trough@ihuxi.UUCP (Chris Scussel) (01/30/85)
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
lindley@ut-ngp.UUCP (John L. Templer) (02/01/85)
> 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 > > Chris Scussel > AT&T Bell Labs > ihnp4!ihuxi!trough Surely you mean P(n)=, as in: oo n --- A * x \ n P(n) = / ------- = A for n >= 0 --- n! n k=0 -- John L. Templer University of Texas at Austin {allegra,gatech,seismo!ut-sally,vortex}!ut-ngp!lindley "and they called it, yuppy love."
dpb@philabs.UUCP (Paul Benjamin) (02/01/85)
> > 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 > > > > Chris Scussel > > AT&T Bell Labs > > ihnp4!ihuxi!trough > > Surely you mean P(n)=, as in: > oo n > --- A * x > \ n > P(n) = / ------- = A for n >= 0 > --- n! n > k=0 > > -- > > John L. Templer > University of Texas at Austin > > {allegra,gatech,seismo!ut-sally,vortex}!ut-ngp!lindley > > "and they called it, yuppy love." Even more surely you mean: oo k --- A * n \ k P(n) = / ------- = A for n >= 0 --- k! n k=0 0 A * 0 0 0 Since P(0) = ------- + (zeros) = 0 (if 0 is defined as 0), 0! we can change the lower limit of summation to 1.
ndiamond@watdaisy.UUCP (Norman Diamond) (02/04/85)
Actually, wouldn't THIS be a little more accurate? (more typographical changes)
oo k
--- A * x
\ k
P(n) = / ------- = A for n >= 0
--- k! n
k=0
-- Norman Diamond
UUCP: {decvax|utzoo|ihnp4|allegra|clyde}!watmath!watdaisy!ndiamond
CSNET: ndiamond%watdaisy@waterloo.csnet
ARPA: ndiamond%watdaisy%waterloo.csnet@csnet-relay.arpa
"Opinions are those of the keyboard, and do not reflect on me or higher-ups."dmcanzi@watdcsu.UUCP (David Canzi) (02/05/85)
How about this version?
oo k
--- A * n
\ k
P(n) = / ------- = A for n >= 0
--- k! n
k=0
Keep trying... sooner or later *somebody* has to get it right.
--
David Canzi
"But lo! men have become the tools of their tools." -- Henry David Thoreaudpb@philabs.UUCP (Paul Benjamin) (02/07/85)
> How about this version? > > oo k > --- A * n > \ k > P(n) = / ------- = A for n >= 0 > --- k! n > k=0 > > Keep trying... sooner or later *somebody* has to get it right. > -- > David Canzi > > "But lo! men have become the tools of their tools." -- Henry David Thoreau Actually, this is identical to my posting of 1 feb 85. Paul Benjamin