A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions.

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ∫^∞ ₀ x^−ρ J_μ ( ax) J_ν ( bx) dx which is discontinuous on the diagonal a = b the integral ∫^∞ ₀ x^−ρ J⁽²⁾ _ν ( a(1 − q) x; q) J⁽¹⁾ _μ ( b(1 − q) x; q) dx is evaluated where J⁽¹⁾ _μ ( x; q) and J⁽²⁾ _μ ( 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^ρ in terms of J⁽²⁾ _{ν+1+2 n} ((1 − q) x; q). Finally, a q-Lommel function is introduced. [ABSTRACT FROM AUTHOR]