*(English)*Zbl 1031.81525

The book concerns the angular momentum coupling and recoupling coefficients, their relation to generalized hypergeometric functions, their $q$-generalizations, polynomial zeros, relation to orthogonal and $q$-orthogonal polynomials, and their numerical computation. The book contains 7 chapters.

In Chapter 1, the authors give the required mathematical notations and results of the theory of ordinary and basic hypergeometric functions as well as of the theory of multiple hypergeometric series. They also modify the statement of Bell’s theorem for the homogeneous multiplicative Diophantine equation and provide a straightforward proof for it.

In Chapters 2-4 the angular momentum coefficients (3-$j$, 6-$j$ and 9-$j$ symbols) and the corresponding sets of hypergeometric functions of unit argument are given. The interrelationships of expressions for these coefficients via the transformation formulas for hypergeometric series are studied. A brief account of the group-theoretical analysis of the 18 terminating ${}_{3}{F}_{2}\left(1\right)$ series is given.

Chapter 5 deals with zeros of the 3-$j$, 6-$j$ and 9-$j$ coefficients, their classification, and the generation of degree-1 zeros. Degree-1 zeros are studied using algorithms based on either closed form expressions or solutions of multiplicative Diophantine equations. Algorithms for the generation of the complete set of polynomial zeros of degree 2 of the above coefficients are also given. After a brief discussion of the physical significance of the polynomial zeros of 6-$j$ coefficients and a tabulation of the 12 generic zeros for which explanations were provided on the basis of realizations of exceptional Lie algebras, the authors show that the coupled $\text{SO}\left(3\right)$ tensor operators of the form ${({T}^{{k}_{1}}\otimes {T}^{{k}_{2}})}_{q}^{k}$ close under commutation, with the 9-$j$ coefficient appearing as a part of the structure constants.

Chapter 6 is devoted to the relation of 3-$j$ and 6-$j$ coefficients to Hahn and Racah polynomials. The recurrence relations for the angular momentum coefficients are derived from those satisfied by Hahn and Racah polynomials.

Chapter 7 concerns the numerical computation of the coefficients. The possible exploitation of the hierarchical formulas for parallel algorithms is studied. Fortran programs for the computation of 3-$j$, 6-$j$ and 9-$j$ coefficients are included.

Note that the book overlaps with Chapters 8 and 14 of *N. Ya. Vilenkin* and *A. U. Klimyk’s* book “Representation of Lie groups and special functions;; [ Vol. 1, Kluwer, Dordrecht (1991; Zbl 0742.22001); Vol. 3 (1992; Zbl 0778.22001)].

##### MSC:

81R05 | Representations of finite-dimensional groups and algebras in quantum theory |

22E70 | Applications of Lie groups to physics; explicit representations |

33C80 | Connections of hypergeometric functions with groups and algebras |

81-01 | Textbooks (quantum theory) |

33D80 | Connections of basic hypergeometric functions with groups, algebras and related topics |