Transmutation operators and a new representation for solutions of perturbed Bessel equations

New representations for an integral kernel of the transmutation operator and a regular solution perturbed Bessel equation form $-u^{\prime\prime}+\left(\frac{\ell(\ell+1)}{x^{2}}+q(x)\right)u=\omega^{2}u$ are obtained. The is represented as Fourier-Jacobi series. Neumann series functions uniformly convergent with respect to $\omega$. For coefficients convenient numerical computation recurrent integration formulas new representation improves ones from arXiv:1609.06679 arXiv:1712.01363 large values $\omega$ $\ell$ non-integer $\ell$. results based on application several ideas classical (transformation) theory, asymptotic solution, connecting decay rate Fourier transform smoothness function, Paley-Wiener theorem some constructive approximation theory. We show that analytical obtained among other possible applications offers simple efficient method able compute sets eigendata nondeteriorating accuracy.