A linear ODE for the Omega function associated with the Euler function E-alpha(z) and the Bernoulli function B-alpha(z) (CROSBI ID 125571)
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Butzer, P.L. ; Poganj, Tibor ; Srivastava, H.M.
engleski
A linear ODE for the Omega function associated with the Euler function E-alpha(z) and the Bernoulli function B-alpha(z)
The authors derive a linear ODE whose particular solution is the Butzer-Flocke-Hauss complete real parameter Omega function $\Omega(w)$, which is associated with the complex-index Bernoulli function $B_\alpha(z)$ and with the complex-index Euler-function $E_\alpha(z)$. This is accomplished here with the aid of an integral representation of the alternating Mathieu series $\widetilde{; ; ; S}; ; ; (w)$. A new integral representation and some two/sided bounding inequalities are also given for the Omega function.
alternating Mathieu series; Butzer-Flocke-Hauss complete Omega function; complex-index Bernoulli function; complex index Euler-function; Intergal representation of Mathieu series; Riemann's Zeta; Bessel function of first kind
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