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 q-Pochhammer symbol and the basic hypergeometric function φ by

They can be reduced to the Bessel function by the continuous limit:

There is a connection formula between the first and second Jackson q-Bessel function (Gasper & Rahman (2004)):

For integer order, the q-Bessel functions satisfy

Properties[edit]

Negative Integer Order[edit]

By using the relations (Gasper & Rahman (2004)):

we obtain

Zeros[edit]

Hahn mentioned that has infinitely many real zeros (Hahn (1949)). Ismail proved that for all non-zero roots of are real (Ismail (1982)).

Ratio of q-Bessel Functions[edit]

The function has complete monotonicity (Ismail (1982)).

Recurrence Relations[edit]

The first and second Jackson q-Bessel function has the following recurrence relations (see Ismail (1982) and Gasper & Rahman (2004)):

Inequalities[edit]

When , the second Jackson q-Bessel function satisfies: (see Zhang (2006).)
For , (see Koelink (1993).)

Generating Function[edit]

The following formulas are the q-analog of the generating function for the Bessel function (see Gasper & Rahman (2004)):

is the q-exponential function.

Alternative Representations[edit]

Integral Representations[edit]

The second Jackson q-Bessel functions has the following integral representations (see Rahman (1987) and Ismail and Zhang (2016)):

is the q-Pochhammer symbol, this representation reduces to the integral representation of the Bessel function with the limit .

Hypergeometric Representations[edit]

The second Jackson q-Bessel function has the following hypergeometric representations (see Koelink (1993), Chen, Ismail, and Muttalib (1994)):

An asymptotic expansion can be obtained as an immediate consequence of the second formula.

For other hypergeometric representations, see Rahman (1987).

Modified q-Bessel Functions[edit]

The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function (Ismail (1981) and Olshanetsky & Rogov (1995)):

There is a connection formula between the modified q-Bessel functions:

Recurrence Relations[edit]

By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained ( also satisfies the same relation) (Ismail (1981)):

For other recurrence relations, see Olshanetsky & Rogov (1995).

Continued Fraction Representation[edit]

The ratio of modified q-Bessel functions form a continued fraction(Ismail (1981)):

Alternative Representations[edit]

Hypergeometric Representations[edit]

The function has the following representation(Ismail & Zhang (2015)):

Integral Representations[edit]

The modified q-Bessel functions have the following integral representations (Ismail (1981)):

See also[edit]

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 0649849 
  • 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 (2): 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 
  • Rahman, M (1987), "An Integral Representation and Some Transformation Properties of q-Bessel Functions", Journal of Mathematical Analysis and Applications, 125: 58–71 
  • Ismail, Mourad E. H.; Zhang, R. (2016), "Integral and Series Representations of q-Polynomials and Functions: Part I", arXiv:1604.08441Freely accessible [math.CA] 
  • Koelink, H. T. (1993), "Hansen-Lommel Orthogonality Relations for Jackson's q-Bessel Functions.", Journal of Mathematical Analysis and Applications, 175: 425–437 
  • Zhang, R. (2006), "Plancherel-Rotach Asymptotics for q-Series", arXiv:math/0612216Freely accessible 
  • Gasper, G.; Rahman, M. (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 
  • Chen, Yang; Ismail, Mourad E. H.; Muttalib, K.A. (1994), "Asymptotics of basic Bessel functions and q-Laguerre polynomials.", Journal of Computational and Applied Mathematics, 54: 263–272 
  • Ismail, Mourad (1981), "The Basic Bessel Functions and Polynomials", SIAM Journal on Mathematical Analysis, 12 (3): 454–468 
  • Olshanetsky, M. A.; Rogov, V. B. (1995), "The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions", arXiv:q-alg/9509013Freely accessible