Application of integral transforms composition method (ITCM) to obtaining transmutations via integral transforms with Bessel functions in kernels

In this paper we apply the Integral Transforms Composition Method (ITCM) in order to derive compositions of integral transforms with Bessel functions in kernels, and obtain norm estimates and other properties of such composition transforms. Exactly, we consider transmutations which are compositions of classical Hankel and $Y$ integral transforms. Norms estimates in $L_2$ for these integral transforms with Bessel functions in kernels and their compositions are obtained. Also boundedness conditions for such transforms in weighted Lebesgue classes are proved. Classical integral transforms are used in this method as basic blocks. The ITCM and transmutations obtained by this method are applied to deriving connection formulas for solutions of singular differential equations.