Inverse of an Infinite Matrix (with factorials)

Question

How to calculate this monstrous expression? $$ \begin{pmatrix} \frac{1}{1!} & \frac{1}{2!} & \frac{1}{3!} & \frac{1}{4!} & \frac{1}{5!}& \cdots\\ 0 & \frac{1}{1!} & \frac{1}{2!} & \frac{1}{3!} & \frac{1}{4!}& \cdots \\ -2 & 0 & \frac{1}{1!} & \frac{1}{2!} & \frac{1}{3!} & \cdots \\ 0 & -3 & 0 & \frac{1}{1!} & \frac{1}{2!} & \cdots \\ 0 & 0 & -4 & 0 & \frac{1}{1!} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}^{-1} \begin{pmatrix}0\\1\\0\\0\\0\\0\\\vdots\end{pmatrix} $$ I don't think trying to find the inverse of this huge matrix (which I am not able to) will be helpful, as we only need the $2^{nd}$ column of the inverse matrix. Any help is appreciated! Thank You!

Answer 1

Using inversion by LDU-decomposition and including Euler-summation for the occuring divergent dot-products I get for the first couple of entries derivatives of the gamma-function $\Gamma(x)$ at argument $1$:

$$ \begin{array}{r|rl} i & \text{num value} & \text{interpretation} \\ \hline 1 & -0.577215664902 & = \Gamma^{(1)} (1) \\ 2 & 1.97809833665 & = \Gamma^{(2)} (1) \\ 3 & -5.44487445649 & = \Gamma^{(3)} (1) \\ 4 & 23.5614740841 & = \Gamma^{(4)} (1) \end{array}$$

So I think, this continues for the other entries of the result-vector and the interpretation, suggested by the approximations, hold in general.

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Using Pari/GP we get the found values in the exponential generating-function for the $\Gamma(1+x)$:

serlaplace(gamma(1+x)-1)
 %168 = -0.577215664902*x + 1.97811199066*x^2
 - 5.44487445649*x^3 + 23.5614740840*x^4
 - 117.839408268*x^5 + 715.067362527*x^6
 - 5019.84887263*x^7 + 40243.6215733*x^8
 - 362526.289115*x^9 + 3627042.41276*x^10
 - 39907084.1514*x^11 + 478943291.765*x^12
 - 6226641351.55*x^13 + 87175633810.7*x^14
 - 1.30765442950 E12*x^15 + O(x^16)

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Appendix Tables

Here are the top-left segments of the LDU-components such that $M=L \cdot D \cdot U$:

    1    .      .      .       .        .  |
    .    1      .      .       .        .  |
   -2    1      1      .       .        .  |
    .   -3    9/5      1       .        .  |      L   
    .    .  -24/5    8/3       1        .  |
    .    .      .  -20/3  185/52        1  |
    -    -      -      -       -        -  +
    1    .      .      .       .        .  |
    .    1      .      .       .        .  |
    .    .    5/6      .       .        .  |      D
    .    .      .    3/4       .        .  |
    .    .      .      .   52/75        .  |
    .    .      .      .       .  203/312  |
    -    -      -      -       -        -  +
    1  1/2    1/6   1/24   1/120    1/720  |
    .    1    1/2    1/6    1/24    1/120  |
    .    .      1    1/2  17/100   13/300  |       U
    .    .      .      1   37/75   38/225  |
    .    .      .      .       1  151/312  |
    .    .      .      .       .        1  |
    -    -      -      -       -        -  +

Here are their inverses, such that $$ M^{-1} = \lim_{dim \to \infty} U^{-1} \underset {\mathfrak E} * ( D^{-1} \cdot L^{-1})$$ where $\underset {\mathfrak E} * $ means doing the divergent dotproducts using Euler-summation

         1     -1/2     1/12       .   -1/600  -1/37440  |
         .        1     -1/2    1/12    1/450  -17/9360  |
         .        .        1    -1/2   23/300    5/1248  |
         .        .        .       1   -37/75  109/1560  |    U^-1
         .        .        .       .        1  -151/312  |
         .        .        .       .        .         1  |
         -        -        -       -        -         -  +
         1        .        .       .        .         .  |
         .        1        .       .        .         .  |
         .        .      6/5       .        .         .  |
         .        .        .     4/3        .         .  |    D^-1
         .        .        .       .    75/52         .  |
         .        .        .       .        .   312/203  |
         -        -        -       -        -         -  +
         1        .        .       .        .         .  |
         .        1        .       .        .         .  |
         2       -1        1       .        .         .  |    L^-1
     -18/5     24/5     -9/5       1        .         .  |
      96/5    -88/5     48/5    -8/3        1         .  |
  -1200/13  1230/13  -600/13  210/13  -185/52         1  |
         -        -        -       -        -         -  +

Because we have the convergent dotproduct $L^{-1} \cdot I_1$ where $I_1 =[0,1,0,0,...]$ we need only the second column of $L^{-1}$ and the part $ \text{rhs}=D^{-1} \cdot L^{-1} \cdot I_1 $ gives, in decimal notation

              .  |
  1.00000000000  |
 -1.20000000000  |
  6.40000000000  |
 -25.3846153846  |
  145.418719212  |
 -930.992018244  |   rhs=D^-1 * L^-1 * I[,1]
  6963.47826087  |  (dotproducts of finite lengthes are exact)
 -58772.5918570  |
  554512.555292  |
 -5779721.10527  |
  65970421.8290  |
  ....           |  

The dot-products of left-multiplication with $U^{-1}$ include Eulersummation to assign the divergent sums of alternating series finite values. We get the following approximations:

-0.577218921840  |
  1.97813788259  |
 -5.44460173003  | fairly good aproximations to coefficients of the
  23.5552716067  | laplace-transformation of the Gamma(1+x)-power series
 -117.723592387  | see above
  712.587241686  |
(-4968.35894817) |  approximation at higher indexes worsen because
( ...   )        |  of finite size of matrices