The zeros of basic Bessel functions, the functions Jv + ax(x), and associated orthogonal polynomials

AbstractThe modified Lommel polynomials are generalized and their orthogonality relation is obtained. As a byproduct we prove that the non-zero roots of Jv + ax(x) are real, simple and interlace with those of Jv + ax − 1(x). The q-Lommel polynomials are shown to play in the theory of basic Bessel function the role played by the Lommel polynomials in the Bessel function theory. The q-Lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support. This is then used to prove that the prositive zeros of x−νJ(2)v(x; q) are real simple and interlace with the zeros of x−ν − 1J(2)v + 1(x; q), when ν > −1. We also establish the complete monotonicity of −1x−12Jv + 1(2)(i√x; q)Jv(2)(i√x; q