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SandBox[Polynomial Sequences: Matrix]
  
last edited 6 years ago by rrogers

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(1) -> --- Functions to facilitate Transforming Polynomial Generating Functions into 
---- Coefficient arrays.  With some examples
--- These are coefficient arrays (A) and can be rendered into a polynomial list
--- by A*[1,x,x^2,...]^T
--- EGF Validation can always done for f(x,t)
--- By doing a Taylor series expansion indexed by t
--- OGF/Power Series is similiar except the factorial n! in t^n/n! 
--- has to be applied to the polynomial.
--- Little error checking.  Most Functions expect UniPotent of Nilpotent
--- Arrays.
---
--- The base ideas come from generating functions like
--- f(t)*exp(x*g(t))
--- From Roman's "Umbral Calculus" and
--- The Matrices of Pascal and Other Greats
--- Lidia Aceto & Donato Trigiante
--- http://www.academia.edu/22095557/The_Matrices_of_Pascal_and_Other_Greats

--- A, somewhat tedious, introduction to the mathematics is at:
---https://www.dropbox.com/s/tb9j7z0nhrdq92k/H-generating-9.pdf?dl=0
--- https://www.dropbox.com/sh/i2f7lehirme848p/AAA3jUgIkLNshPK88HuOR4MJa?dl=0
--- H-generating-9.pdf
---
---
---)Clear all
--- Complex here to accomidate cos(),sin() sequences
FPFI ==> Fraction(Polynomial(Fraction(Complex(Integer))))
Type: Void
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---
---
--- Lower strict Jordan
--- When used on the left of variables x it divides by x
--- with 1/x =0
--- When used on the right of the coefficient matrix it shifts
--- the columns to the left by one.
--- Really a horizontal shift of the coefficient array and 
--- Should be applied A*SJ_lower*[x...]^T
---
SJ_lower(dim) ==
        J : Matrix(Polynomial(Fraction(Integer))) := new(dim,dim,0)
        for i in 1..(dim-1) repeat J(i+1,i):=1
        J
Type: Void
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---
---
--- Upper strict Jordan
--- When used on the left of variables x it multiplies by x
--- With last entry 0
--- When used on the right of coefficient matrix it shifts
--- the columns to the right by one.
---
SJ_upper(dim) ==
        J : Matrix(FPFI) := new(dim,dim,0)
        for i in 1..(dim-1) repeat J(i,i+1):=1
        J
Type: Void
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---
---
--- Matrix exponential limited for now
--- Returns exp(Mat)
---
Exp_Matrix(Mat) ==
        n:=nrows(Mat)
        Mat_ret :Matrix(FPFI) := Mat^0
        for i in 1..(n) repeat Mat_ret := Mat_ret +(Mat^i)/factorial(i)
        Mat_ret
Type: Void
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---
---
--- Id matrix
--- Exp_Matrix defined below
---
Id_M(dim) ==
  exp_tmp: Matrix(Integer) := new(dim,dim,0)
  exp :=Exp_Matrix(exp_tmp)
  exp
Type: Void
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---
---
--- H and Pascals matrix
---
Gen_H(dim) ==
  H : Matrix(FPFI) := new(dim,dim,0)
  for i in 1..(dim-1) repeat H(i+1,i):=i
  H
Type: Void
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---
---
---Generate Pascal Matrices with a scalar (x) coefficient
--- exp(x*H)
---
Gen_Pascal(x,dim) ==
        H : Matrix(FPFI):=x*Gen_H(dim)
        Exp_Matrix(H)
Type: Void
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---
---
---H Left handed Psuedo inverse H_PIl
--- H_PIl*H ~ I
--- inverts H except for the last row.
---
Gen_H_PIl(dim) == 
        H_PIl : Matrix(FPFI) := new(dim,dim,0)
        for i in 1..(dim-1) repeat H_PIl(i,i+1):=1/i
        H_PIl
Type: Void
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---
---
--- Calculate Roman's integral equation
--- (exp(x*H)-1)/H
--- Which must be done in proper order
--- Note that in order for the last row
--- to be correct the expansion must be done
--- to dim+1 and then is trimmed to dim.
