Sonin-type expansions for solutions to a generalized Bessel equation of order \(m\) and their application (Q2773565)

The author proves theorems which express solutions NEWLINE\[NEWLINE U_{\nu_s}^{(s)}(z)=\sum\limits_{k=0}^{\infty} \frac{(z/m)^{\nu_s +mk}}{[(m-1)k+s-1]!\Gamma [k+\nu_s-(s-1)+1]}, NEWLINE\]NEWLINE where \(\nu_s=p+m(s-1)/(m-1)\), \(s=1,\dots, m-1\), and NEWLINE\[NEWLINE U_{\nu_m}^{(m)}(z)=\sum\limits_{k=0}^{\infty} \frac{(z/m)^{\nu_m +mk}}{k!\Gamma [(m-1)k+\nu_m+1]}, NEWLINE\]NEWLINE where \(\nu_m=-(m-1)p\), to the generalized Bessel equation of order \(m\), NEWLINE\[NEWLINE \Bigl[(m-1)\frac{d}{dz}+\frac{\nu +1}{z}\Bigr]\dots \Bigl[(m-1)\frac{d}{dz}+\frac{\nu +m-1}{z}\Bigr] \Bigl(\frac{d}{dz}-\frac{\nu}{z}\Bigr)U(z)=U(z), NEWLINE\]NEWLINE where \(\nu=-(m-1)p\), \(p\) is a complex parameter, as a Sonin--Neumann type series and then applies them to derivation of more complicated relations for products of such solutions, the so-called Bateman-type multiplication theorems.