Computation of Sums of Natural Powers of the Inverses of Roots of an Equation Connected with a Spectral Problem

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We consider equations arising in the oblique derivative spectral problem of the form$$ az{J}_v^{\prime }(z)+b{J}_v(z)=0,\kern1em z\in \mathbb{C}, $$ where ν, a, b ∈ ℂ are parameters such that |a|+|b| > 0 and Jν(z) is the Bessel function. For roots of the equation we prove summation relations. The results obtained agree with the theory of Rayleigh sums which are calculated in terms of zeros of the Bessel functions.