Lithuanian Journal of Physics article / Lietuvos fizikos žurnalo straipsnis

EXPRESSION OF SPECIAL STRETCHED 9j COEFFICIENTS IN TERMS OF ₅F₄ HYPERGEOMETRIC SERIES
Jean-Christophe Pain^a,b
^aCEA, DAM, DIF, F-91297 Arpajon, France
^bCEA, Laboratoire Matière en Conditions Extrêmes, Université Paris-Saclay, F-91680 Bruyères-le-Châtel, France
Email: jean-christophe.pain@cea.fr

Received 3 October 2024; revised 16 October 2024; accepted 22 October 2024

The Clebsch–Gordan coefficients or Wigner 3j symbols are known to be proportional to a ₃F₂(1) hypergeometric series, and Racah 6j coefficients to ₄F₃(1). In general, however, non-trivial 9j symbols cannot be expressed as ₅F₄. In this paper, we show, using the Dougall–Ramanujan identity, that special stretched 9j symbols can be reformulated as ₅F₄(1) hypergeometric series.
Keywords: angular momentum theory, stretched 9j coefficients, hypergeometric functions

TAM TIKRŲ IŠTEMPTŲJŲ 9j KOEFICIENTŲ REIŠKIMAS ₅F₄ HIPERGEOMETRINE EILUTE
Jean-Christophe Pain^a,b

^aAtominės energijos komisariato tyrimų centras, Arpažonas, Prancūzija
^bAtominės energijos komisariato Medžiagų ekstremaliomis sąlygomis laboratorija Paryžiaus Saklė universitete, Bruyères-le-Châtel, Prancūzija

References / Nuorodos

[1] D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988),
https://doi.org/10.1142/0270
[2] S.J. Ališauskas and A.P. Jucys, Weight lowering operators and the multiplicity-free isoscalar factors of the group R₅, J. Math. Phys. 12, 594–605 (1971)
https://doi.org/10.1063/1.1665626
[3] S.J. Ališauskas and A.P. Jucys, Erratum: Weight lowering operators and the multiplicity-free isoscalar factors of the group R₅, J. Math. Phys. 13, 575 (1972)
https://doi.org/10.1063/1.1666016
[4] A.P. Jucys and A.A. Bandzaitis, Angular Momentum Theory in Quantum Physics (Mokslas, Vilnius, 1977) [in Russian]
[5] K. Srinivasa Rao, V. Rajeswari, and C.B. Chiu, A new Fortran program for the 9j angular momentum coefficient, Comput. Phys. Commun. 56, 231–248 (1989),
https://doi.org/10.1016/0010-4655(89)90021-0
[6] H. Rosengren, On the triple sum formula for Wigner 9j symbols, J. Math. Phys. 39, 6730–6744 (1998),
https://doi.org/10.1063/1.532634
[7] H.M. Srivastava, Generalized Neumann expansions involving hypergeometric functions, Proc. Cambridge Philos. Soc. 63, 425–429 (1967),
https://doi.org/10.1017/S0305004100041359
[8] G. Lauricella, Sulle funzioni ipergeometriche a piu variabili, Rend. Circ. Mat. Palermo 7, 111–158 (1893),
https://doi.org/10.1007/BF03012437
[9] J. Van der Jeugt, S.N. Pitre, and K.S. Rao, Multiple hypergeometric functions and 9-j coefficients, J. Phys. A: Math. Gen. 27, 5251–5264 (1994),
https://doi.org/10.1088/0305-4470/27/15/023
[10] A.C.T. Wu, Structure of Ališauskas-Jucys form of the 9j coefficients, J. Math. Phys. 14, 1222–1223 (1973),
https://doi.org/10.1063/1.1666469
[11] I.M. Gel'fand, M.I. Graev, and N.Ya. Vilenkin, Generalized Functions, Volume 5: Integral Geometry and Representation Theory (Academic, New York, 1966),
https://doi.org/10.1016/C2013-0-12222-2
[12] A.A. Bandzaitis, A. Karosiene, and A.P. Yutsis, On the calculation of 9j coefficients, Liet. Fiz. Rink. 4, 457–466 (1964), [in Russian]
[13] R.T. Sharp, Stretched X-coefficients, Nucl. Phys. A 95, 222–228 (1967),
https://doi.org/10.1016/0375-9474(67)90163-7
[14] K. Srinivasa Rao and J. Van der Jeugt, Stretched 9-j coefficients and summation theorems, J. Phys. A: Math. Gen. 27, 3083–3090 (1994),
https://doi.org/10.1088/0305-4470/27/9/022
[15] L. Zamick and S.J.Q. Robinson, Zeros of 6-j symbols: Atoms, nuclei, and bosons, Phys. Rev. C 84, 044325 (2011),
https://doi.org/10.1103/PhysRevC.84.044325
[16] J.-C. Pain, Special six-j and nine-j symbols for a single-j shell, Phys. Rev. C 84, 047303 (2011),
https://doi.org/10.1103/PhysRevC.84.047303
[17] J.-C. Pain, A note on recursive calculations of particular 9j coefficients, Lith. J. Phys. 51, 194–198 (2011),
https://doi.org/10.3952/lithjphys.51303
[18] A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, NJ, 1957),
https://doi.org/10.1515/9781400884186
[19] B. Kleszyk and L. Zamick, Analytical and numerical calculations for the asymptotic behavior of unitary 9j coefficients, Phys. Rev. C 89, 044322 (2014),
https://doi.org/10.1103/PhysRevC.89.044322
[20] K.S. Rao and J. Van der Jeugt, Transformations of single and double hypergeometric series from the triple sum series for the 9-j coefficient, Int. J. Theor. Phys. 37, 891–905 (1998),
https://doi.org/10.1023/A:1026632900221
[21] J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25, 114–132 (1907),
https://doi.org/10.1017/S0013091500033642
[22] G. Gasper and M. Rahman, Basic Hypergeometric Series (Cambridge University Press, 2004),
https://doi.org/10.1017/CBO9780511526251
[23] A.C. Dixon, Summation of a certain series, Proc. London Math. Soc. 35, 284–291 (1902),
https://doi.org/10.1112/plms/s1-35.1.284
[24] Wolfram Research, Inc., Mathematica, Version 14.0 (Champaign, IL, 2024),
https://functions.wolfram.com/
On the website, the formula https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric5F4/03/02/01/0001/
contains a typographical error: the parameter n + 1/2 should be n/2 + 1.
[25] J.G. Wills, On the evaluation of angular momentum coupling coefficients, Comput. Phys. Commun. 2, 381–382 (1971),
https://doi.org/10.1016/0010-4655(71)90030-0
[26] V. Bretz, An improved method for calculation of angular momentum coupling coefficients, Acta Phys. Acad. Sci. Hungaricae 40, 255–259 (1976),
https://doi.org/10.1007/BF03157502
[27] K. Srinivasa Rao, Computation of angular momentum coefficients using sets of generalised hypergeometric functions, Comput. Phys. Commun. 22, 297–302 (1981),
https://doi.org/10.1016/0010-4655(81)90063-1