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Daisuke Kanaizumi edited this page Oct 15, 2019 · 41 revisions

Welcome to the wiki! For details about the q-Bessel functions, see the following pages.


Swarttouw's PhD Thesis includes many properties (recurrence and difference-reccurence relations, generating functions, orthogonality relations, integral representations, addition formulas and a q-Neumann function).

Previous Research

Previously, the Bessel function was verified by using

  • Leibniz Criterion (Yamamoto-Matsuda,2005)
  • Numerical Integration (kv library)
  • Asymptotic Expansion (Oishi, 2008)
  • Hypergeometric Function (Johansson, 2015)
    Rigorous computation of the q-Bessel function came from these ideas. Johansson (2015) and kv library also verified the Airy function.

Zeros of Bessel Functions

Three methods are known.

  • Using Newton's method to the Bessel function (see "Numerical Methods for Physicists" by A. L. Garcia)
  • Using Newton's method to the ratio of Bessel functions (see "Numerical Methods for Special Functions" by Gil, Segura, Temme)
  • Using iteration including \arctan(ratio of Bessel functions) (see "Numerical Methods for Special Functions" by Gil, Segura, Temme)

Zeros of the Bessel function are applied to numerical integration (Ogata-Sugihara, Frappier-Olivier, Grozev-Rahman).

References

  • Frappier, C., & Olivier, P. (1993). A quadrature formula involving zeros of Bessel functions. Mathematics of Computation, 60(201), 303-316.
  • Grozev, G. R., & Rahman, Q. I. (1995). A quadrature formula with zeros of Bessel functions as nodes. Mathematics of Computation, 64(210), 715-725.

Properties of the Zeros of q-Bessel Functions (When \nu>-1)

Hahn (1949) proved that Jackson's 2nd q-Bessel function has infinite number of zeros on the real axis. Ismail (1982) proved that all zeros of Jackson's 2nd q-Bessel function are on the real axis and they are simple. The same properties were proved for the Hahn-Exton q-Bessel function by Koelink-Swarttouw (1994).

References

  • Ismail, M. E. (1982). Journal of Mathematical Analysis and Applications, 86(1), 1-19.
  • Koelink, H., Swarttouw, R. (1994). On the Zeros of the Hahn-Exton q-Bessel Function and Associated q-Lommel Polynomials, Journal of Mathematical Analysis and Applications, 186, 690-710.

Methods to Obtain Zeros of q-Bessel Functions

I couldn't use the Newton method, Krawczyk method, interval Newton method because I couldn't compute the derivative of the q-Bessel functions. Instead, I used the q-Krawczyk method and interval q-Newton method.

Related papers

  • Neumaier, A. (1984). An interval version of the secant method. BIT Numerical Mathematics, 24(3), 366-372.
  • Rajkovi´c, P., Stankovi´c, M., Marinkovi´c D., (2002). Mean Value Theorems in q-Calculus. Matematiˇcki Vesnik, 54(3-4), 171-178.
  • Alefeld, G. (1994). Inclusion Methods for Systems of Nonlinear Equations in: J. Herzberger (Ed.), Topics in Validated Computations, Studies in Computational Mathematics, Elsevier, Amsterdam, 7-26.

Zeros of Analytic Functions

For methods to obtain zeros of analytic functions, see Delves-Lyness method, Burniston-Siewart method, Ioakimidis-Anastasselou (1985), Kravanja-Cools-Haegemans (1998) and Johnson-Tucker (2009).