---
Gen_RI(x,dim) ==
        RI := Gen_Pascal(x,dim+1)
        RI := RI-RI^0
        RI := Gen_H_PIl(dim+1)*RI
        RI := subMatrix(RI,1,dim,1,dim)
        RI
Type: Void
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---
---

--- Log of particular matrix
--- Requires A - I be nilpotent and square; 
--- i.e. A is Unipotent/Lower unittriangular
---
Ln_Uni_Matrix(A) ==
        n := nrows(A)
        m := ncols(A)
        if n=m then
                        A1 := A-A^0
                        A_ret : Matrix(FPFI) := new(n,n,0)
                        if (A1^n = A_ret) then
                                        for i in 1..n repeat A_ret:=A_ret+(-A1)^i/i
                                else
                                        output("Ln_nil_matrix entered with bad matrix")
                else
                        output("Ln_nil_matrix entered with non-square matrix")
                        output(A)
        -A_ret
Type: Void
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---
--- (A^B) A a matrix, B either integer or matrix
--- A must be Unipotent
---
Exp_Matrix_AB(A,B) ==
        Exp_Matrix(Ln_Uni_Matrix(A)*B)
Type: Void
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---
--- Find p in P^-1* A* P = J
--- J is really a shift
--- A must be Stricly Lower Triangular 
---
ST_J(A) ==
        n:=nrows(A)
        m:=ncols(A)
        P : Matrix(FPFI) := new(n,n,0)
        if (n=m) and (A^n = P) then
                        P(1,1):=1
                        for i in 1..(n-1) repeat P(1..n,i+1):= A*P(1..n,i)
                else
                        output("Input to ST_H not strictly triangular")
        P
Type: Void
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---
--- Same as ST_H except the equation is P^-1*A *P=H
---
ST_H(A) ==
        n:=nrows(A)
        m:=ncols(A)
        P : Matrix(     FPFI) := new(n,n,0)
        if (n=m) and (A^n = P) then
                        P(1,1):=1
                        for i in 1..(n-1) repeat P(1..n,i+1):= A*P(1..n,i)/(i)
                else
                        output("Input to ST_H not strictly triangular")
        P
Type: Void
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---
---
--- Start of Examples
---
dim:=5
\label{eq1}5(1)
Type: PositiveInteger?
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H:=Gen_H(dim)
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Compiling function Gen_H with type PositiveInteger -> Matrix(
      Fraction(Polynomial(Fraction(Complex(Integer)))))
\label{eq2}\left[ \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 & 0 \ 0 & 0 & 3 & 0 & 0 \ 0 & 0 & 0 & 4 & 0 (2)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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GP :=Gen_Pascal(1,dim)
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Compiling function Exp_Matrix with type Matrix(Fraction(Polynomial(
      Fraction(Complex(Integer))))) -> Matrix(Fraction(Polynomial(
      Fraction(Complex(Integer)))))
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Compiling function Gen_Pascal with type (PositiveInteger, 
      PositiveInteger) -> Matrix(Fraction(Polynomial(Fraction(Complex(
      Integer)))))
\label{eq3}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 1 & 1 & 0 & 0 & 0 \ 1 & 2 & 1 & 0 & 0 \ 1 & 3 & 3 & 1 & 0 \ 1 & 4 & 6 & 4 & 1 (3)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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Id := GP^0
\label{eq4}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 1 (4)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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--- Enough is enough in taylor stream
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)set streams calculate 4

t := series 't
\label{eq5}t(5)
Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)
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--- To change a coefficient array to the corresponding polynomial array
XC := matrix[[1],[x],[x^2],[x^3],[x^4]]
\label{eq6}\left[ \begin{array}{c} 1 \ x \ {{x}^{2}} \ {{x}^{3}} \ {{x}^{4}} (6)
Type: Matrix(Polynomial(Integer))
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---
---
--- Appell Sequences
--- f(t)*exp(x*t)
---
--- Euler Polynmials from WikiPedia
--- 2/(e^t +1) exp(x*t)
---
Euler_A:=2*((GP+1)^-1)
\label{eq7}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ -{\frac{1}{2}}& 1 & 0 & 0 & 0 \ 0 & - 1 & 1 & 0 & 0 \ {\frac{1}{4}}& 0 & -{\frac{3}{2}}& 1 & 0 \ 0 & 1 & 0 & - 2 & 1 (7)
Type: SquareMatrix?(5,Fraction(Integer))
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--- Test
Euler_A*XC
\label{eq8}\left[ \begin{array}{c} 1 \ {x -{\frac{1}{2}}} \ {{{x}^{2}}- x} \ {{{x}^{3}}-{{\frac{3}{2}}\ {{x}^{2}}}+{\frac{1}{4}}} \ {{{x}^{4}}-{2 \ {{x}^{3}}}+ x} (8)
Type: Matrix(Polynomial(Fraction(Integer)))
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---
--- Bernoulli Polynomials from WikiPedia
--- (t/(e^t -1))* exp(x*t)
--- We recognize the this is Roman's integral formula
--- and we have 0/0 for t/H=0 and requires special
--- handling as noted above.
