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From: trough@ihuxi.UUCP (Chris Scussel)
Newsgroups: net.math
Subject: Self-generating power series?
Message-ID: <1155@ihuxi.UUCP>
Date: Wed, 30-Jan-85 08:19:01 EST
Article-I.D.: ihuxi.1155
Posted: Wed Jan 30 08:19:01 1985
Date-Received: Thu, 31-Jan-85 01:45:47 EST
Organization: AT&T Bell Labs, Naperville, IL
Lines: 21

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