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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 (1906a, 1906b, 1905a, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

Definition

The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function ${\displaystyle \phi }$ by

${\displaystyle J_{\nu }^{(1)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{2}\phi _{1}(0,0;q^{\nu +1};q,-x^{2}/4),\quad |x|<2,}$

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{0}\phi _{1}(;q^{\nu +1};q,-x^{2}q^{\nu +1}/4),\quad x\in \mathbb {C} ,}$

${\displaystyle J_{\nu }^{(3)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{1}\phi _{1}(0;q^{\nu +1};q,qx^{2}/4),\quad x\in \mathbb {C} .}$

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

${\displaystyle \lim _{q\to 1}J_{\nu }^{(k)}(x(1-q);q)=J_{\nu }(x),\ k=1,2,3.}$

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

${\displaystyle J_{\nu }^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }J_{\nu }^{(1)}(x;q),\ |x|<2.}$

For integer order, the q-Bessel functions satisfy

${\displaystyle J_{n}^{(k)}(-x;q)=(-1)^{n}J_{n}^{(k)}(x;q),\ n\in \mathbb {Z} ,\ k=1,2,3.}$

Properties

Negative Integer Order

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

${\displaystyle (q^{m+1};q)_{\infty }=(q^{m+n+1};q)_{\infty }(q^{m+1};q)_{n},}$

${\displaystyle (q;q)_{m+n}=(q;q)_{m}(q^{m+1};q)_{n},\ m,n\in \mathbb {Z} ,}$

we obtain

${\displaystyle J_{-n}^{(k)}(x;q)=(-1)^{n}J_{n}^{(k)}(x;q),\ k=1,2.}$

Zeros

Hahn mentioned that ${\displaystyle J_{\nu }^{(2)}(x;q)}$ has infinitely many real zeros (Hahn (1949)). Ismail proved that for ${\displaystyle \nu >-1}$ all non-zero roots of ${\displaystyle J_{\nu }^{(2)}(x;q)}$ are real (Ismail (1982)).

Ratio of q-Bessel Functions

The function ${\displaystyle -ix^{-1/2}J_{\nu +1}^{(2)}(ix^{1/2};q)/J_{\nu }^{(2)}(ix^{1/2};q)}$ is a completely monotonic function (Ismail (1982)).

Recurrence Relations

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

${\displaystyle q^{\nu }J_{\nu +1}^{(k)}(x;q)={\frac {2(1-q^{\nu })}{x}}J_{\nu }^{(k)}(x;q)-J_{\nu -1}^{(k)}(x;q),\ k=1,2.}$

${\displaystyle J_{\nu }^{(1)}(x{\sqrt {q}};q)=q^{\pm \nu /2}\left(J_{\nu }^{(1)}(x;q)\pm {\frac {x}{2}}J_{\nu \pm 1}^{(1)}(x;q)\right).}$

Inequalities

When ${\displaystyle \nu >-1}$, the second Jackson q-Bessel function satisfies: ${\displaystyle \left|J_{\nu }^{(2)}(z;q)\right|\leq {\frac {(-{\sqrt {q}};q)_{\infty }}{(q;q)_{\infty }}}\left({\frac {|z|}{2}}\right)^{\nu }\exp \left\{{\frac {\log \left(|z|^{2}q^{\nu }/4\right)}{2\log q}}\right\}.}$ (see Zhang (2006).)

For ${\displaystyle n\in \mathbb {Z} }$, ${\displaystyle \left|J_{n}^{(2)}(z;q)\right|\leq {\frac {(-q^{n+1};q)_{\infty }}{(q;q)_{\infty }}}\left({\frac {|z|}{2}}\right)^{n}(-|z|^{2};q)_{\infty }.}$ (see Koelink (1993).)

Generating Function

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

${\displaystyle \sum _{n=-\infty }^{\infty }t^{n}J_{n}^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }e_{q}(xt/2)e_{q}(-x/2t),}$

${\displaystyle \sum _{n=-\infty }^{\infty }t^{n}J_{n}^{(3)}(x;q)=e_{q}(xt/2)E_{q}(-qx/2t).}$

${\displaystyle e_{q}}$ is the q-exponential function.

Alternative Representations

Integral Representations

The second Jackson q-Bessel function has the following integral representations (see Rahman (1987) and Ismail & Zhang (2018a)):

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{2\nu };q)_{\infty }}{2\pi (q^{\nu };q)_{\infty }}}(x/2)^{\nu }\cdot \int _{0}^{\pi }{\frac {\left(e^{2i\theta },e^{-2i\theta },-{\frac {ixq^{(\nu +1)/2}}{2}}e^{i\theta },-{\frac {ixq^{(\nu +1)/2}}{2}}e^{-i\theta };q\right)_{\infty }}{(e^{2i\theta }q^{\nu },e^{-2i\theta }q^{\nu };q)_{\infty }}}\,d\theta ,}$

${\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty },\ \Re \nu >0,}$

where ${\displaystyle (a;q)_{\infty }}$ is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit ${\displaystyle q\to 1}$.

