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Bessel–Struve intertwining operator and generalized Taylor series on the real line

In this paper, we consider a differential operator ℓα, α > −1/2, called Bessel–Struve operator. We construct and study transmutation operator between ℓα and the second derivative operator d2/dx 2. We establish for the Bessel–Struve operator a Taylor formula with integral remainder. We apply these results to expand as Taylor series the translation operator associated with ℓα. We provide an analyticity criterion for functions on ℝ involving ℓα.