Eigensequences

[Gottfried Helms]

In  a recent treatise of the pascal matrix and its
connections to the bernoulli-numbers I restated
their connection as an eigenvector/eigensequence-
relation.

Say P is the lower triangular matrix of binomial-
coefficients, Pj the matrix, where the column-signs
alternate and B the vector of bernoulli-numbers
(b_1 = +1/2)
then
   Pj * B = B

Now in connection with the subject of divergent
summmation I investigated powerseries, say
  V(x) = [1,x,x^2,x^3,... ]
in connection with Pj, like

  ??? = Pj * V(x)      // binomial-transform
  ??? = V(x)~ * Pj

(using ~ for the transpose)

Now since

   Pj * B = B

also

   V(x) * Pj * B = V(x) * B

and the effect of the coefficients of Pj
disappear, if I consider powerseries with
the bernoulli-numbers as coefficients:

 f(x) = b0 + b1*x + b2*x^2 + b3*x^3 + ...
      =    b0
         + (1*b0 - 1*b1)*x
         + (1*b0 - 2*b1 + 1*b2)*x^2
         + ...

---------
Now incidently I also found, that, using J as a diagonal-
unit-matrix with units with alternating signs
 J = diag(1,-1,1,-1,...)

such that also

 Pj  = P * J

then a matrix G, containing the bernoulli-numbers
in the first column (and shifted and scaled
in a certain manner in the follwing columns)

 Pj = G * J * G^-1

and thus, for instance, the above eigensequence-
property of the bernoulli-numbers can be extended
to an infinite set of coefficients arising from

   V(x) * Pj * G = V(x) * G * J

in the infinite number of columns of the result.

The second column, for instance adds to the above
f(x) the function g(x)

 g(x) =    1*b0*x + 2*b1*x^2 + 3*b2*x^3 + 4*b3*x^4 + ...
      = -  (    1* 0
             + (1* 0 - 1*1*b0)*x
             + (1* 0 - 2*1*b0 + 1*2*b1)*x^2
             + (1* 0 - 3*1*b0 + 3*2*b1 -  1*3*b2)*x^3
             + ...
      = -  (
               ( - 1*b0)*x
             + ( - 2*b0 + 2*b1)*x^2
             + ( - 3*b0 + 6*b1 -  3*b2)*x^3
             + ...

or, using x=1, the said eigensequence-operation, here
with the associated eigenvalue -1.

-------------------------------------
Even more interesting seems to me, that the said matrix
G can be decomposed into an eigensystem, where

  G = St2 * Z(1) * St2^-1

with St2 = matrix of stirling-numbers of 2'nd kind
     Z(s)= diag(1,  1/2^s,  1/3^s, 1/4^s,... )
the diagonal-matrix of the zeta-summands of power s.

That means, that even the bernoulli-numbers have
eigensequences related to them (even infinite many),
namely the columns of St2 , associated with the
eigenvalues (1,1/2,1/3,...) such that

  V(x) * G * St2*Z(1) = V(x)*St2

and for instance,

  f(x) = 1 + 1*x + 1*x^2 + 1*x^3 + ....  // taken from V(x)*St2

       =       1*b0                     // taken from V(x) * G * St2 * Z(1)
          + (1/2*b0 + b1)*m
          + (1/3*b0 + b1 +     b2 )*m^2
          + (1/4*b0 + b1 + 3/2*b2 +      b3 )*m^3
          + (1/5*b0 + b1 +   2*b2 +    2*b3 +     b4)*m^4
          + (1/6*b0 + b1 + 5/2*b2 + 10/3*b3 + 5/2*b4 +      b5)*m^5
           ...

which may look kind trivial, since the involved Stirling-
coefficients are all 1
Other coefficients occur in the next columns 1,2,3,..., for instance
in column 1:

   g(x)  =.  x + S2_1*x^2 + S3_1*x^3 + S4_1*x^4 + S5_1*x^5 + ....
         =              b0 * x
             ( 2/3*S2_1*b0 +      2*b1 ) * x^2
             ( 1/2*S3_1*b0 + 2*S2_1*b1 + 3     *b2 )*m^3
             ( 2/5*S4_1*b0 + 2*S3_1*b1 + 4*S2_1*b2 +    4     *b3 ) *m^4
             ( 1/3*S5_1*b0 + 2*S4_1*b1 + 5*S3_1*b2 + 20/3*S2_1*b3  +  5*b4) *m^5

and so on for the next columns.
With the involved Stirling-numbers evaluated we have then for this 2'nd
column:

   g(x)  =   .    2*x + 6*x^2 + 14*x^3 + 30*x^4  +  62*x^5
         =          2*b0 * x
            +   (   4*b0 +  4*b1)*x^2
            +   (   7*b0 + 12*b1 +  6*b2) *x^3
            +   (  12*b0 + 28*b1 + 24*b2 +  8*b3) *m^4
            +   (62/3*b0 + 60*b1 + 70*b2 + 40*b3 + 10*b4))))*m^5
              ...
which, if the bernoulli-numbers are inserted is obvious.
Using x=1 and written as sequences, we have another
eigensequence-expression and implicitely infinitely many.

This description was animated by Neil's article in the
integer-sequences-journal, which I came across recently -
though I do not know, if the above is actually of interest
related to the article (my current context is originally
the study of divergent summation, where these identities
popped up)

Regards-

Gottfried Helms