---
Bern_Wiki_A := Gen_RI(1,dim)^-1
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Compiling function Gen_H_PIl with type PositiveInteger -> Matrix(
      Fraction(Polynomial(Fraction(Complex(Integer)))))
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Compiling function Gen_RI with type (PositiveInteger, 
      PositiveInteger) -> Matrix(Fraction(Polynomial(Fraction(Complex(
      Integer)))))
\label{eq9}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ -{\frac{1}{2}}& 1 & 0 & 0 & 0 \ {\frac{1}{6}}& - 1 & 1 & 0 & 0 \ 0 &{\frac{1}{2}}& -{\frac{3}{2}}& 1 & 0 \ -{\frac{1}{30}}& 0 & 1 & - 2 & 1 (9)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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---
--- Bernoulli of order a from Roman
------ (t/(e^t -1))^a * exp(x*t)
---- This is the same as above exponenitated
---
Bern_Roman_A := Exp_Matrix_AB(Bern_Wiki_A,a)
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Compiling function Ln_Uni_Matrix with type Matrix(Fraction(
      Polynomial(Fraction(Complex(Integer))))) -> Matrix(Fraction(
      Polynomial(Fraction(Complex(Integer)))))
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Compiling function Exp_Matrix_AB with type (Matrix(Fraction(
      Polynomial(Fraction(Complex(Integer))))), Variable(a)) -> Matrix(
      Fraction(Polynomial(Fraction(Complex(Integer)))))
\label{eq10}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ -{{\frac{1}{2}}\ a}& 1 & 0 & 0 & 0 \ {{{\frac{1}{4}}\ {{a}^{2}}}-{{\frac{1}{12}}\ a}}& - a & 1 & 0 & 0 \ {-{{\frac{1}{8}}\ {{a}^{3}}}+{{\frac{1}{8}}\ {{a}^{2}}}}&{{{\frac{3}{4}}\ {{a}^{2}}}-{{\frac{1}{4}}\ a}}& -{{\frac{3}{2}}\ a}& 1 & 0 \ {{{\frac{1}{16}}\ {{a}^{4}}}-{{\frac{1}{8}}\ {{a}^{3}}}+{{\frac{1}{4 8}}\ {{a}^{2}}}+{{\frac{1}{120}}\ a}}&{-{{\frac{1}{2}}\ {{a}^{3}}}+{{\frac{1}{2}}\ {{a}^{2}}}}&{{{\frac{3}{2}}\ {{a}^{2}}}-{{\frac{1}{2}}\ a}}& -{2 \ a}& 1 (10)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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---
----
---- From 
----Some Identities for Euler and Bernoulli Polynomials
----and Their Zeros ---equation 4.
