×

Monic non-commutative orthogonal polynomials. (English) Zbl 1152.05389

Summary: Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint distribution of operators on a Fock space.

MSC:

05E35 Orthogonal polynomials (combinatorics) (MSC2000)
46Nxx Miscellaneous applications of functional analysis

Software:

MOPS
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Michael Anshelevich, Orthogonal polynomials with a resolvent-type generating function, math.CO/0410482, accepted for publication by Transactions of the AMS, 2006. · Zbl 1146.05058
[2] -, Free Meixner states, math.CO/0702158, accepted for publication by Communications in Mathematical Physics, 2007. · Zbl 1133.33001
[3] T. Banks and T. Constantinescu, Orthogonal polynomials in several non-commuting variables. II, math.FA/0412528, 2004.
[4] T. Banks, T. Constantinescu, and J. L. Johnson, Relations on non-commutative variables and associated orthogonal polynomials, Operator theory, systems theory and scattering theory: multidimensional generalizations, Oper. Theory Adv. Appl., vol. 157, Birkhäuser, Basel, 2005, pp. 61 – 90. · Zbl 1075.42008 · doi:10.1007/3-7643-7303-2_2
[5] Ioana Dumitriu, Alan Edelman, and Gene Shuman, MOPS: multivariate orthogonal polynomials (symbolically), J. Symbolic Comput. 42 (2007), no. 6, 587 – 620. · Zbl 1122.33019 · doi:10.1016/j.jsc.2007.01.005
[6] Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. · Zbl 0964.33001
[7] P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980), no. 2, 125 – 161. , https://doi.org/10.1016/0012-365X(80)90050-3 Philippe Flajolet, Combinatorial aspects of continued fractions, Ann. Discrete Math. 9 (1980), 217 – 222. Combinatorics 79 (Proc. Colloq., Univ. Montréal, Montreal, Que., 1979), Part II.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.