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In the first part of this paper, we introduce the notion of "cyclic stratum"of a Frobenius manifold $M$. This is the set of points of the extended manifold$\mathbb C^*\times M$ at which the unit vector field is a cyclic vector for theisomonodromic system defined by the flatness condition of the extended deformedconnection. The study of the geometry of the complement of the cyclic stratumis addressed. We show that at points of the cyclic stratum, the isomonodromicsystem attached to $M$ can be reduced to a scalar differential equation, calledthe "master differential equation" of $M$. In the case of Frobenius manifoldscoming from Gromov-Witten theory, namely quantum cohomologies of smoothprojective varieties, such a construction reproduces the notion of quantumdifferential equation.  In the second part of the paper, we introduce two multilinear transforms,called "Borel-Laplace $(\boldsymbol \alpha,\boldsymbol\beta)$-multitransforms",on spaces of Ribenboim formal power series with exponents and coefficients inan arbitrary finite dimensional $\mathbb C$-algebra $A$. When $A$ isspecialized to the cohomology of smooth projective varieties, the integralforms of the Borel-Laplace $(\boldsymbol\alpha,\boldsymbol\beta)$-multitransforms are used in order to rephrase theQuantum Lefschetz Theorem. This leads to explicit Mellin-Barnes integralrepresentations of solutions of the quantum differential equations for a wideclass of smooth projective varieties, including Fano complete intersections inprojective spaces.  In the third and final part of the paper, as an application, we show how touse the new analytic tools, introduced in the previous parts, in order to studythe quantum differential equations of Hirzebruch surfaces. This finally leadsto the proof of Dubrovin Conjecture for all Hirzebruch surfaces.

Cyclic stratum of Frobenius manifolds, Borel-Laplace  $(\boldsymbol\alpha,\boldsymbol\beta)$-multitransforms, and integral  representations of solutions of Quantum Differential Equations