On certain new connections between Legendre and Bessel functions. (Q5924150)

Es werden die folgenden Formeln abgeleitet: \[ \begin{matrix} \r&\,\l\\ 2z & \int\limits _0^1\,P_n\,(1-2y^2)\,J_0\,(2yz)\,y\,dy=J_{2n+1}\,(2z),\\ 8z & \int\limits _0^1\,P_n\,(1-2y^4)\,J_0\,(2yz)\,I_0\,(2yz)\,y^3\,dy= \dfrac {d}{dz}\,\bigl(J_{2n+1}\,(2z)\,I_{2n+1}\,(2z)\bigr),\\ & \int\limits _0^1\,P_n\,(1-2y^4)\,\dfrac {d}{dy}\,\{y^2\, \bigl(\text{ber}_1^2\,(2yz)+\text{bei}_1^2\,(2yz)\bigr)\} \,dy\\ &\hfill \llap{\(=(-1)^n\,\bigl(\text{ber}_{2n+1}^2\,(2z)+\text{bei}_{2n+1}^2\,(2z)\bigr)\)}\\ & \int\limits _0^1\,P_n\,(1-2y^4)\,\dfrac {d}{dy}\, \bigl(y^2\,J_1\,(2yz)\,I_1\,(2yz)\bigr)\,dy=J_{2n+1}\,(2z)\,I_{2n+1}\,(2z). \end{matrix} \]