Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zerosстатья
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Аннотация:Modified Bessel functions of the first kind Iv (z) (Infeld functions) where v > −1 are considered. Due to the constraint on the parameter v, all zeros of the function Iv (z) except z = 0 are simple, located on the imaginary axis by symmetric pairs and form an infinite countable set. On the basis on previous research of the authors dealing with general Bessel functions of the first kind Jv(z), a question about reciprocal expansion 1/Iv(z) in series of simple fractions of a certain structure (Krein's series) is studied. The general formulas to calculate of special infinite sums containing degrees of Infeld function zeros are an important consequence of obtained expansion in simple fractions of the value 1/Iv(z) with any v > −1. The possibility of concrete definition of established summation relationships at different values of parameters and their connection with analogous relationships for the Bessel functions of the first kind Jv (z) is discussed.