Conference rusure::math

216.0. "Self Generating Power Series" by HARE::STAN () Fri Feb 01 1985 13:18

Newsgroups: net.math
Path: decwrl!decvax!bellcore!allegra!ulysses!mhuxr!mhuxb!mhuxn!mhuxm!mhuxj!houxm!ihnp4!ihuxi!trough
Subject: Self-generating power series?
Posted: Wed Jan 30 05:19:01 1985


Can anyone find a (nontrivial) power series that, evaluated over the
nonnegative integers, gives its own coefficients? That is,

                   oo        n
                  ---  A  * x
                  \     n
        P(x)  =   /    -------    =  A    for  n >= 0
                  ---    n!           n
                  k=0

Trivially A  = 0 does it.  P(0) = A  , so that's one for free.
           n                       0
The series for sin and cos seem close, but off by a factor of pi/2 with
respect to the power series variable, and I've not been able to see any
way to fix it up. By the way, this doesn't have any importance that I
know of, other than recreational. Enjoy!

				Chris Scussel
				AT&T Bell Labs
				ihnp4!ihuxi!trough
				

--------------------------
Comment from Stan: I believe he means

                   oo        k
                  ---  A  * x
                  \     k
        P(x)  =   /    -------
                  ---    k!  
                  k=0


Find the A   such that P(n) = A   for all integral n >= 0.
          k		       n


C'mon folks!  This one's *easy*!