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In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$where $X$ is a spectrally positive or negative L\'evy process with infinitevariation. As a consequence we derive a generalization of the well-knownformula for the supremum distribution of Wiener process that is we obtain$\Prob(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha \Prob(Z_{\alpha}(t)\geq u)$for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative L\'evy process with$1<\alpha\leq 2$ which also stems from Kendall's identity for the firstcrossing time. Our proof uses a formula for the supremum distribution of aspectrally positive L\'evy process which follows easily from the elementarySeals formula.

Explicit formula for the supremum distribution of a spectrally negative stable process