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Journal ArticleDOI

Canonical forms of the generalized hypergeometric series for the 3j and the 6j symbol

S. Datta Majumdar1
01 Jan 1984-Czechoslovak Journal of Physics (Kluwer Academic Publishers)-Vol. 34, Iss: 1, pp 15-21
TL;DR: In this article, expressions for the 3j and the 6j symbols involving generalized hypergeometric series, valid for all admissible values of the j's and the re's, are derived and it is shown that all such expressions are deducible from two basic forms by simple substitutions.
Abstract: Expressions for the 3j and the 6j symbol involving generalized hypergeometric series, valid for all admissible values of the j's and the re's, are derived and it is shown that, for a symbol of either type, all such expressions are deducible from two basic forms by simple substitutions.
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Journal ArticleDOI
TL;DR: The contribution of Raynal to angular-momentum theory is highly valuable as discussed by the authors, and we recall the main aspects of his work related to Wigner $3j$ symbols.
Abstract: The contribution of Jacques Raynal to angular-momentum theory is highly valuable. In the present article, I intend to recall the main aspects of his work related to Wigner $3j$ symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the $3j$ coefficients were initially characterized by the number of terms of the series minus one, which is the degree of the coefficient. A detailed study of the zeros of the $3j$ coefficient with respect to the degree $n$ for $J = a + b + c \leq 240$ ($a$, $b$ and $c$ being the angular momenta in the first line of the $3j$ symbol) by Raynal revealed that most zeros of high degree had small magnetic quantum numbers. This led him to define the order $m$ to improve the classification of the zeros of the $3j$ coefficient. Raynal did a search for the polynomial zeros of degree 1 to 7 and found that the number of zeros of degree 1 and 2 are infinite, though the number of zeros of degree larger than 3 decreases very quickly as the degree increases. Based on Whipple's transformations of hypergeometric $_3F_2$ functions with unit argument, Raynal generalized the Wigner $3j$ symbols to any arguments and pointed out that there are twelve sets of ten formulas (twelve sets of 120 generalized $3j$ symbols) which are equivalent in the usual case. In this paper, we also discuss other aspects of the zeros of $3j$ coefficients, such as the role of Diophantine equations and powerful numbers, or the alternative approach involving Labarthe patterns.

Cites background from "Canonical forms of the generalized ..."

  • ...This minimum length corresponds to the number of terms when the coefficient is rearranged as a generalized hypergeometric series [32–34]....

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Journal ArticleDOI
TL;DR: The contribution of Raynal to angular-momentum theory is highly valuable as mentioned in this paper, and we recall the main aspects of his work related to Wigner 3j symbols.
Abstract: The contribution of Jacques Raynal to angular-momentum theory is highly valuable. In the present article, I intend to recall the main aspects of his work related to Wigner 3j symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the 3j coefficients were initially characterized by the number of terms of the series minus one, which is the degree of the coefficient. A detailed study of the zeros of the 3j coefficient with respect to the degree n for $$J=a+b+c\le 240$$ (a, b and c being the angular momenta in the first line of the 3j symbol) by Raynal revealed that most zeros of high degree had small magnetic quantum numbers. This led him to define the order m to improve the classification of the zeros of the 3j coefficient. Raynal did a search for the polynomial zeros of degree 1 to 7 and found that the number of zeros of degree 1 and 2 are infinite, though the number of zeros of degree larger than 3 decreases very quickly as the degree increases. Based on Whipple’s symmetries of hypergeometric $$_3F_2$$ functions with unit argument, Raynal generalized the Wigner 3j symbols to any arguments and pointed out that there are twelve sets of ten formulas (twelve sets of 120 generalized 3j symbols) which are equivalent in the usual case. In this paper, we also discuss other aspects of the zeros of 3j coefficients, such as the role of Diophantine equations and powerful numbers, or the alternative approach involving Labarthe patterns.
References
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Book
01 Jul 1965
TL;DR: Koornwinder as discussed by the authors gave identitity (2.5) with N = 0 and formulas (5.3), 5.3, and 5.4) substituted.
Abstract: This also gives in the paper T. H. Koornwinder, Orthogonal polynomials with weight function (1− x)α(1 + x)β + Mδ(x + 1) + Nδ(x− 1), Canad. Math. Bull. 27 (1984), 205–214 the identitity (2.5) with N = 0 and formulas (5.3), (5.4) substituted. p.95, §10.4, formula (7): second line: replace in denominator (v + n− 1)(w + n− 1) by Γ(v + n− 1)Γ(w + n− 1); third line: replace in denominator Γ(v + n− 1) by (v + n− 1); fifth line: replace in denominator Γ(w + n− 1) by (w + n− 1).

1,562 citations

Journal ArticleDOI
TL;DR: In this paper, the 3j coefficient is expressed as a function of five new parameters which have unique properties and satisfy simple validity criteria, and display the symmetry properties of the function in a particularly transparent manner.
Abstract: The 3j coefficient is expressed as a function of five new parameters which have unique properties. They are completely independent, satisfy simple validity criteria, and display the symmetry properties of the function in a particularly transparent manner. By means of the new parameters, the known 72‐element symmetry group is reduced to an eight‐element group, and the absolute symmetries are separated in a clear way from those which contain a phase factor.

8 citations