zbMATH — the first resource for mathematics

A uniform asymptotic expansion of the single variable Bell polynomials. (English) Zbl 1025.41020
Bell polynomials $$P_n(z)$$ of a single variable are defined by the generating function $\exp(z(e^u-1))=1+\sum_{n=1}^\infty P_n(z) u^n/n!$ A representation in terms of Stirling numbers $$S(n,k)$$ of the second kind reads $$P_n(z)=\sum_{k=0}^n S(n,k)z^k$$. This paper discusses in detail the asymptotic behaviour of $$P_n(z)$$ for large values of $$n$$. The expansions are in terms of Airy-type expansions and hold uniformly with respect to $$z$$ in certain intervals of the negative axis. The surprising result is that the Cauchy-type integral that follows from the generating function has to be written, for $$z$$ in certain intervals on the negative axis, as a finite sum of integrals that each are expanded in terms of Airy-type expansions. Furthermore, one expansion is derived, also in terms of the Airy functions, completed with error bounds, holding uniformly for $$z\in (-\infty,-\delta]$$ for positive $$\delta$$.

MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33E20 Other functions defined by series and integrals
Full Text:
References:
  Bleistein, N., Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities, Comm. pure appl. math., 19, 353-370, (1966) · Zbl 0145.15801  Carlitz, L., Single variable Bell polynomials, Collect. math., 14, 13-25, (1962) · Zbl 0109.02906  Chester, C.; Friedman, B.; Ursell, F., An extension of the method of steepest descents, Proc. Cambridge philos. soc., 53, 599-611, (1957) · Zbl 0082.28601  Comtet, L., Advanced combinatorics, (1974), Reidel Dordrecht, Netherlands  Elbert, C., Strong asymptotics for the generating polynomials of the Stirling numbers of the second kind, J. approx. theory, 109, 198-217, (2001) · Zbl 0977.05003  Elbert, C., Weak asymptotics of the generating polynomials of the Stirling numbers of the second kind, J. approx. theory, 109, 218-228, (2001) · Zbl 0974.30005  Frenzen, C.L.; Wong, R., Uniform asymptotic expansion of Laguerre polynomials, SIAM J. math. anal., 19, 1232-1248, (1988) · Zbl 0654.33004  Lubinsky, D.S.; Sidi, A., Strong asymptotics for polynomials biorthogonal to powers of $$logx$$, Analysis, 14, 341-379, (1994) · Zbl 0814.30022  Olde Daalhuis, A.B.; Temme, N.M., Uniform Airy-type expansions of integrals, SIAM J. math. anal., 25, 304-321, (1994) · Zbl 0799.41028  Olver, F.W.J., Asymptotics and special functions, (1974), Academic Press New York · Zbl 0303.41035  Soni, K.; Soni, R.P., A system of polynomials associated with the Chester, Friedman, and ursell technique, (), 417-440 · Zbl 0723.41028  Szegö, G., Orthogonal polynomials, (1975), American Mathematical Society Providence, RI · JFM 65.0278.03  Titchmarsh, E.C., The theory of functions, (1939), Oxford University Press London · Zbl 0022.14602  Wong, R., Asymptotics approximations of integrals, (1989), Academic Press Boston · Zbl 0679.41001  R. Wong, Y.-Q. Zhao, Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials, preprint. · Zbl 1045.33009
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.