Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

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Consider the set of unitary operators on a complex separable Hilbert space \mathcal{H} , denoted as \mathcal{U}(\mathcal{H}) . Consider 1<p<\infty . We establish that a function f defined on the unit circle \mathbb{T} is n times continuously Fréchet \mathcal{S}^{p} -differentiable at every point in \mathcal{U}(\mathcal{H}) if and only if f\in C^{n}(\mathbb{T}) . Take a function U \colon\R\rightarrow\mathcal{U}(\mathcal{H}) such that the function t\in\R\mapsto U(t)-U(0) takes values in \mathcal{S}^{p} and is n times continuously \mathcal{S}^{p} -differentiable on \R . Consequently, for f\in C^{n}(\mathbb{T}) , we prove that f is n times continuously Gâteaux \mathcal{S}^{p} -differentiable at U(t) . We provide explicit expressions for both types of derivatives of f in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the n -th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and \mathcal{S}^{p} -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.