# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. (English) Zbl 1110.30021
Summary: It has been known for some time that the existing asymptotic methods for integrals and differential equations are not applicable in the case of Stieltjes-Wigert polynomials with degree going to infinity. Using the recently introduced nonlinear steepest descent method for Riemann-Hilbert problems, here we not only derive an asymptotic expansion for these polynomials, but we also show that the result holds uniformly in the complex plane except for a sector containing the real axis from $-\infty$ to $\frac {1}{4}$. Furthermore, we give an asymptotic formula for the zeros of these polynomials, which approximates the true values of the zeros closely.

##### MSC:
 30E25 Boundary value problems, complex analysis 33C45 Orthogonal polynomials and functions of hypergeometric type 42C05 General theory of orthogonal functions and polynomials
Full Text:
##### References:
 [1] Chihara, T. S.: An introduction to orthogonal polynomials. (1978) · Zbl 0389.33008 [2] Christiansen, J. S.: The moment problem associated with the Stieltjes -- wigert polynomials. J. math. Anal. appl. 277, 218-245 (2003) · Zbl 1019.44005 [3] Deift, P.: Orthogonal polynomials and random matrices: A Riemann Hilbert approach. Courant lecture notes in mathematics 3 (1999) · Zbl 0997.47033 [4] Deift, P.; Zhou, X.: A steepest descent method for oscillatory Riemann -- Hilbert problems, applications for the mkdv equation. Ann. math. 137, 295-368 (1993) · Zbl 0771.35042 [5] Deift, P.; Kriecherbauer, T.; Mclaughlin, K. T. -R.; Venakis, S.; Zhou, X.: Strong asymptotics with respect to exponential weights. Comm. pure appl. Math. 52, 1491-1552 (1999) · Zbl 1026.42024 [6] Fokas, A. S.; Its, A. R.; Kitaev, A. V.: Discrete Painlevé equations and their appearance in quantum gravity. Comm. math. Phys. 142, 313-344 (1991) · Zbl 0742.35047 [7] Ismail, M. E. H.: Asymptotics of q orthogonal polynomials and a q-Airy function. Int. math. Res. notices 18, 1063-1088 (2005) · Zbl 1072.33014 [8] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report no. 98-17, TU-Delft, 1998 [9] Olver, F. W. J.: Asymptotics and special functions. (1974) · Zbl 0303.41035 [10] Szegő, G.: Orthogonal polynomials. Colloquium publications 23 (1975) [11] Wigert, S.: Sur LES polynômes orthogonaux et l’approximation des fonctions continues. Arkiv för matematik, astronomi och fysik 17 (1923) · Zbl 49.0296.01 [12] Wang, Z.; Wong, R.: Asymptotic expansions for second-order linear difference equations with a turning point. Numer. math. 94, 147-194 (2003) · Zbl 1030.39016 [13] Wang, Z.; Wong, R.: Linear difference equations with transition points. Math. comp. 74, 629-653 (2005) · Zbl 1083.41022 [14] Wong, R.: Asymptotic approximations of integrals. (1989) · Zbl 0679.41001