some reciproce transform

[Wouter Meeussen]

hi all,

the Taylor series of 1/g[z] produces a kind of transform, if 
the k'th derivative of g[z] at z=0 is written as f[k]

Expand[(Rest[CoefficientList[Series[1/g[z], {z, 0, 4}], z]] /. 
          Derivative[k_][g][0] :> g[k] /. g[0] -> 1 )  Range[4]!]

or equivalently as

recip[n_Integer, 
    f_Symbol:f] := (Table[
      Plus @@ (Apply[ Multinomial[##]multiplicity[{##}]((-f[#] &) /@ w[##]) &,
               Partitions[m], {1}] /. w -> Times), {m, n}]) 
with
multiplicity[alist : {Integer__}] := 
  Apply[Multinomial, Length /@ Split[Sort[alist]]]; multiplicity[{}] := 1;


the first 5 lines are: (recip[5] = )
{   -f[1], 
   2*f[1]^2 - f[2], 
  -6*f[1]^3 + 6*f[1]*f[2] - f[3], 
  24*f[1]^4 - 36*f[1]^2*f[2] + 6*f[2]^2 + 8*f[1]*f[3] -f[4], 
-120*f[1]^5 + 240*f[1]^3*f[2] - 90*f[1]*f[2]^2 - 
       60*f[1]^2*f[3] + 20*f[2]*f[3] + 10*f[1]*f[4] - f[5]}

(abreviated: re = recip[12])

This transform is its own inverse : in the following I give

:: it = re /. f[k_] :> some_function_of_k , 
:: re /. f[k_] :> it[[k]]
::
:: the transformed function (replace f[k] by the function with argument k)
:: the reverse transformation back to the original function_of_k 
   (replace f[k] by the k'th part of list "it")

the transform of k! is {-1, 0 ... } :

it = re /. f[k_] :> k!
Expand[re /. f[k_] :> it[[k]]  ]

{-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} 
{1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600}


it = re /. f[k_] :> -k!
Expand[re /. f[k_] :> it[[k]]  ]

{1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600,
40874803200, 980995276800}
{-1, -2, -6, -24, -120, -720, -5040, -40320, -362880, -3628800, -39916800,
-479001600}

the transform of (1-k) is A000166 :
 Subfactorial or rencontres numbers,or derangements:permutations
 of n elements with no fixed points :1,0,1,2,9,44,265

it = re /. f[k_] :> 1 - k
Expand[re /. f[k_] :> it[[k]]  ]

{0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841}
{0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11}

this one is not familiar :

it = re /. f[k_] :> 1 + k
Expand[re /. f[k_] :> it[[k]]  ]

{-2, 5, -16, 65, -326, 1957, -13700, 109601, -986410, 9864101, -108505112, \
1302061345}
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

but this is:

it = re /. f[k_] :> BernoulliB[k]
Expand[re /. f[k_] :> it[[k]]  ]

{1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 
  1/12, 1/13}
{-(1/2), 1/6, 0, -(1/30), 0, 1/42, 0, -(1/30), 0, 5/66, 
  0, -(691/2730)}

and this one:

it = re /. f[k_] :> EulerE[k] 
Expand[re /. f[k_] :> it[[k]]  ]

{0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1}
{0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765}

these again are strange:

it = re /. f[k_] :> (-1)^k Fibonacci[k]
Expand[re /. f[k_] :> it[[k]]  ]

{1, 1, 2, 7, 25, 102, 531, 3141, 20218, 146215, 1174889, 10225678}
{-1, 1, -2, 3, -5, 8, -13, 21, -34, 55, -89, 144}

it = re /. f[k_] :> ( cat[k])
Expand[re /. f[k_] :> it[[k]]  ]

{-1, 0, 1, 2, -2, -28, -65, 338, 3262, 4352, -113082, -879140}
{1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012}


If it's not new, then please let me know.


Cheers.


Wouter.


Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be



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