Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A combinatorial interpretation of the Legendre-Stirling numbers


Authors: George E. Andrews and Lance L. Littlejohn
Journal: Proc. Amer. Math. Soc. 137 (2009), 2581-2590
MSC (2000): Primary 05A05, 05A15, 33C45; Secondary 34B24, 34L05, 47E05
Published electronically: February 17, 2009
MathSciNet review: 2497469
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Abstract: The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order Legendre differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown by Littlejohn and Wellman, are the coefficients of integral powers of the Laguerre differential expression. An open question regarding the Legendre-Stirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.


References [Enhancements On Off] (What's this?)

  • 1. Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
  • 2. G. E. Andrews, W. Gawronski, and L. L. Littlejohn, Some properties of the Legendre-Stirling numbers, in preparation.
  • 3. A. Bruder, L. L. Littlejohn, D. Tuncer, and R. Wellman, Left-definite theory with applications to orthogonal polynomials, J. Comput. Appl. Math., to appear.
  • 4. Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
  • 5. W. N. Everitt, Legendre polynomials and singular differential operators, Ordinary and partial differential equations (Proc. Fifth Conf., Univ. Dundee, Dundee, 1978) Lecture Notes in Math., vol. 827, Springer, Berlin, 1980, pp. 83–106. MR 610812
  • 6. W. N. Everitt, L. L. Littlejohn, and R. Wellman, The left-definite spectral theory for the classical Hermite differential equation, J. Comput. Appl. Math. 121 (2000), no. 1-2, 313–330. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780053, 10.1016/S0377-0427(00)00335-6
  • 7. W. N. Everitt, L. L. Littlejohn, and R. Wellman, Legendre polynomials, Legendre-Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148 (2002), no. 1, 213–238. On the occasion of the 65th birthday of Professor Michael Eastham. MR 1946196, 10.1016/S0377-0427(02)00582-4
  • 8. W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, and 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), no. 1, 29–56. MR 2347735, 10.1016/j.cam.2006.10.045
  • 9. L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations 181 (2002), no. 2, 280–339. MR 1907144, 10.1006/jdeq.2001.4078
  • 10. M. A. Naĭmark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968. MR 0262880
  • 11. Ȧke Pleijel, On Legendre’s polynomials, New developments in differential equations (Proc. 2nd Scheveningen Conf., Scheveningen, 1975) North-Holland, Amsterdam, 1976, pp. 175–180. North-Holland Math. Studies, Vol. 21. MR 0454152
  • 12. Ȧke Pleijel, On the boundary condition for the Legendre polynomials, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 397–408. MR 0492509
  • 13. Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, 10.1007/BF01474161

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Additional Information

George E. Andrews
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16801
Email: andrews@math.psu.edu

Lance L. Littlejohn
Affiliation: Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
Email: Lance_Littlejohn@baylor.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09814-1
Keywords: Legendre-Stirling numbers, Stirling numbers of the second kind, Legendre polynomials, left-definite theory, self-adjoint operator
Received by editor(s): September 2, 2008
Received by editor(s) in revised form: October 21, 2008
Published electronically: February 17, 2009
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.