On Groups which are Linear and Homogeneous in both Variables and Parameters*
William R. Burnside · Proceedings of the London Mathematical Society · 1902
206 Prof. W. Bnrnside on Groups luhich are Linear and [Nov.13, which expresses the Bessel ftmction of the second hind in terms of the Legendre function.Since the most general solution of Bessel's differential equation of order n consists of a linear combination of /" (z) and Y n (z), it follows that this can be expressed as a linear combination of p Jo ""* and z 1 sin zt P,,.b (t) dt.Jo But this result, which is thus obtained for the case in which n is integral, is the same as the result already obtained for the case in which n is not integral; and therefore we see that in all cases the general solution of BesseVs equation is a linear combination of a* \ cosztP n .h (t)dt J o , .(V.) and s 1 I sin zt P fl .j(t) dt Jo On Groups -which are Linear and Homogeneous in both Variables and Parameters.*By W. BURNSIDE.