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Ratios of Bessel functions and roots of αJ v (x)+xJ v ' (x)=0. (English) Zbl 0938.33002

The author shows that ζJ ν (νζ)/J ν+1 (νζ) is stricty decreasing in ν on any interval not containing a singularity and obtaining an upper bound on its derivative:

νJ ν (νζ) J ν+1 (νζ)-2 ν 2 ·

He also shows that x v[Jν(x) J ν+1 (x)]2 where xJ ν (x)/Jν+1(x) is strictly increasing in ν. The graph of J ν (x)/J ν+1 (x) as a function of x is illustrated in figures for various values of ν. These results generalize and sharpen previously known results, and allow the deduction of a more complete description, including monotonicity and multiplicity, of the positive roots of the equation αJν(x)+xJ ν ' (x)=0 for all real ν and α.

33C10Bessel and Airy functions, cylinder functions, 0 F 1