References

  • Delves, L. M., & Lyness, J. N. (1967). A numerical method for locating the zeros of an analytic function. Mathematics of computation, 21(100), 543-560.
  • Burniston, E. E., & Siewert, C. E. (1973, January). The use of Riemann problems in solving a class of transcendental equations. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 73, No. 1, pp. 111-118). Cambridge University Press.
  • Ioakimidis, N. I., & Anastasselou, E. G. (1985). A new, simple approach to the derivation of exact analytical formulae for the zeros of analytic functions. Applied mathematics and computation, 17(2), 123-127.
  • Kravanja, P., Cools, R., & Haegemans, A. (1998). Computing zeros of analytic mappings: A logarithmic residue approach. BIT Numerical Mathematics, 38(3), 583-596.
  • Johnson, T., & Tucker, W. (2009). Enclosing all zeros of an analytic function—A rigorous approach. Journal of Computational and Applied Mathematics, 228(1), 418-423.

Papers about Delves-Lyness method

  • Li, T. Y. (1983). On locating all zeros of an analytic function within a bounded domain by a revised Delves/Lyness method. SIAM Journal on Numerical Analysis, 20(4), 865-871.
  • Ioakimidis, N. I., & Anastasselou, E. G. (1985). A modification of the Delves-Lyness method for locating the zeros of analytic functions. Journal of Computational Physics, 59(3), 490-492.

Papers about Burniston-Siewart method

  • Anastasselou, E. G., & Ioakimidis, N. I. (1984). A generalization of the Siewert–Burniston method for the determination of zeros of analytic functions. Journal of Mathematical Physics, 25(8), 2422-2425.
  • Ioakimidis, N. I. (1985). Application of the generalized Siewert-Burniston method to locating zeros and poles of meromorphic functions. Zeitschrift für angewandte Mathematik und Physik ZAMP, 36(5), 733-742.

Importance of Computing q-Functions

Computation of q-functions will be a powerful tool to eliminate fake formulas from scientific publications. For example, one paper claimed that a q-analog of the Gaussian multiplication formula for the gamma function was found. However, computation results of the left-hand side and right-hand side did not match. Verified computation should have easily prevented the appearance of this mistake. The mentioned paper is also blamed by Koepf-Marinkovic-Rajkovic (2016).

Other Motivations

Computation of (q-) special functions may be applied to digital signal processing. Refer Oaku-Shiraki-Takayama (2003) for details.

q-analog

q-special functions are q-analogs (q-extensions, q-deformations) of special functions. q-analogs are generalizations adding a new parameter q to the original. It reduces to the original with the continuous limit q\to 1. q-analogs are defined in a form that matches q-analysis (where q-derivatives, q-integrals, q-Pochhammer symbols appear). Modifications just using the parameter q are not regarded as q-analogs (For example, the damped Newton method is not a q-analog of the Newton method).

Significance of q-analogs

q-analogs are defined for various mathematical objects (such as special functions, mathematical constants, differential equations, probability distributions etc.). Since q-derivative is defined as difference, a q-analog of an ODE, PDE is a discretization.

Mathematica and q-functions

Mathematica is capable to compute q-Pochhammer symbols, q-gamma function and the q-hypergeometric function since version 7 (2008). A visualization of the q-Pochhammer symbol by Mathematica can be seen by here.

Computational Methods for the Infinite q-Pochhammer Symbol


See Sokal (2002) or Gabutti-Allasia (2008) for details of the algorithm.

  • Gabutti, B., & Allasia, G. (2008). Evaluation of q-gamma function and q-analogues by iterative algorithms. Numerical Algorithms, 49(1-4), 159.

Further Reading


  • Zhang, R. (2008). Plancherel-Rotach Asymptotics for Certain Basic Hypergeometric Series, Advances in Mathematics 217, 1588-1613.
  • Zhang, R. (2008). On asymptotics of q-Gamma functions. Journal of Mathematical Analysis and Applications, 339(2), 1313-1321. (arXiv Version)

DE formula

DE (Double Exponential) formula is a trapezoidal integration formula combined with DE variable transformation, and it works to compute infinite interval integration. It was proposed by Takahashi-Mori (1974). This method is also used in the Mathematica function NIntegrate, and verification of some special functions such as gamma function and modified bessel function (Yamanaka-Okayama-Oishi). For more details, see tanh-sinh quadrature.

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