Opuscula Math. 29, no. 3 (2009), 313-329
http://dx.doi.org/10.7494/OpMath.2009.29.3.313

Opuscula Mathematica

# A double index transform with a product of Macdonald's functions revisited

Semyon B. Yakubovich

Abstract. We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions $$K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)$$, where $$(x, y) \in \mathbb{R}_+ \times \mathbb{R}_+$$ and $$i \tau, \tau \in \mathbb{R}_+$$ is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function $$K_{i \tau}(t)$$ as a kernel.

Keywords: Macdonald function, index transform, Kontorovich-Lebedev transform, double Mellin transform, Plancherel theorem, Parseval equality.

Mathematics Subject Classification: 44A20, 33C10, 35C15.

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