The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants. (English) Zbl 1390.15018
Summary: We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of J. Grommer [J. Reine Angew. Math. 144, 114–165 (1914; JFM 45.0650.02)] and N. Tschebotareff [Math. Ann. 99, 660–686 (1928; JFM 54.0351.02)] allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz’s theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.
MSC:
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
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