Expansions of the Kurepa functions

Gradimir Milovanovi\'c

Abstract: The Taylor series expansions of the Kurepa function $K(a+z)$, $a\ge 0$, and numerical determination of their coefficients $b_\nu(a)$ for $a=0$ and $a=1$ are given. An asymptotic behaviour of $b_\nu(a)$ as well as that $|b_\nu(a)/b_{\nu+1}(a)|\sim a+1$, when $\nu\to\infty$, are shown. Using this fact, a transformation of series with much faster convergence is done. Numerical values of coefficients in such a transformed series for $a=0$ and $a=1$ are given with $30$ decimal digits. Also, the Chebyshev expansions of $K(1+z)$ and $1/K(1+z)$ are obtained.

Classification (MSC2000): 33B15, 64D20

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