查看“︁杰克逊q贝塞尔函数”︁的源代码

'''杰克逊q贝塞尔函数'''是英国数学家杰克逊在20世纪初创立的3个特殊函数，它们是[[贝塞尔函数]]的q模拟。定义如下

杰克逊q贝塞尔函数是通过[[阶乘幂]]和[[基本超几何函数]]定义的：

 

:<math> 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) </math>
:<math> 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^2q^{\nu +1}/4) </math>
:<math> 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) </math>

==参考文献==

*{{Citation | last1=Ismail | first1=Mourad E. H. | title=The zeros of basic Bessel functions, the functions J<sub>ν +ax</sub>(x), and associated orthogonal polynomials | doi=10.1016/0022-247X(82)90248-7 | mr=649849 | year=1982 | journal=Journal of Mathematical Analysis and Applications | issn=0022-247X | volume=86 | issue=1 | pages=1–19}}
*{{Citation | last1=Jackson | first1=F. H. | title=On generalized functions of Legendre and Bessel | year=1903 | journal= Transactions of the Royal Society of Edinburgh | volume=41 | pages=1–28}}
*{{Citation | last1=Jackson | first1=F. H. | title= Theorems relating to a generalization of the Bessel functions | year=1903 | journal= Transactions of the Royal Society of Edinburgh | volume=41 | pages= 105–118}}
*{{Citation | last1=Jackson | first1=F. H. | title=Theorems relating to a generalization of  Bessel's function. | jfm=36.0513.02 | year=1904 | journal=Transactions of the Royal Society of Edinburgh | volume=41 | pages=399–408 | doi=10.1017/s0080456800034475}}
*{{Citation | last1=Jackson | first1=F. H. | title=The Application of Basic Numbers to Bessel's and Legendre's Functions  | doi=10.1112/plms/s2-2.1.192  | year=1905 | journal=Proceedings of the London Mathematical Society | volume=2 | issue=1 | pages=192–220}}
*{{Citation | last1=Jackson | first1=F. H. | title=The Application of Basic Numbers to Bessel's and Legendre's Functions (Second paper) | doi=10.1112/plms/s2-3.1.1  | year=1905 | journal=Proceedings of the London Mathematical Society | volume=3 | issue=1 | pages=1–23}}

[[Category:特殊函数]]
[[Category:Q-模拟]]
{{q超几何函数}}
[[Category:Q超几何多项式]]