publication . Preprint . 2014

On the asymptotic normality of the Legendre-Stirling numbers of the second kind

Gawronski, Wolfgang; Littlejohn, Lance L.; Neuschel, Thorsten;
Open Access English
  • Published: 03 Aug 2014
Abstract
For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.
Subjects
arxiv: Mathematics::History and OverviewComputer Science::Symbolic ComputationMathematics::Number TheoryMathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, Mathematics - Classical Analysis and ODEs
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26 references, page 1 of 2

[1] G. E. Andrews, L. L. Littlejohn, A combinatorial interpretation of the LegendreStirling numbers, Proc. Amer. Math. Soc. 137 (2009) 2581-2590.

[2] G. E. Andrews, W. Gawronski, L. L. Littlejohn, The Legendre-Stirling numbers, Discrete Math. 311 (2011) 1255-1272. [OpenAIRE]

[3] G. E. Andrews, E. S. Egge, W. Gawronski, L. L. Littlejohn, The Jacobi-Stirling numbers, J. Comb. Theory, Ser. A 120 (2013) 288-303. [OpenAIRE]

[4] E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory, Ser. A 15 (1973) 91-111. [OpenAIRE]

[5] L. Carlitz, D. C. Kurtz, R. Scoville, O. P. Stackelberg, Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie verw. Geb. 23 (1972) 47-54.

[6] G. Cheon, J. Jung, r-Whitney numbers of Dowling lattices, Discrete Math. 312 (2012), 2337-2348.

[7] A. Claesson, S. Kitaev, K. Ragnarsson, B. E. Tenner, Boolean complexes of Ferrers graphs, Australas. J. Comb. 48 (2010) 159-173.

[8] L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Co., Boston, MA, 1974.

[9] E. S. Egge, Legendre-Stirling permutations, European J. Combin. 31 (2010) 1735- 1750.

[10] F. G. M. Eisenstein, Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenha¨ngenden Doppelreihen, Crelles J. 35 (1847).

[11] W. N. Everitt, L. L. Littlejohn, R. Wellman, The left-definite spectral theory for the classical Hermite differential equation, J. Comput. Appl. Math. 121 (2000) 313-330. [OpenAIRE]

[12] W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, G. J. Yoon, JacobiStirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression, J. Comput. Appl. Math. 208 (2007), 29-56. [OpenAIRE]

[13] P. Flajolet, R. Sedgewick, Analytic combinatorics, Cambridge University Press, New York, 2009.

[14] W. Gawronski, T. Neuschel, Euler-Frobenius numbers, Integral Transforms Spec. Funct. 24 (2013) 817-830.

[15] W. Gawronski, L. L. Littlejohn and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, Stud. Appl. Math. 133 (2014) 1- 17. [OpenAIRE]

26 references, page 1 of 2
Abstract
For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.
Subjects
arxiv: Mathematics::History and OverviewComputer Science::Symbolic ComputationMathematics::Number TheoryMathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, Mathematics - Classical Analysis and ODEs
Download from
26 references, page 1 of 2

[1] G. E. Andrews, L. L. Littlejohn, A combinatorial interpretation of the LegendreStirling numbers, Proc. Amer. Math. Soc. 137 (2009) 2581-2590.

[2] G. E. Andrews, W. Gawronski, L. L. Littlejohn, The Legendre-Stirling numbers, Discrete Math. 311 (2011) 1255-1272. [OpenAIRE]

[3] G. E. Andrews, E. S. Egge, W. Gawronski, L. L. Littlejohn, The Jacobi-Stirling numbers, J. Comb. Theory, Ser. A 120 (2013) 288-303. [OpenAIRE]

[4] E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory, Ser. A 15 (1973) 91-111. [OpenAIRE]

[5] L. Carlitz, D. C. Kurtz, R. Scoville, O. P. Stackelberg, Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie verw. Geb. 23 (1972) 47-54.

[6] G. Cheon, J. Jung, r-Whitney numbers of Dowling lattices, Discrete Math. 312 (2012), 2337-2348.

[7] A. Claesson, S. Kitaev, K. Ragnarsson, B. E. Tenner, Boolean complexes of Ferrers graphs, Australas. J. Comb. 48 (2010) 159-173.

[8] L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Co., Boston, MA, 1974.

[9] E. S. Egge, Legendre-Stirling permutations, European J. Combin. 31 (2010) 1735- 1750.

[10] F. G. M. Eisenstein, Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenha¨ngenden Doppelreihen, Crelles J. 35 (1847).

[11] W. N. Everitt, L. L. Littlejohn, R. Wellman, The left-definite spectral theory for the classical Hermite differential equation, J. Comput. Appl. Math. 121 (2000) 313-330. [OpenAIRE]

[12] W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, G. J. Yoon, JacobiStirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression, J. Comput. Appl. Math. 208 (2007), 29-56. [OpenAIRE]

[13] P. Flajolet, R. Sedgewick, Analytic combinatorics, Cambridge University Press, New York, 2009.

[14] W. Gawronski, T. Neuschel, Euler-Frobenius numbers, Integral Transforms Spec. Funct. 24 (2013) 817-830.

[15] W. Gawronski, L. L. Littlejohn and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, Stud. Appl. Math. 133 (2014) 1- 17. [OpenAIRE]

26 references, page 1 of 2
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