A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions |
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Authors: | Rahman Mizan |
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Institution: | (1) School of Mathematics and Statistics, Carleton University, Ottawa, Ont, K1S 5B6 |
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Abstract: | In an attempt to find a q-analogue of Weber and Schafheitlin's integral ![int](/content/j1v26p8v54860607/xxlarge8747.gif)
0
x
–
J
(ax) J
(bx) dx which is discontinuous on the diagonal a = b the integral ![int](/content/j1v26p8v54860607/xxlarge8747.gif)
0
x
–
J
(2)
(a(1 – q)x; q)J
(1)
(b(1 – q)x; q) dx is evaluated where J
(1)
(x; q) and J
(2)
(x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x
in terms of J
(2)
+1+2n
((1 – q)x; q). Finally, a q-Lommel function is introduced. |
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Keywords: | Weber-Schafheitlin integral q-Bessel functions Ramanujan integral basic hypergeometric series q-integrals q-Lommel function q-Neumann series |
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