Meixner polynomials of the second kind and quantum algebras representing \(\mathrm{su}(1,1)\). (English) Zbl 1307.33005
Summary: We show how Viennot’s combinatorial theory of orthogonal polynomials may be used to generalize some recent results of A.Hodges and C. V. Sukumar [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 463, No. 2086, 2401–2414 (2007; Zbl 1132.81347); 463, No. 2086, 2415–2427 (2007; Zbl 1132.81348)] on the matrix entries in powers of certain operators in a representation of \(\mathrm{su}(1,1)\). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and van der Jeugt (see J. Van der Jeugt [J. Math. Phys. 38, 2728–2740 (1997; Zbl 0897.17005)]; H. T. Koelink and J. Van der Jeugt [SIAM J. Math. Anal. 29, 794–822 (1998; Zbl 0977.33013)]; W. Groenevelt and E. Koelink [J. Phys. A, Math. Gen. 35, No. 1, 65–85 (2002; Zbl 1002.33006)]), for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this ‘coincidence’, using X. G. Viennot’s “A combinatorial theory of orthogonal polynomials.” (French) [Lecture Notes UQAM, 217 p., Publication du LACIM, Université du Québec à Montréal (1984), réed. (1991), see http://web.mac.com/xgviennot].
MSC:
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
Online Encyclopedia of Integer Sequences:
Tangent (or ”Zag”) numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).
Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.
Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.
Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.
Infinitesimal generator matrix for a diagonally-shifted Pascal matrix, binomial(n+m,k+m), for m=1, related to Laguerre(n,x,m).
Infinitesimal generator for a diagonally-shifted Lah matrix, unsigned A105278, related to n! Laguerre(n,-x,1).
Triangle read by rows, giving the numbers T(n,m) = binomial(n+1, m+1); or, Pascal’s triangle A007318 with its left-hand edge removed.
References:
| [1] | DUKE MATH J 26 pp 1– (1959) |
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