Analytical evaluation of some Bessel function integrals related to Sonineʹs finite integrals

In this note integrals of the typeIij(t;k1,k2,k3)=∫0tek3τρi(τ;k1)ρj(t−τ;k2) dτ, i,j=0,1,withρ0(t;k)=I0(2kt), ρ1(t;k)=ktI1(2kt),will be evaluated. It is possible to expand Iij in a threefold power expansion in k1t, k2t, k3t, see (25). Moreover, it is also possible to expand these integrals into a single sum with Bessel functions In and Laguerre polynomials Ln, see (45) and (46).