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  • 1
    Electronic Resource
    Electronic Resource
    The multiple sum formulas for 12j coefficients of SU(2) and uq(2) (2002)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 43 (2002), S. 1547-1568 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The expressions for 12j coefficients of the both kinds (without and with braiding) of the SU(2) group and the quantum algebra uq(2) are considered. Using Dougall's summation formula of the very well-poised hypergeometric 5F4(1) series and its q-generalization, several fourfold sum formulas [with each sum related to the balanced 5F4(1) or 5φ4 series] for the q-12j coefficients of the second kind (without braiding) are derived. Applying q-generalizations of rearrangement formulas of the very well-poised hypergeometric 6F5(−1) series [which correspond to a new expression for the Clebsch–Gordan coefficients of SU(2) and uq(2)], the new expressions with five sums [of the 4F3(1) and 3F2(1) or 4φ3 and 3φ2 type] are derived for the q-12j coefficients of the first kind (with braiding) instead of the usual expansions in terms of q-6j coefficients. Stretched and doubly stretched q-12j coefficients [as triple, double, or single sums, related to composed or separate hypergeometric 4F3(1) and 5F4(1) or 4φ3 and 5φ4 series and, particularly, to q-9j or q-6j coefficients] are considered.© 2002 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1436305
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  • 2
    Electronic Resource
    Electronic Resource
    Towards the canonical tensor operators of uq(3). II. The denominator function problem (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 5939-5955 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The explicit denominator (normalization) function of the canonical tensor operators of the quantum algebra uq(3), corresponding to the maximal null space case is derived ab initio in terms of double basic hypergeometric series, which cannot be obtained as any q-extension of the SU(3) denominator polynomial Gb″1(Δ,x) in terms of multiple (double or triple) balanced hypergeometric series, introduced by Biedenharn, Louck, and their collaborators (although their q=1 versions are shown being equivalent). The corresponding orthonormal seed isoscalar factors of the coupling (Wigner–Clebsch–Gordan) coefficients of uq(3) and SU(3) with multiple irreducible representations are presented. Conjectured expression of the q-polynomials [which ratios appear in the uq(3) and (new) SU(3) denominator functions for an arbitrary value of the canonical multiplicity label t of the repeating irreducible representations] in terms of multiple partition dependent q-series (extension of the maximal and minimal null space versions) is presented and considered. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533063
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  • 3
    Electronic Resource
    Electronic Resource
    Towards the canonical tensor operators of uq(3). I. The maximal null space case (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 5719-5746 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Generalizing the SU(3) canonical tensor operator concept (Biedenharn and Louck) to the quantum algebra uq(3), the Wigner–Clebsch–Gordan coefficients of uq(3) with repeating irreducible representations are considered. Extremal projectors of the quantum algebra uq(3) in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq(3) canonical isoscalar factors. Explicit expressions of the uq(3) isofactors, corresponding to the maximal null space case of the uq(3) unit canonical tensor operators, and their normalization factors (denominator functions) are presented. The transposition and conjugation phase factors for the SU(3) and uq(3) canonical isofactors are correlated with phases and zeros of boundary isofactors. Invariance of the canonical isofactors (or absence of such invariance) under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance (or numerical degeneracy) of the usual splitting (distinctive) conditions. Some oversights of previous publications are disclosed. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531737
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  • 4
    Electronic Resource
    Electronic Resource
    On biorthogonal and orthonormal Clebsch–Gordan coefficients of SU(3): Analytical and algebraic approaches (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 2351-2366 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Minimal biorthogonal systems of the Clebsch–Gordan (Wigner) coefficients of SU(3)&supuline;U(2) are discussed as well as the dual coupled bases. The closed system of analytical expressions for the dual isofactors (reduced Wigner coefficients) and the overlaps of coupled states is obtained with the help of analytical inversion symmetry. The Regge-type symmetry of the overlaps and the boundary orthonormal isofactors (orthogonalization coefficients) is discovered. The polynomial structure of the alternative complete algebraic systems of the orthonormal SU(3) isofactors (characterized by the null spaces, symmetries, and additional selection rules and obtained by means of the Hecht or Gram–Schmidt process) is considered. The realizations of the external "missing label'' operators of the third and the fourth orders in the minimal coupled bases, which lead to preferable algorithms to evaluate the orthonormal SU(3) coupling coefficients satisfying different symmetry properties, are presented. With the help of the 6j coefficients of SU(2) or inverted truncated SU(2) recoupling matrices, the biorthogonal systems associated with the SU(3) canonical tensor operators are expanded in terms of minimal ones.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528119
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  • 5
    Electronic Resource
    Electronic Resource
