Asymptotic analysis of the Hermite polynomials from their differential-difference equation

Journal of Difference Equations and Applications

Volumn 13, Issue 12, 2007, Pages 1115-1128

  Dominici, Diego ^a  

^a STATE UNIVERSITY OF NEW YORK   (United States)

Author keywords

Asymptotic analysis; Differential difference equations; Discrete WKB method; Hermite polynomials; Orthogonal polynomials; Ray method

Indexed keywords

EID: 36248943043     PISSN: 10236198     EISSN: 15635120     Source Type: Journal    
DOI: 10.1080/10236190701458824     Document Type: Article

References (42)

1. Ahmed, S., 1978, Systems of nonlinear equations for the zeros of Hermite polynomials. Letters al Nuovo Cimento (2), 22(9), 367-370.
2. Area, I., Dimitrov, D.K., Godoy, E. and Ronveaux, A., 2004, Zeros of Gegenbauer and Hermite polynomials and connection coefficients. Mathematics of Computation, 73(248), 1937-1951 (electronic).
3. Calogero, F., 1977, On the zeros of Hermite polynomials. Letters al Nuovo Cimento (2), 20(14), 489-490.
4. Calogero, F. and Perelomov, A.M., 1978, Asymptotic density of the zeros of Hermite polynomials of diverging order, and related properties of certain singular integral operators. Letters al Nuovo Cimento (2), 23(18), 650-652.
5. Costin, O. and Costin, R., 1996, Rigorous WKB for finite-order linear recurrence relations with smooth coefficients. SIAM Journal on Mathematical Analysis, 27(1), 110-134.
6. Dehesa, J.S., Domínguez-Adame, F., Amola, E.R, and Zarzo, A., 1991, Hydrogen atom and orthogonal polynomials. Orthogonal Polynomials and Their Applications (Erice, 1990) Volume 9 of IMACS Ann. Comput. Appl. Math., pages 223-229 (Basel: Baltzer).
7. Deift, P.A., 1999, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach Volume 3 of Courant Lecture Notes in Mathematics (New York: New York University Courant Institute of Mathematical Sciences).
8. Dette, H. and Studden, W.J., 1995, Some new asymptotic properties for the zeros of Jacobi, Laguerre, and Hermite polynomials. Constructive Approximation, 11(2), 227-238.
9. Dingle, R.B. and Morgan, G.J., 1967, WKB methods for difference equations. I, II. Applied Scientific Research, 18, 221-237 238-245, 1968.
10. Dominici, D., Asymptotic analysis of the Askey-scheme II: from. Charlier to Hermite. Submitted, arXiv:math.CA/0508264 2005.
11. Dosdale, T., Duggan, G. and Morgan, G.J., 1974, Asymptotic solutions to differential-difference equations. Journal of Physics A, 7, 1017-1026.
12. Geronimo, J.S., Bruno, O. and Van Assche, W., 2004, WKB and turning point theory for second-order difference equations. Spectral Methods for Operators of Mathematical Physics Volume 154 of Oper. Theory Adv. Appl., pages 101-138 (Basel: Birkhäuser).
13. Geronimo, J.S. and Smith, D.T., 1992, WKB (Liouville-Green) analysis of second order difference equations and applications. Journal of Approximation Theory, 69(3), 269-301.
14. Greenwood, R.E. and Miller, J.J., 1948, Zeros of the Hermite polynomials and weights for Gauss' mechanical quadrature formula. Bulletin of the American Mathematical Society, 54, 765-769.
15. Hannabuss, K., 1997, An Introduction to Quantum Theory Volume 1 of Oxford Graduate Texts in Mathematics (New York: The Clarendon Press Oxford University Press).
16. Heflinger, L.O., 1956, The asymptotic behaviour of the Hermite polynomials. Procceings of the Koninklijke Nederlandse Akademie van Wetenschappen Series A. Indagationes Mathematicae, 18, 255-264.
17. Koekoek, R. and Swarttouw, R.F., 1998, The Askey-scheme of hypergeometric orthogonal, polynomials and its q-analogue, Technical. Report 98-17, Delft University of Technology, 1998. Email: http://aw.twi.tudelft.nl/ koekoek/askey/.
18. Lange, C.G. and Miura, R.M., 1982, Singular perturbation analysis of boundary value problems for differential-difference equations. SIAM Journal on Applied Mathematics, 42(3), 502-531.
19. Lange, CG. and Miura, R.M., 1985, Singular perturbation analysis of boundary value problems for differential-difference equations. II. Rapid oscillations and resonances. SIAM Journal on Applied Mathematics, 45(5), 687-707.
