniform treatment of Darboux’s method.

References

Xiao-Xi Bai and Yu-Qiu Zhao, A uniform asymptotic expansion for Jacobi polynomials via uniform treatment of Darboux’s method, J. Approx. Theory 148 (2007), no. 1, 1–11. MR 2356572, DOI 10.1016/j.jat.2007.02.001
Charles F. Dunkl, The Poisson kernel for Heisenberg polynomials on the disk, Math. Z. 187 (1984), no. 4, 527–547. MR 760053, DOI 10.1007/BF01174188
Charles F. Dunkl, Boundary value problems for harmonic functions on the Heisenberg group, Canad. J. Math. 38 (1986), no. 2, 478–512. MR 833580, DOI 10.4153/CJM-1986-024-9
A. Erdélyi, Uniform asymptotic expansion of integrals, Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968) Gordon and Breach, New York, 1970, pp. 149–168. MR 0336181
Jerry L. Fields, A uniform treatment of Darboux’s method, Arch. Rational Mech. Anal. 27 (1967), 289–305. MR 220984, DOI 10.1007/BF00281716
G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376. MR 315267, DOI 10.1090/S0002-9904-1973-13171-4
G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
George Gasper, Orthogonality of certain functions with respect to complex valued weights, Canadian J. Math. 33 (1981), no. 5, 1261–1270. MR 638380, DOI 10.4153/CJM-1981-095-3
Peter C. Greiner, Spherical harmonics on the Heisenberg group, Canad. Math. Bull. 23 (1980), no. 4, 383–396. MR 602590, DOI 10.4153/CMB-1980-057-9
Roger Howe, On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 2, 821–843. MR 578375, DOI 10.1090/S0273-0979-1980-14825-9
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
N. M. Temme, Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986), no. 4, 369–376. MR 892162, DOI 10.1007/BF01893438
Sundaram Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Mathematics, vol. 159, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1633042, DOI 10.1007/978-1-4612-1772-5
R. Wong, Asymptotic approximations of integrals, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. MR 1016818
R. Wong and Yu-Qiu Zhao, On a uniform treatment of Darboux’s method, Constr. Approx. 21 (2005), no. 2, 225–255. MR 2107940, DOI 10.1007/s00365-004-0562-9