[seqfan] Idle fiddling with log(PADIC) and catalan-numbers

[Gottfried Helms]

I was experimenting with Pari/GP to learn about
its padic-facilities. In the user manual you get
warned:
 " if x is a p_adic, you may compute y=log(x), but
 "  be aware, exp(y) =/= x .

So to see the effect, I tried

  x = 3 + O(7^20)    \\ means 3 in p_adic base 7 up to precision of 7^20

 then
  y = log(x)
  z = exp(y)

 z = 1 + 7 + 7^2 + 2*7^3 + 5*7^4 + 2*7^6 + 5*7^7 + 3*7^9 + 4*7^12 + 5*7^13 + 2*7^14 + 3*7^16 + 7^17 + 7^18 + 7^19 + O(7^20)

Looking only at the coefficients of powers of 7

  zvec   = [1, 1, 1, 2, 5, 0, 2, 5, 0, 3, 0, 0, 4, 5, 2, 0, 3, 1, 1, 1]

The leading 5 numbers set a signal for the catalan -numbers,

  A000108= [   1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]

 and the next (=0) can be 14 (mod 7), and the next (=2) is not 42 (mod 7), but well,
 it's 42 (mod 7) + 14/7, so...

 We have even an exact match at the beginning :
  A120588= [1, 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]

           comment:  "G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, starting with [1,1,1]"
           (Hi, Paul :-) )

So let's try this and express this as p_adic

  s = sum(k=1,#A120588, A120588[k]*7^(k-1) )
  s = s + O(7^20)    \\ to convert this into a p-adic

Compare s and z
 s = 1 + 7 + 7^2 + 2*7^3 + 5*7^4 + 2*7^6 + 5*7^7 + 3*7^9 + 4*7^12 + 5*7^13 + 2*7^14 + 3*7^16 +    3*7^17 + 5*7^18 + 3*7^19 + O(7^20)
 z = 1 + 7 + 7^2 + 2*7^3 + 5*7^4 + 2*7^6 + 5*7^7 + 3*7^9 + 4*7^12 + 5*7^13 + 2*7^14 + 3*7^16 +      7^17 + 7^18 + 7^19 + O(7^20)

 matches up to 7^16; matches improve if I use higher p-adic-precision and more terms of A120588.

 ------

Well - that's frappant. *How* is padic-log, padic-exp working? (At least in Pari/GP)

And - can we find similar representation for other log/exp's?

Here are the coefficients of the exp(log(x + O(7^20))) for x=1,2,3,4,5,6:

1: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
2: [1, 5, 0, 6, 5, 2, 1, 1, 5, 6, 4, 4, 2, 1, 4, 2, 4, 5, 0, 3]
3: [1, 1, 1, 2, 5, 0, 2, 5, 0, 3, 0, 0, 4, 5, 2, 0, 3, 1, 1, 1] -> catalan-numbers,A120588
4: [1, 3, 5, 1, 2, 1, 4, 4, 3, 0, 4, 4, 1, 4, 5, 1, 5, 2, 5, 0]
5: [1, 1, 5, 5, 2, 3, 3, 0, 1, 4, 1, 2, 0, 0, 0, 4, 6, 6, 4, 2]
6: [1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]


Gottfried Helms