Jackson q-Bessel function

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In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1903, 1903b, 1905, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

Definition[edit]

The three Jackson q-Bessel functions are given in terms of the Pochhammer symbol and the basic hypergeometric function φ by

References[edit]

  • Ismail, Mourad E. H. (1982), "The zeros of basic Bessel functions, the functions Jν +ax(x), and associated orthogonal polynomials", Journal of Mathematical Analysis and Applications, 86 (1): 1–19, doi:10.1016/0022-247X(82)90248-7, ISSN 0022-247X, MR 649849 
  • Jackson, F. H. (1903), "On generalized functions of Legendre and Bessel", Transactions of the Royal Society of Edinburgh, 41: 1–28 
  • Jackson, F. H. (1903), "Theorems relating to a generalization of the Bessel functions", Transactions of the Royal Society of Edinburgh, 41: 105–118 
  • Jackson, F. H. (1904), "Theorems relating to a generalization of Bessel's function.", Transactions of the Royal Society of Edinburgh, 41: 399–408, doi:10.1017/s0080456800034475, JFM 36.0513.02 
  • Jackson, F. H. (1905), "The Application of Basic Numbers to Bessel's and Legendre's Functions", Proceedings of the London Mathematical Society, 2 (1): 192–220, doi:10.1112/plms/s2-2.1.192 
  • Jackson, F. H. (1905), "The Application of Basic Numbers to Bessel's and Legendre's Functions (Second paper)", Proceedings of the London Mathematical Society, 3 (1): 1–23, doi:10.1112/plms/s2-3.1.1