Results 1 - 10 of 327
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[en] A multivariable generalization is presented for all the discrete families of the Askey tableau. This significantly extends the multivariable Hahn polynomials introduced by Karlin and McGregor. The latter are recovered as a limit case from a family of multivariable Racah polynomials
[en] Several families of multivariable, biorthogonal, partly continuous and partly discrete, Wilson polynomials are presented. These yield limit cases that are purely continuous in some of the variables and purely discrete in the others, or purely discrete in all the variables. The latter are referred to as the multivariable biorthogonal Racah polynomials. Interesting further limit cases include the multivariable biorthogonal Hahn and dual Hahn polynomials
[en] A set of hypergeometric orthogonal polynomials, a set of biorthogonal rational functions generalizing them, and some new three-term relations for hypergeometric series containing properties of these functions are exhibited. The orthogonal polynomials depend on four free parameters, and their orthogonality relations include as special or limiting cases the orthogonalities for the classical polynomials, the Hahn and dual Hahn polynomials, Pollaczek's polynomials orthogonal on an infinite interval, and the 6-j symbols of angular momentum in quantum mechanics. Their properties include a second-order difference equation and a Rodrigues-type formula involving a divided difference operator
[en] Basing on evaluation of the Racah coefficients for SUq(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answers for all 5-strand knots with 9 crossings. In fact, the 7-strand formulas are sufficient to reproduce all the HOMFLY polynomials from the katlas.org: they are all described at once by a simple explicit formula with a very transparent structure. Moreover, would the formulas for the relevant SUq(3) Racah coefficients remain true for all other quantum groups, the paper provides a complete description of the fundamental HOMFLY polynomials for all braids with any number of strands.
[en] It has been suggested that the existence of one-dimensional irreps of a group leads to symmetries in the Racah algebra of the group. The familiar 2jm symbol arises as a special case of these symmetries where the one-dimensional irrep is the identity irrep. The general result is derived and examples are given for the symmetric groups and for the point groups. These examples show that these new symmetries are more complicated than the previous suggestions imply. (author)
[en] The Wigner-Racah algebra of an arbitrary (finite or compact continuous) group is presented in an original way that constitutes a straightforward extension of the corresponding algebra of the rotation group. Illustrative examples are given around the rotation group and the octahedral group. The adaptation of the Wigner-Racah algebra of the double rotation group to one of its subgroups G is discussed in detail. Special emphasis is put on the case where G corresponds to the octahedral group
[en] For a system with three identical nucleons in a single-j shell, the states can be written as the angular-momentum coupling of a nucleon pair and the odd nucleon. The overlaps between these nonorthonormal states form a matrix that coincides with the one derived by Rowe and Rosensteel [Phys. Rev. Lett. 87, 172501 (2001)]. The propositions they state are related to the eigenvalue problems of the matrix and dimensions of the associated subspaces. In this work, the propositions are proven from the symmetric properties of the 6j symbols. Algebraic expressions for the dimension of the states, eigenenergies, as well as conditions for conservation of seniority can be derived from the matrix.