On zeros of the combination of products of Bessel functions

In this paper, the function f_ν(t)=J_ν(t)I_-ν(t)+I_ν(t)J_-ν(t),0<ν<1, Ret >0, is investigated. Such functions were little studied in the literature. It is proved that more general functions f_ν,μ⁽¹⁾,⁽²⁾ (t)=J_ν(t)I_-μ(t)±I_μ(t)J_-ν(t) have a countable set of real zeros and a countable set of pure imaginary zeros. The proof uses the well-known Sturm theorem for second-order differential equations. The statement is applied to specific examples. In the case ν = 1/2, the function f_1/2(x) = J_1/2(x)I_-1/2(x) + I_1/2(x)J_-1/2(x) is reduced to an elementary function f_1/2(x) = 2/

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