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In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.

SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS