傑克遜q貝索函數

傑克遜q貝索函數是英國數學家傑克遜在20世紀初創立的3個特殊函數，它們是貝索函數的q模擬。定義如下

傑克遜q貝索函數是通過階乘冪和基本超幾何函數定義的：

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

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

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

參考文獻

• Ismail, Mourad E. H., The zeros of basic Bessel functions, the functions J_{ν +ax}(x), and associated orthogonal polynomials, Journal of Mathematical Analysis and Applications, 1982, 86 (1): 1–19, ISSN 0022-247X, MR 0649849, doi:10.1016/0022-247X(82)90248-7 
• Jackson, F. H., On generalized functions of Legendre and Bessel, Transactions of the Royal Society of Edinburgh, 1903, 41: 1–28 
• Jackson, F. H., Theorems relating to a generalization of the Bessel functions, Transactions of the Royal Society of Edinburgh, 1903, 41: 105–118 
• Jackson, F. H., Theorems relating to a generalization of Bessel's function., Transactions of the Royal Society of Edinburgh, 1904, 41: 399–408, JFM 36.0513.02, doi:10.1017/s0080456800034475 
• Jackson, F. H., The Application of Basic Numbers to Bessel's and Legendre's Functions, Proceedings of the London Mathematical Society, 1905, 2 (1): 192–220, doi:10.1112/plms/s2-2.1.192 
• Jackson, F. H., The Application of Basic Numbers to Bessel's and Legendre's Functions (Second paper), Proceedings of the London Mathematical Society, 1905, 3 (1): 1–23, doi:10.1112/plms/s2-3.1.1