A combinatorial interpretation of the LegendreStirling numbers
Authors:
George E. Andrews and Lance L. Littlejohn
Journal:
Proc. Amer. Math. Soc. 137 (2009), 25812590
MSC (2000):
Primary 05A05, 05A15, 33C45; Secondary 34B24, 34L05, 47E05
Published electronically:
February 17, 2009
MathSciNet review:
2497469
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Abstract: The LegendreStirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical secondorder 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 LegendreStirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.
 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
(29 #4914)
 2.
G. E. Andrews, W. Gawronski, and L. L. Littlejohn, Some properties of the LegendreStirling numbers, in preparation.
 3.
A. Bruder, L. L. Littlejohn, D. Tuncer, and R. Wellman, Leftdefinite 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 (57 #124)
 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
(82m:34020)
 6.
W.
N. Everitt, L.
L. Littlejohn, and R.
Wellman, The leftdefinite spectral theory for the classical
Hermite differential equation, J. Comput. Appl. Math.
121 (2000), no. 12, 313–330. Numerical
analysis in the 20th century, Vol. I, Approximation theory. MR 1780053
(2001m:34065), http://dx.doi.org/10.1016/S03770427(00)003356
 7.
W.
N. Everitt, L.
L. Littlejohn, and R.
Wellman, Legendre polynomials, LegendreStirling numbers, and the
leftdefinite 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
(2003k:34089), http://dx.doi.org/10.1016/S03770427(02)005824
 8.
W.
N. Everitt, K.
H. Kwon, L.
L. Littlejohn, R.
Wellman, and G.
J. Yoon, JacobiStirling numbers, Jacobi polynomials, and the
leftdefinite analysis of the classical Jacobi differential
expression, J. Comput. Appl. Math. 208 (2007),
no. 1, 29–56. MR 2347735
(2008k:33046), http://dx.doi.org/10.1016/j.cam.2006.10.045
 9.
L.
L. Littlejohn and R.
Wellman, A general leftdefinite theory for certain selfadjoint
operators with applications to differential equations, J. Differential
Equations 181 (2002), no. 2, 280–339. MR 1907144
(2003e:47047), http://dx.doi.org/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
(41 #7485)
 11.
Ȧke
Pleijel, On Legendre’s polynomials, New developments in
differential equations (Proc. 2nd Scheveningen Conf., Scheveningen, 1975),
NorthHolland, Amsterdam, 1976, pp. 175–180. NorthHolland Math.
Studies, Vol. 21. MR 0454152
(56 #12403)
 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
(58 #11623)
 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, http://dx.doi.org/10.1007/BF01474161
 1.
 M. Abramowitz and I. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972. MR 0167642 (29:4914)
 2.
 G. E. Andrews, W. Gawronski, and L. L. Littlejohn, Some properties of the LegendreStirling numbers, in preparation.
 3.
 A. Bruder, L. L. Littlejohn, D. Tuncer, and R. Wellman, Leftdefinite theory with applications to orthogonal polynomials, J. Comput. Appl. Math., to appear.
 4.
 L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Co., Dordrecht, 1974. MR 0460128 (57:124)
 5.
 W. N. Everitt, Legendre polynomials and singular differential operators, Lecture Notes in Mathematics, Vol. 827, SpringerVerlag, BerlinNew York, 1980, 83106. MR 610812 (82m:34020)
 6.
 W. N. Everitt, L. L. Littlejohn, and R. Wellman, The leftdefinite spectral theory for the classical Hermite differential equation, J. Comput. Appl. Math., 121 (2000), 313330. MR 1780053 (2001m:34065)
 7.
 W. N. Everitt, L. L. Littlejohn, and R. Wellman, Legendre polynomials, LegendreStirling numbers, and the leftdefinite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math., 148 (2002), 213238. MR 1946196 (2003k:34089)
 8.
 W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, and G. J. Yoon, JacobiStirling numbers, Jacobi polynomials, and the leftdefinite analysis of the classical Jacobi differential expression, J. Comput. Appl. Math., 208 (2007), 2956. MR 2347735 (2008k:33046)
 9.
 L. L. Littlejohn and R. Wellman, A general leftdefinite theory for certain selfadjoint operators with applications to differential equations, J. Differential Equations, 181(2) (2002), 280339. MR 1907144 (2003e:47047)
 10.
 M. A. Naımark, Linear differential operators. II, Frederick Ungar Publishing Co., New York, 1968. MR 0262880 (41:7485)
 11.
 Å. Pleijel, On Legendre's polynomials, Mathematics Studies 21, NorthHolland Publishing Co., Amsterdam, 1976, pp. 175180. MR 0454152 (56:12403)
 12.
 Å. Pleijel, On the boundary condition for the Legendre polynomials, Annales AcademiæScientiarum Fennicæ, Series A I Mathematica, 2 (1976), 397408. MR 0492509 (58:11623)
 13.
 H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Mathematische Annalen, 68 (1910), 220269. MR 1511560
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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 767987328
Email:
Lance_Littlejohn@baylor.edu
DOI:
http://dx.doi.org/10.1090/S0002993909098141
PII:
S 00029939(09)098141
Keywords:
LegendreStirling numbers,
Stirling numbers of the second kind,
Legendre polynomials,
leftdefinite theory,
selfadjoint 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.
