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The function appearing in the title is examined to determine the number of roots and their general location. It is found that the roots appear in complex conjugate pairs, that the number of roots depends on      n n      , and that, with the exception of a positive real root arising for      a &gt; 0 a &gt; 0      , all roots lie in the left half-plane.

On the roots of 𝑓_{𝑛}(𝑧)=𝑎+𝐾_{𝑛+1}(𝑧)/𝑧𝐾_{𝑛}(𝑧)