publication . Preprint . 2018

On certain combinatorial expansions of the Legendre-Stirling numbers

Ma, Shi-Mei; Ma, Jun; Yeh, Yeong-Nan;
Open Access English
  • Published: 28 May 2018
Abstract
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
Subjects
free text keywords: 05A05, 05A15, Mathematics - Combinatorics
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18 references, page 1 of 2

[1] G.E. Andrews, E.S Egge, W. Gawronski, L.L. Littlejohn, The Jacobi-Stirling numbers, J. Combin. Theory Ser. A, 120(1) (2013), 288-303.

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

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

[4] W.Y.C. Chen, Context-free grammars, differential operators and formal power series, Theoret. Comput. Sci., 117 (1993), 113-129.

[5] W.Y.C. Chen, A.M. Fu, Context-free grammars for permutations and increasing trees, Adv. in Appl. Math., 82 (2017), 58-82.

[6] E.S. Egge, Legendre-Stirling permutations, European J. Combin., 31(7) (2010), 1735-1750.

[7] W.N. Everitt, L.L. Littlejohn, R. Wellman, Legendre polynomials, Legendre-Stirling numbers, and the leftdefinite analysis of the Legendre differential expression, J. Comput. Appl. Math., 148 (1) (2002), 213-238. [OpenAIRE]

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

[9] W. Gawronski, L.L. Littlejohn, T. Neuschel, On the asymptotic normality of the Legendre-Stirling numbers of the second kind, European J. Combin., 49 (2015), 218-231.

[10] Y. Gelineau, J. Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, Electron. J. Combin., 17 (2010), #R70.

[11] I.M. Gessel, Z. Lin, J. Zeng, Jacobi-Stirling polynomials and P -partitions, European J. Combin., 33 (2012), 1987-2000.

[12] Z. Lin, J. Zeng, Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials, Adv. in Appl. Math., 53 (2014), 12-27.

[13] S.-M. Ma, Some combinatorial arrays generated by context-free grammars, European J. Combin., 34 (2013), 1081-1091.

[14] S.-M. Ma, J. Ma, Y.-N. Yeh, B.-X. Zhu, Context-free grammars for several polynomials associated with Eulerian polynomials, Electron. J. Combin., 25(1) (2018), #P1.31.

[15] M. Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J. Number Theory, 151(2015), 223-229.

18 references, page 1 of 2
Abstract
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
Subjects
free text keywords: 05A05, 05A15, Mathematics - Combinatorics
Download from
18 references, page 1 of 2

[1] G.E. Andrews, E.S Egge, W. Gawronski, L.L. Littlejohn, The Jacobi-Stirling numbers, J. Combin. Theory Ser. A, 120(1) (2013), 288-303.

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

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

[4] W.Y.C. Chen, Context-free grammars, differential operators and formal power series, Theoret. Comput. Sci., 117 (1993), 113-129.

[5] W.Y.C. Chen, A.M. Fu, Context-free grammars for permutations and increasing trees, Adv. in Appl. Math., 82 (2017), 58-82.

[6] E.S. Egge, Legendre-Stirling permutations, European J. Combin., 31(7) (2010), 1735-1750.

[7] W.N. Everitt, L.L. Littlejohn, R. Wellman, Legendre polynomials, Legendre-Stirling numbers, and the leftdefinite analysis of the Legendre differential expression, J. Comput. Appl. Math., 148 (1) (2002), 213-238. [OpenAIRE]

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

[9] W. Gawronski, L.L. Littlejohn, T. Neuschel, On the asymptotic normality of the Legendre-Stirling numbers of the second kind, European J. Combin., 49 (2015), 218-231.

[10] Y. Gelineau, J. Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, Electron. J. Combin., 17 (2010), #R70.

[11] I.M. Gessel, Z. Lin, J. Zeng, Jacobi-Stirling polynomials and P -partitions, European J. Combin., 33 (2012), 1987-2000.

[12] Z. Lin, J. Zeng, Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials, Adv. in Appl. Math., 53 (2014), 12-27.

[13] S.-M. Ma, Some combinatorial arrays generated by context-free grammars, European J. Combin., 34 (2013), 1081-1091.

[14] S.-M. Ma, J. Ma, Y.-N. Yeh, B.-X. Zhu, Context-free grammars for several polynomials associated with Eulerian polynomials, Electron. J. Combin., 25(1) (2018), #P1.31.

[15] M. Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J. Number Theory, 151(2015), 223-229.

18 references, page 1 of 2
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