## The properties, inequalities and numerical approximation of modified Bessel functions

Juri M. Rappoport

### Abstract

Some new properties of kernels of modified Kontorovitch–Lebedev integral transforms –- modified Bessel functions of the second kind with complex order $K_{\frac{1}{2}+i\beta }(x)$ are presented. Inequalities giving estimations for these functions with argument $x$ and parameter $\beta$ are obtained. The polynomial approximations of these functions as a solutions of linear differential equations with polynomial coefficients and their systems are proposed.

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### Key words

Chebyshev polynomials, modified Bessel functions, Lanczos Tau method, Kontorovich-Lebedev integral transforms

### AMS subject classifications

33C10, 33F05, 65D20

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