Ferreira, M.Luchko, Yu.Rodrigues, M. M.Vieira, N.2021-03-182021-12-012020-12-06Ferreira, M., Luchko, Yu., Rodrigues, M.M., Vieira, N., Eigenfunctions of the time‐fractional diffusion‐wave operator. Math Meth Appl Sci. 2021; 44(2): 1713–1743. https://doi.org/10.1002/mma.68740170-42141099-1476http://hdl.handle.net/10400.8/5523In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time-fractional diffusion-wave operator with the time-fractional derivative of order $\beta \in ]1,2[$ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier-Bessel transform and as double contour integrals of the Mellin-Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any $\beta \in ]1,2[$ and as Kamp\'{e} de F\'{e}riet and Lauricella series in two variables for the rational values of $\beta$. The limit cases $\beta=1$ (diffusion operator) and $\beta=2$ (wave operator) as well as an intermediate case $\beta=\frac{3}{2}$ are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order $\beta$ and the spatial dimension $n$.engTime-fractional diffusion-wave operator Eigenfunctions; Caputo fractional derivatives; Generalized hypergeometric series.EigenfunctionsCaputo fractional derivativesGeneralized hypergeometric seriesjournal articlehttps://doi.org/10.1002/mma.6874