How can I produce Lagrange inversion coefficients?

Question

I am new to Mathematica. I want to produce the Lagrange inversion coefficients by using Bell polynomials.

In the following image you can see expressions for the first seven coefficients $A_1$ to $A_7$ using Bell Polynomials. There is also a general expression in term of Bell polynomials $B_{n-1,\,k}$

I want to produce coefficients higher than $A_7$. I don't know how many times I should go further; it will depend on how quickly I get convergence to my result.

I believe it shouldn't be hard to produce coefficients higher than $A_7$, but I don't have enough knowledge of Mathematica yet to do myself.

Answer 1

You could use InverseSeries. Here is an example polynomial to invert:

poly = Series[Sum[Subscript[a,i]x^i,{i,0,10}],{x,0,10}];
poly //TeXForm

$a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5+a_6 x^6+a_7 x^7+a_8 x^8+a_9 x^9+a_{10} x^{10}+O\left(x^{11}\right)$

Using InverseSeries:

InverseSeries[poly][[3]] //Simplify //Column //TeXForm

$\begin{array}{l} \frac{1}{a_1} \\ -\frac{a_2}{a_1^3} \\ \frac{2 a_2^2-a_1 a_3}{a_1^5} \\ -\frac{5 a_2^3-5 a_1 a_3 a_2+a_1^2 a_4}{a_1^7} \\ \frac{14 a_2^4-21 a_1 a_3 a_2^2+6 a_1^2 a_4 a_2+3 a_1^2 a_3^2-a_1^3 a_5}{a_1^9} \\ \frac{-42 a_2^5+84 a_1 a_3 a_2^3-28 a_1^2 a_4 a_2^2+7 a_1^2 \left(a_1 a_5-4 a_3^2\right) a_2+a_1^3 \left(7 a_3 a_4-a_1 a_6\right)}{a_1^{11}} \\ \frac{132 a_2^6-330 a_1 a_3 a_2^4+120 a_1^2 a_4 a_2^3-36 a_1^2 \left(a_1 a_5-5 a_3^2\right) a_2^2+8 a_1^3 \left(a_1 a_6-9 a_3 a_4\right) a_2+a_1^3 \left(-12 a_3^3+8 a_1 a_5 a_3+a_1 \left(4 a_4^2-a_1 a_7\right)\right)}{a_1^{13}} \\ \frac{-429 a_2^7+1287 a_1 a_3 a_2^5-495 a_1^2 a_4 a_2^4+165 a_1^2 \left(a_1 a_5-6 a_3^2\right) a_2^3-45 a_1^3 \left(a_1 a_6-11 a_3 a_4\right) a_2^2+3 a_1^3 \left(55 a_3^3-30 a_1 a_5 a_3+3 a_1 \left(a_1 a_7-5 a_4^2\right)\right) a_2+a_1^4 \left(-45 a_4 a_3^2+9 a_1 a_6 a_3+a_1 \left(9 a_4 a_5-a_1 a_8\right)\right)}{a_1^{15}} \\ \frac{1430 a_2^8-5005 a_1 a_3 a_2^6+2002 a_1^2 a_4 a_2^5-715 a_1^2 \left(a_1 a_5-7 a_3^2\right) a_2^4+220 a_1^3 \left(a_1 a_6-13 a_3 a_4\right) a_2^3-55 a_1^3 \left(26 a_3^3-12 a_1 a_5 a_3+a_1 \left(a_1 a_7-6 a_4^2\right)\right) a_2^2+10 a_1^4 \left(66 a_4 a_3^2-11 a_1 a_6 a_3+a_1 \left(a_1 a_8-11 a_4 a_5\right)\right) a_2+a_1^4 \left(55 a_3^4-55 a_1 a_5 a_3^2+5 a_1 \left(2 a_1 a_7-11 a_4^2\right) a_3+a_1^2 \left(5 a_5^2+10 a_4 a_6-a_1 a_9\right)\right)}{a_1^{17}} \\ \frac{-4862 a_2^9+19448 a_1 a_3 a_2^7-8008 a_1^2 a_4 a_2^6+3003 a_1^2 \left(a_1 a_5-8 a_3^2\right) a_2^5-1001 a_1^3 \left(a_1 a_6-15 a_3 a_4\right) a_2^4+286 a_1^3 \left(35 a_3^3-14 a_1 a_5 a_3+a_1 \left(a_1 a_7-7 a_4^2\right)\right) a_2^3-66 a_1^4 \left(91 a_4 a_3^2-13 a_1 a_6 a_3+a_1 \left(a_1 a_8-13 a_4 a_5\right)\right) a_2^2+11 a_1^4 \left(-91 a_3^4+78 a_1 a_5 a_3^2+6 a_1 \left(13 a_4^2-2 a_1 a_7\right) a_3+a_1^2 \left(-6 a_5^2-12 a_4 a_6+a_1 a_9\right)\right) a_2+a_1^5 \left(286 a_4 a_3^3-66 a_1 a_6 a_3^2+11 a_1 \left(a_1 a_8-12 a_4 a_5\right) a_3+a_1 \left(-22 a_4^3+11 a_1 a_7 a_4+11 a_1 a_5 a_6-a_1^2 a_{10}\right)\right)}{a_1^{19}} \\ \end{array}$

Answer 2

It is unfortunate that the OP neglected to provide the source for his expressions. Nevertheless, as noted, one can use the (matrix) Bell polynomials to perform Lagrangian inversion.

As an example:

With[{n = 5},
     (x - a[0])/a[1] +
     Sum[(x - a[0])^m/(m! (m - 1)! a[1]^m)
         BellY[Table[{(m + k - 2)!, -(k - 1)! a[k]/a[1]}, {k, 2, m}]],
         {m, 2, n}] + O[x, a[0]]^(n + 1)]

which yields

   (x - a[0])/a[1] -
   (a[2] (x - a[0])^2)/a[1]^3 +
   ((2 a[2]^2 - a[1] a[3]) (x - a[0])^3)/a[1]^5 +
   ((-5 a[2]^3 + 5 a[1] a[2] a[3] - a[1]^2 a[4]) (x - a[0])^4)/a[1]^7 +
   ((14 a[2]^4 - 21 a[1] a[2]^2 a[3] + 3 a[1]^2 a[3]^2 +
    6 a[1]^2 a[2] a[4] - a[1]^3 a[5]) (x - a[0])^5)/a[1]^9 +
   O[x - a[0]]^6

One could compare this with the result of InverseSeries[a[0] + Sum[a[k] x^k, {k, n}] + O[x]^(n + 1)], but a slicker way to check is to use ComposeSeries[]:

With[{n = 5}, 
     ComposeSeries[%, a[0] + Sum[a[k] x^k, {k, n}] + O[x]^(n + 1)] // Simplify]
   x + O[x]^6

(See as well the discussion in Charalambides, and this related question.)