# zbMATH — the first resource for mathematics

Asymptotic estimates for Laguerre polynomials. (English) Zbl 0688.33007
A summary is given of recent results concerning the asymptotic behaviour of the Laguerre polynomials $$L_ n^{(\alpha)}(x)$$. First the results are summarized of a paper of Frenzen and Wong in which $$n\to \infty$$ and $$\alpha >-1$$ is fixed. Two different expansions are needed in that case, one with a J-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which $$\alpha$$ is not necessarily fixed. 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 zeroes of the approximant with the related zeros of the Laguerre polynomial.
Reviewer: N.M.Temme

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
##### Keywords:
Laguerre polynomial
Full Text:
##### References:
 [1] M. A. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions. Nat. Bur. Stand. Appl. Math. Ser. 55, Washington D.C. (1964). [2] G. B. Baumgartner, Jr., Uniform asymptotic approximations for the Whittaker functionM ?, ? (z). Ph.D-Thesis, Illinois Institute of Technology, Chicago 1980. [3] F. Calogero, Asymptotic behaviour of the zeros of the generalized Laguerre polynomialL n ? (x) ???and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento23, 101-102 (1978). [4] T. M. Dunster,Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal.20, 744-760 (1989). · Zbl 0673.33003 [5] A. Erdélyi,Bateman Manuscript project, Higher Transcendental Functions. Vol. II, McGraw-Hill, New York 1953. [6] A. Erdélyi,Asymptotic forms for Laguerre polynomials. J. Indian Math. Soc.24, 235-250 (1960). · Zbl 0105.05401 [7] C. L. Frenzen and R. Wong,Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal.19, 1232-1248 (1988). · Zbl 0654.33004 [8] H. Hochstadt,The Functions of Mathematical Physics. Wiley-Interscience, New York 1971. · Zbl 0217.39501 [9] W. Magnus, F. Oberhettinger and R. P. Soni,Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin 1966. · Zbl 0143.08502 [10] F. W. J. Olver,Asymptotics and Special Functions. Academic Press, New York 1974. · Zbl 0303.41035 [11] 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). · Zbl 0446.33008 [12] G. Szegö,Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ., Vol. 23, New York 1958.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.