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 (1906a, 1906b, 1905a, 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 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 is a completely monotonic function (Ismail (1982)).
Recurrence Relations
[edit]The first and second Jackson q-Bessel function have 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 function has the following integral representations (see Rahman (1987) and Ismail & Zhang (2018a)):
where is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in 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:
For statistical applications, see Kemp (1997).
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 (2018b)):
Integral Representations
[edit]The modified q-Bessel functions have the following integral representations (Ismail (1981)):
See also
[edit]References
[edit]- 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 (3): 263–272, doi:10.1016/0377-0427(92)00128-v
- Gasper, G.; Rahman, M. (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
- Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten, 2 (1–2): 4–34, doi:10.1002/mana.19490020103, ISSN 0025-584X, MR 0030647
- Ismail, Mourad E. H. (1981), "The Basic Bessel Functions and Polynomials", SIAM Journal on Mathematical Analysis, 12 (3): 454–468, doi:10.1137/0512038
- 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
- Ismail, M. E. H.; Zhang, R. (2018a), "Integral and Series Representations of q-Polynomials and Functions: Part I", Analysis and Applications, 16 (2): 209–281, arXiv:1604.08441, doi:10.1142/S0219530517500129, S2CID 119142457
- Ismail, M. E. H.; Zhang, R. (2018b), "q-Bessel Functions and Rogers-Ramanujan Type Identities", Proceedings of the American Mathematical Society, 146 (9): 3633–3646, arXiv:1508.06861, doi:10.1090/proc/13078, S2CID 119721248
- Jackson, F. H. (1906a), "I.—On generalized functions of Legendre and Bessel", Transactions of the Royal Society of Edinburgh, 41 (1): 1–28, doi:10.1017/S0080456800080017
- Jackson, F. H. (1906b), "VI.—Theorems relating to a generalization of the Bessel function", Transactions of the Royal Society of Edinburgh, 41 (1): 105–118, doi:10.1017/S0080456800080078
- Jackson, F. H. (1906c), "XVII.—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. (1905a), "The Application of Basic Numbers to Bessel's and Legendre's Functions", Proceedings of the London Mathematical Society, 2, 2 (1): 192–220, doi:10.1112/plms/s2-2.1.192
- Jackson, F. H. (1905b), "The Application of Basic Numbers to Bessel's and Legendre's Functions (Second paper)", Proceedings of the London Mathematical Society, 2, 3 (1): 1–23, doi:10.1112/plms/s2-3.1.1
- Kemp, A. W. (1997), "On Modified q-Bessel Functions and Some Statistical Applications", in N. Balakrishnan (ed.), Advances in Combinatorial Methods and Applications to Probability and Statistics, pp. 451–463, doi:10.1007/978-1-4612-4140-9_27, ISBN 978-1-4612-4140-9, S2CID 124998083
- Koelink, H. T. (1993), "Hansen-Lommel Orthogonality Relations for Jackson's q-Bessel Functions", Journal of Mathematical Analysis and Applications, 175 (2): 425–437, doi:10.1006/jmaa.1993.1181
- Olshanetsky, M. A.; Rogov, V. B. (1995), "The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions", arXiv:q-alg/9509013
- Rahman, M. (1987), "An Integral Representation and Some Transformation Properties of q-Bessel Functions", Journal of Mathematical Analysis and Applications, 125: 58–71, doi:10.1016/0022-247x(87)90164-8
- Zhang, R. (2006), "Plancherel-Rotach Asymptotics for q-Series", arXiv:math/0612216