20. Lange, C.G. and Miura, R.M., 1985, Singular perturbation analysis of boundary value problems for differential-difference equations. III. Turning point problems. SIAM Journal on Applied Mathematics, 45(5), 708-734.
21. Lange, C.G. and Miura, R.M., 1991, Singular perturbation analysis of boundary value problems for differential-difference equations. IV. A nonlinear example with layer behavior. Studies in Applied Mathematics, 84(3), 231-273.
22. Lange, C.G. and Miura, R.M., 1994, Singular perturbation analysis of boundary value problems for differential-difference equations. V. Small shifts with layer behavior. SIAM Journal on Applied Mathematics, 54(1), 249-272.
23. Lange, C.G. and Miura, R.M., 1994, Singular perturbation analysis of boundary value problems for differential-difference equations. VI. Small shifts with rapid oscillations. SIMM Journal on Applied Mathematics, 54(1), 273-283.
24. Lebedev, N.N., 1972, Special Functions and Their Applications (New York: Dover Publications Inc.).
25. López, J.L. and Temme, N.M., 1999, Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. Journal of Mathematical Analysis and Applications, 239(2), 457-477.
26. McLaughlin, K., 2004, Introduction to asymptotics for orthogonal polynomials via Riemann-Hilbert methods. Laredo Lectures on Orthogonal Polynomials and Special Functions Adv. Theory Spec. Funct. Orthogonal Polynomials, pages 85-109 (Hauppauge, NY: Nova Science Publishers).
27. Olver, F.W.J., 1959, Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. Journal of Research of the National Bureau of Standards Section B, 63B, 131-169.
28. Pittaluga, G. and Sacripante, L., 1993, Bounds for the zeros of Hermite polynomials. Annals of Numerical Mathematics, 2(1-4), 371-379 Special functions (Torino, 1993).
29. -(x2/2)). Commentarii Mathematicl Helvetia, 1(1), 227-254.
30. Ricci, P.E., 1995, Improving the asymptotics for the greatest zeros of Hermite polynomials. Computers and mathematics with applications, 30(3-6), 409-416.
31. [31J Rosser, J.B., 1950, Note on zeros of the Hermite polynomials and weights for Gauss' mechanical quadrature formula. Proceedings of the American Mathematical Society, 1, 388-389.
32. Salzer, H.E., Zucker, R. and Capuano, R., 1952, Table of the zeros and weight factors of the first twenty Hermite polynomials. Journal of Research of the Notional Bureau of Standards, 48, 111-116.
33. Subramanian, P.R., 1993, Zeros of the Hermite polynomials and the simple Laguerre polynomials are irrational. Proceedings of the Third Annual Conference of Vijnana Parishad of India and the National Symposium of Ancient Science in India (Srinagar, 1993), Vol. 1, pp. 49-57.
34. Subramanian, P.R., 1995, Nonzero zeros of the Hermite polynomials are irrational. Fibonacci Quarterly, 33(2), 131-134.
35. Szego, G., 1975, Orthogonal Polynomials, 4th ed. (Providence, RI: American Mathematical Society).
36. Temme, N.M. and López, J.L., 1999, The role of Hermite polynomials in asymptotic analysis. Special Functions (Hong Kong, 1999) (River Edge, NJ: World Scientific Publishing), pp. 339-350.
37. Voznyuk, A.V., 1987, Asymptotic formulas for parabolic cylindrical functions and for Hermite polynomials for large values of the argument. Vychislitel'naya i Prikladnaya Matematika (Kiev), 125(63), 19-24.
38. v(va) via a difference equation. Numerische Mathematik, 91(1), 147-193.
39. Wang, Z. and Wong, R., 2003, Asymptotic expansions for second-order linear difference equations with a turning point. Numerische Mathematik, 94(1), 147-194.
40. Wang, Z. and Wong, R., 2005, Linear difference equations with transition points. Mathematics of Computation, 74(250), 629-653 (electronic).
41. Wilmott, P., 1985, A note on the WKB method for difference equations. IMA Journal of Applied Mathematics, 34(3), 295-302.
42. Wyman, M., 1963, The asymptotic behaviour of the Hermite polynomials. Canadian Journal of Mathematics, 15, 332-349.