${\displaystyle J_{\nu }^{(2)}(z;q)={\frac {(z/2)^{\nu }}{\sqrt {2\pi \log q^{-1}}}}\int _{-\infty }^{\infty }{\frac {\left({\frac {q^{\nu +1/2}z^{2}e^{ix}}{4}};q\right)_{\infty }\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{ix};q)_{\infty }}}\,dx.}$

Hypergeometric Representations

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

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(x/2)^{\nu }}{(q;q)_{\infty }}}\ _{1}\phi _{1}(-x^{2}/4;0;q,q^{\nu +1}),}$

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(x/2)^{\nu }({\sqrt {q}};q)_{\infty }}{2(q;q)_{\infty }}},\ f(x,a;q):=(iax;{\sqrt {q}})_{\infty }\ _{3}\phi _{2}\left({\begin{matrix}a,&-a,&0\\-{\sqrt {q}},&iax\end{matrix}};{\sqrt {q}},{\sqrt {q}}\right).}$

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

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

${\displaystyle I_{\nu }^{(j)}(x;q)=e^{i\nu \pi /2}J_{\nu }^{(j)}(x;q),\ j=1,2.}$

${\displaystyle K_{\nu }^{(j)}(x;q)={\frac {\pi }{2\sin(\pi \nu )}}\left\{I_{-\nu }^{(j)}(x;q)-I_{\nu }^{(j)}(x;q)\right\},\ j=1,2,\ \nu \in \mathbb {C} -\mathbb {Z} ,}$

${\displaystyle K_{n}^{(j)}(x;q)=\lim _{\nu \to n}K_{\nu }^{(j)}(x;q),\ n\in \mathbb {Z} .}$

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

${\displaystyle I_{\nu }^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }I_{\nu }^{(1)}(x;q).}$

For statistical applications, see Kemp (1997).

Recurrence Relations

By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained (${\displaystyle K_{\nu }^{(j)}(x;q)}$ also satisfies the same relation) (Ismail (1981)):

${\displaystyle q^{\nu }I_{\nu +1}^{(j)}(x;q)={\frac {2}{z}}(1-q^{\nu })I_{\nu }^{(j)}(x;q)+I_{\nu -1}^{(j)}(x;q),\ j=1,2.}$

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

Continued Fraction Representation

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

${\displaystyle {\frac {I_{\nu }^{(2)}(z;q)}{I_{\nu -1}^{(2)}(z;q)}}={\cfrac {1}{2(1-q^{\nu })/z+{\cfrac {q^{\nu }}{2(1-q^{\nu +1})/z+{\cfrac {q^{\nu +1}}{2(1-q^{\nu +2})/z+\ddots }}}}}}.}$

Alternative Representations

Hypergeometric Representations

The function ${\displaystyle I_{\nu }^{(2)}(z;q)}$ has the following representation (Ismail & Zhang (2018b)):

${\displaystyle I_{\nu }^{(2)}(z;q)={\frac {(z/2)^{\nu }}{(q,q)_{\infty }}}{}_{1}\phi _{1}(z^{2}/4;0;q,q^{\nu +1}).}$

Integral Representations

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

${\displaystyle I_{\nu }^{(2)}(z;q)=\left(z^{2}/4;q\right)_{\infty }\left({\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\cos \nu \theta \,d\theta }{\left(e^{i\theta }z/2;q\right)_{\infty }\left(e^{-i\theta }z/2;q\right)_{\infty }}}-{\frac {\sin \nu \pi }{\pi }}\int _{0}^{\infty }{\frac {e^{-\nu t}\,dt}{\left(-e^{t}z/2;q\right)_{\infty }\left(-e^{-t}z/2;q\right)_{\infty }}}\right),}$

${\displaystyle K_{\nu }^{(1)}(z;q)={\frac {1}{2}}\int _{0}^{\infty }{\frac {e^{-\nu t}\,dt}{\left(-e^{t/2}z/2;q\right)_{\infty }\left(-e^{-t/2}z/2;q\right)_{\infty }}},\ |\arg z|<\pi /2,}$

${\displaystyle K_{\nu }^{(1)}(z;q)=\int _{0}^{\infty }{\frac {\cosh \nu \,dt}{\left(-e^{t/2}z/2;q\right)_{\infty }\left(-e^{-t/2}z/2;q\right)_{\infty }}}.}$

See also

• q-Bessel polynomials

References

• 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 J_ν+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

• Special functions
• Q-analogs