In taking over a worksheet I had written several years ago using MAPLE V Release 4, to the Release 5 version, I was puzzled at not reproducing my earlier results. I traced it to the fact that in the older version, I obtained
>taylor(BesselI(0,p+x),x,2); 2 BesselI(0, p) + BesselI(1, p) x + O(x )
which is fine, considering that the superscript 2 goes with O(x) whereas in the newer version I find
>taylor(BesselI(0,p+x),x,2); 2 BesselI(0, p) + O(x )
which does not have the linear term. The Taylor series for BesselJ (0,p+x) and BesselY(0,p+x) are correct in Release 5, but BesselK(0,p+x) has the same problem as BesselI(0,p+x).
Any suggestions, other than that I keep doing BesselI and BesselK with Release 4?
Yesterday I sent you a message concerning an error in the first term of the Taylor series expansion of the modified BesselI(0,p+x) and BesselK(0,p+x). In fact, it’s worse than I thought, since the linear term is wrong for all BesselI(n,p+x) (I didn’t prove this, of course). It looks as though you are using the relation between the derivative and the difference of the two adjacent BesselI, n-1 and n+1, whereas for the
modified Bessel functions it is the sum which appears. Check out Magnus and Oberhettinger p. 19. I hope you can get this fixed, because I think Release 5 is much more fun to work with than Release 4, which has all the Bessel functions right. Strangely enough, when I ask MAPLE V Release 5 to calculate the derivative using diff, all is well for all of the various Bessel
functions.
It is corrected with Maple 6. (U. Klein)