q Bessel Formulas

This page is about Jackson's 2nd q-Bessel function and the Hahn-Exton q-Bessel function.

Definitions

Integral Representations

• Rahman, M. (1987). An integral representation and some transformation properties of q-Bessel functions. Journal of Mathematical Analysis and Applications, 125, 58-71.
  
  
  
• Ismail, M. E., & Zhang, R. (2018). Integral and series representations of q-polynomials and functions: Part I. Analysis and Applications, 16(02), 209-281.

Asymptotic Expansions

• Koelink, H. (1993). Hansen-Lommel Orthogonality Relations for Jackson's q-Bessel Functions. Journal of Mathematical Analysis and Applications, 175, 425-437.
  
  
• Chen, Y., Ismail, M., Muttalib, K. (1994). Asymptotics of Basic Bessel Functions and q-Laguerre Polynomials, Journal of Computational and Applied Mathematics, 54, 263-272.
  
  
• Daalhuis, A. (1994). Asymptotic Expansions for q-Gamma, q-Exponential, and q-Bessel functions. Journal of Mathematical Analysis and Applications, 186, 896-913.

Modified q-Bessel

Definition

• 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/9509013.

Asymptotic Expansions

• Ismail, M. E. H.; Zhang, R. (2018), "q-Bessel Functions and Rogers-Ramanujan Type Identities", Proceedings of the American Mathematical Society, 146 (9): 3633–3646, arXiv:1508.06861.

Further Reading

• Kemp, A. W. (1997). On Modified q-Bessel Functions and Some Statistical Applications. In Advances in Combinatorial Methods and Applications to Probability and Statistics (pp. 451-463). Birkhäuser Boston.

Related Hypergeometric Identities

• Koelink, H. (1993). Hansen-Lommel Orthogonality Relations for Jackson's $q$-Bessel Functions. Journal of Mathematical Analysis and Applications, 175, 425-437.

Confluent Basic Hypergeometric Series

• Koornwinder, T., Swarttouw, R. (1992). On $q$-Analogues of the Fourier and Hankel Transforms. Transactions of the American Mathematical Society, 333, 445-461.
  
• El-Guindy, A., & Ismail, M. E. (2016). On Certain Generalizations of Rogers-Ramanujan Type Identities. arXiv preprint arXiv:1602.00316. Theorem 3.1