H.-K. Hwang. Asymptotic expansions for the Stirling numbers of the rst kind. Journal of Combinatorial Theory, series A, 71 (1995), 343-351. Home/Search Document Not in Database Summary Related Articles Check |
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A Poisson * geometric convolution law for the number of components .. - Hwang (1995) (Correct)
.... aX, where : m=X and L 2 : sup jzj a jF (z)j. Proof. Sketch) Expand F at z = a: F (z) F ( F ( z ) z ) zt (1 t) dt; substitute this formula into I and estimate the integral 1 (z ) zt (1 t) dt dz by Laplace s method. For details, cf. 4] [17] or [36, pp. 230 231] An extension to asymptotic expansion involving Laguerre polynomials as coecients is established in [17] 4 Proof of the Theorems Adapting a number theoretic convention, we shall use the symbols and O interchangeably as is convenient. 4.1 Theorems 1, 2 and 3 First of ....
....(z ) zt (1 t) dt; substitute this formula into I and estimate the integral 1 (z ) zt (1 t) dt dz by Laplace s method. For details, cf. 4] 17] or [36, pp. 230 231] An extension to asymptotic expansion involving Laguerre polynomials as coecients is established in [17]. 4 Proof of the Theorems Adapting a number theoretic convention, we shall use the symbols and O interchangeably as is convenient. 4.1 Theorems 1, 2 and 3 First of all, by (1) we can write (w; z) where (w; z) exp A : Since the radius of convergence of C is equal to 1, ....
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H.-K. Hwang. Asymptotic expansions for the Stirling numbers of the rst kind. Journal of Combinatorial Theory, series A, 71 (1995), 343-351.
Asymptotics Of Multivariate Sequences, Part I: Smooth Points .. - Pemantle, Wilson (Correct)
....theorems of Flajolet Odlyzko (1990) to handle functions that are products of powers with powers of logs. Recent work of Bender and Richmond (Bender Richmond 1996, Bender Richmond 1999) extends the applicability of the central limit results to many problems of combinatorial interest; see also (Hwang 1995, Hwang 1998b) where more precise asymptotics are given, and Hwang (1998a) which extends some results to the combinatorial schemes of Flajolet Soria (1993) This does not exhaust the recent work on the problem of multivariable coecient extraction, but does circumscribe it. The present paper, ....
....are not sucient. Second, our methods obtain automatically a full asymptotic expansion of a r1 ; r d in decreasing powers of the indices r j . This is certainly not inherent in the existing results, whose relatively short proofs involve inversion of the characteristic function (see however Hwang (1995) and Hwang (1996) for something in this direction) The expansion to n terms is completely e ective in terms of the rst n partial derivatives of 1=F at z, as is the error bound. Third, these results explicitly cover the case where the pole at z has order greater than 1. The behavior in this case ....
Hwang, H.-K. (1995), `Asymptotic expansions for the Stirling numbers of the rst kind', J. Combin. Theory Ser. A 71(2), 343{ 351.
Asymptotics Of Multivariate Sequences, Part I: Smooth Points .. - Pemantle, Wilson (Correct)
....theorems of Flajolet Odlyzko (1990) to handle functions that are products of powers with powers of logs. Recent work of Bender and Richmond (Bender Richmond 1996, Bender Richmond 1999) extends the applicability of the central limit results to many problems of combinatorial interest; see also (Hwang 1995, Hwang 1998b) where more precise asymptotics are given, and Hwang (1998a) which extends some results to the combinatorial schemes of Flajolet Soria (1993) This does not exhaust the recent work on the problem of multivariable coecient extraction, but does circumscribe it. The present paper, ....
....are not sucient. Second, our methods obtain automatically a full asymptotic expansion of a r1 ; r d in decreasing powers of the indices r j . This is certainly not inherent in the existing results, whose relatively short proofs involve inversion of the characteristic function (see however Hwang (1995) and Hwang (1996) for something in this direction) The expansion to n terms is completely e ective in terms of the rst n partial derivatives of 1=F at z, as is the error bound. ASYMPTOTICS OF MULTIVARIATE SEQUENCES I 5 Third, these results explicitly cover the case where the pole at z has order ....
Hwang, H.-K. (1995), `Asymptotic expansions for the Stirling numbers of the rst kind', J. Combin. Theory Ser. A 71(2), 343{ 351.
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