    The triple sum formulas for 9j coefficients of SU(2) and uq(2) (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 7589-7610 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Seven different triple sum formulas for 9j coefficients of the quantum algebra uq(2) are derived, using for these purposes the usual expansion of q-9j coefficients in terms of q-6j coefficients and recently derived summation formula of twisted q-factorial series (resembling the very well-poised basic hypergeometric 5φ4 series) as a q-generalization of Dougall's summation formula of the very well-poised hypergeometric 4F3(−1) series. This way for q=1 Rosengren's second proof of the SU(1,1) case is adapted for the SU(2) case to derive the known triple sum formula of Ališauskas and Jucys, as well as six new independent triple sum formulas for the Wigner 9j coefficients of the angular momentum theory. The mutual rearrangement possibilities of the derived triple sum formulas by means of the Chu–Vandermonde summation formulas are considered and applied to derive several versions of double sum formulas for the stretched q-9j coefficients, which give new rearrangement and summation formulas of special Kampé de Fériet functions and their q-generalizations. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1312198
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  • 6
    Electronic Resource
    Electronic Resource
    Orthogonalization coefficients for the Elliott–Draayer states of the two parametric irreps of SU(n)&supuline;SO(n) (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 569-575 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Explicit algebraic-polynomial expressions for the direct and inverse orthogonalization coefficients of the projected (Elliott–Draayer) basis states of SU(n)&supuline;SO(n) are constructed in the case of two parametric irreducible representation (irreps). They cover the general SU(3)&supuline;SO(3) case as well as the two parametric covariant irreps of SU(4) restricted to the SU(2)×SU(2) subgroup of spin and isospin. The obtained orthonormal bases are equivalent to those constructed by means of the ordered Gram–Schmidt process. The orthogonalization coefficients are expressed, up to explicitly given elementary factors, in terms of the numerator and denominator polynomials related with those that appeared in the orthogonalization coefficients of the paracanonical coupling coefficients (isofactors) of SU(3) and with Aλ functions of Louck, Biedenharn, and Lohe.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529396
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  • 7
    Electronic Resource
    Electronic Resource
    Explicit orthonormal Clebsch–Gordan coefficients of SU(3) (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 1325-1332 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The construction of the explicit algebraic-polynomial expressions for the nonmultiplicity-free orthonormal Clebsch–Gordan (Wigner) coefficients of SU(3)&supuline;U(2) is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the Hecht or Gram–Schmidt process. The direct and inverse orthogonalization coefficients (the first of them being equivalent to the boundary orthonormal isofactors) are expressed, up to explicitly given multiplicative factors, in terms of the numerator and denominator polynomials related with the auxiliary Aλ function of Louck, Biedenharn, and Lohe that appears as a fragment of the denominator G-functions of canonical SU(3) tensor operators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528770
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  • 8
    Electronic Resource
    Electronic Resource
    Explicit canonical tensor operators and orthonormal coupling coefficients of SU(3) (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 1983-2004 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The canonical unit SU(3) tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators, with the renormalization factors expressed in terms of the denominator functions of Biedenharn, Gustafson, Lohe, Louck, and Milne. The matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of Asherova and Smirnov. The self-conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique. The corresponding extreme isoscalar factors of the Clebsch–Gordan (Wigner) coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of SU(3) with its considerable simplication for the boundary values of parameters. The general isofactors are also expanded in the different ways in terms of their boundary values. The new classes of the generalized hypergeometric series are used as constructive elements of the SU(3) and SU(2) representation theory functions and their properties are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529622
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  • 9
    Electronic Resource
    Electronic Resource
    Explicit orthogonalization of some biorthogonal bases for SU(n)&supuline;SO(n) and Sp(4)&supuline;U(2) (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3296-3312 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Mutually related explicit algebraic-polynomial expressions of the orthogonalization coefficients are proposed for the biorthogonal [polynomial (Bargmann–Moshinsky), stretched, antistretched, quasistretched, and their dual] bases of the two parametric (mixed tensor and covariant) irreducible representations (irreps) of SU(n) restricted to SO(n), as well as for the projected (Smirnov–Tolstoy) and dual bases of the five-dimensional quasispin. The orthogonalization coefficients of the essentially simplified Gram–Schmidt process are expressed, up to explicitly given elementary factors, in terms of the numerator and denominator polynomials, represented as compositions of the generalized hypergeometric coefficients 〈3F2(...)||μ(approximately-greater-than) and 〈1F0(...)||ν(approximately-greater-than) and equivalent in the diagonal (denominator) and boundary cases to the Aλ(cabde) functions of Biedenharn and Louck. The distribution of zeros and the symmetry properties of the introduced polynomials are crucial for the conjectured solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529932
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  • 10
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    The multiple sum formulas for 12j coefficients of SU(2) and uq(2) (2002)
    American Institute of Physics
    Details
    Publication Date: 2002-03-01
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
    Published by American Institute of Physics
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