On the Macdonald-type function and its relation with index transforms and orthogonal polynomials

Abstract

We continue to investigate properties of the function $M_{\hat{I}}½(z)$ which is associated with the Macdonald function $K_{\hat{I}}½(z)$ in terms of the corresponding Fourier integral. In particular, recurrence relations for this function and its derivatives are obtained, involving properties of the associated Laguerre polynomials. Multiple orthogonal polynomials related to the scaled Macdonald-type weights $\hat{\ddot{I}}{}_{\hat{I}}½(x)=2x^{\hat{I}½/2}{M}_{\hat{I}}½\left(2\sqrt{x}\right),x>0$ are investigated.