Asymptotic estimates for Laguerre polynomials
- 249 Downloads
We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsLn(α)(x). First we summarize the results of a paper of Frenzen and Wong in whichn→∞ and α>−1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which α is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.
Unable to display preview. Download preview PDF.
- M. A. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions. Nat. Bur. Stand. Appl. Math. Ser. 55, Washington D.C. (1964).Google Scholar
- G. B. Baumgartner, Jr., Uniform asymptotic approximations for the Whittaker functionM δ, μ (z). Ph.D-Thesis, Illinois Institute of Technology, Chicago 1980.Google Scholar
- F. Calogero, Asymptotic behaviour of the zeros of the generalized Laguerre polynomialL nα(x) α→∞and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento23, 101–102 (1978).Google Scholar
- T. M. Dunster,Uniform asymptotic expansions for Whittaker's confluent hypergeometric functions. SIAM J. Math. Anal.20, 744–760 (1989).Google Scholar
- A. Erdélyi,Bateman Manuscript project, Higher Transcendental Functions. Vol. II, McGraw-Hill, New York 1953.Google Scholar
- A. Erdélyi,Asymptotic forms for Laguerre polynomials. J. Indian Math. Soc.24, 235–250 (1960).Google Scholar
- C. L. Frenzen and R. Wong,Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal.19, 1232–1248 (1988).Google Scholar
- H. Hochstadt,The Functions of Mathematical Physics. Wiley-Interscience, New York 1971.Google Scholar
- W. Magnus, F. Oberhettinger and R. P. Soni,Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin 1966.Google Scholar
- F. W. J. Olver,Asymptotics and Special Functions. Academic Press, New York 1974.Google Scholar
- F. W. J. Olver,Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions. Proc. Royal Soc. Edinburgh84A, 213–234 (1980).Google Scholar
- G. Szegö,Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ., Vol. 23, New York 1958.Google Scholar