--- by 
----Taekyun Kim and Cheon Seoung Ryoo 
----Frobenius–Euler polynomials of order r
---------- (1-u)/(e^t -u))^r * exp(x*t)
----
Frob_Euler_term:=((1-u)*Id_M(dim))*((Gen_Pascal(1,dim)-u*Id_M(dim))^-1);
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Compiling function Exp_Matrix with type Matrix(Integer) -> Matrix(
      Fraction(Polynomial(Fraction(Complex(Integer)))))
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Compiling function Id_M with type PositiveInteger -> Matrix(Fraction
      (Polynomial(Fraction(Complex(Integer)))))
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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Frob_Euler:=Exp_Matrix_AB(Frob_Euler_term,r)
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Compiling function Exp_Matrix_AB with type (Matrix(Fraction(
      Polynomial(Fraction(Complex(Integer))))), Variable(r)) -> Matrix(
      Fraction(Polynomial(Fraction(Complex(Integer)))))
\label{eq11}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ {\frac{r}{u - 1}}& 1 & 0 & 0 & 0 \ {\frac{{r \ u}+{{r}^{2}}}{{{u}^{2}}-{2 \ u}+ 1}}&{\frac{2 \ r}{u - 1}}& 1 & 0 & 0 \ {\frac{{r \ {{u}^{2}}}+{{\left({3 \ {{r}^{2}}}+ r \right)}\ u}+{{r}^{3}}}{{{u}^{3}}-{3 \ {{u}^{2}}}+{3 \ u}- 1}}&{\frac{{3 \ r \ u}+{3 \ {{r}^{2}}}}{{{u}^{2}}-{2 \ u}+ 1}}&{\frac{3 \ r}{u - 1}}& 1 & 0 \ {\frac{{r \ {{u}^{3}}}+{{\left({7 \ {{r}^{2}}}+{4 \ r}\right)}\ {{u}^{2}}}+{{\left({6 \ {{r}^{3}}}+{4 \ {{r}^{2}}}+ r \right)}\ u}+{{r}^{4}}}{{{u}^{4}}-{4 \ {{u}^{3}}}+{6 \ {{u}^{2}}}-{4 \ u}+ 1}}&{\frac{{4 \ r \ {{u}^{2}}}+{{\left({{12}\ {{r}^{2}}}+{4 \ r}\right)}\ u}+{4 \ {{r}^{3}}}}{{{u}^{3}}-{3 \ {{u}^{2}}}+{3 \ u}- 1}}&{\frac{{6 \ r \ u}+{6 \ {{r}^{2}}}}{{{u}^{2}}-{2 \ u}+ 1}}&{\frac{4 \ r}{u - 1}}& 1 (11)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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---
---
--- Hermite Polynomials wiki version
--- Probabilistic version
---exp(-(t^2)/2) *exp(x*t)
---
Herm_Wiki_A := Exp_Matrix(-H^2/2)
\label{eq12}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ - 1 & 0 & 1 & 0 & 0 \ 0 & - 3 & 0 & 1 & 0 \ 3 & 0 & - 6 & 0 & 1 (12)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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--- Roman's version
Herm_Roman_A := Exp_Matrix(-v*H^2/2)
\label{eq13}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ - v & 0 & 1 & 0 & 0 \ 0 & -{3 \ v}& 0 & 1 & 0 \ {3 \ {{v}^{2}}}& 0 & -{6 \ v}& 0 & 1 (13)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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--- Hermite/Gould Hopper of order 3
--- exp(y*t^3)*exp(x*t)
HGH_A := Exp_Matrix(y*H^3)
\label{eq14}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 \ {6 \ y}& 0 & 0 & 1 & 0 \ 0 &{{24}\ y}& 0 & 0 & 1 (14)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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---
---
--- Associated Sequences requires Jordan transform
--- exp(x*f(t)) -> SJ_H(f(H))
---
--- Lower Factorial Exponential
--- Roman PP 57
--- exp(x*Ln(t+1))
---
Lower_Fact_Exp_A := ST_H(Ln_Uni_Matrix(H+1))
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Compiling function Ln_Uni_Matrix with type SquareMatrix(5,Integer)
       -> Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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Compiling function ST_H with type Matrix(Fraction(Polynomial(
      Fraction(Complex(Integer))))) -> Matrix(Fraction(Polynomial(
      Fraction(Complex(Integer)))))
\label{eq15}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & - 1 & 1 & 0 & 0 \ 0 & 2 & - 3 & 1 & 0 \ 0 & - 6 &{11}& - 6 & 1 (15)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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--- Just to verify
x:='x;
Type: Variable(x)
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1
\label{eq16}1(16)
Type: PositiveInteger?
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x
\label{eq17}x(17)
Type: Variable(x)
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x*(x-1)
\label{eq18}{{x}^{2}}- x(18)
Type: Polynomial(Integer)
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x*(x-1)*(x-2)
\label{eq19}{{x}^{3}}-{3 \ {{x}^{2}}}+{2 \ x}(19)
Type: Polynomial(Integer)
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x*(x-1)*(x-2)*(x-3)
\label{eq20}{{x}^{4}}-{6 \ {{x}^{3}}}+{{11}\ {{x}^{2}}}-{6 \ x}(20)
Type: Polynomial(Integer)
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---
---  Exponential Polynomials 
--- Roman PP 64
--- Also http://mathworld.wolfram.com/BellPolynomial.html
--- exp(x*(exp(t)-1))
---
Exp_A := ST_H(GP-1)
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Compiling function ST_H with type SquareMatrix(5,Integer) -> Matrix(
      Fraction(Polynomial(Fraction(Complex(Integer)))))
\label{eq21}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & 1 & 1 & 0 & 0 \ 0 & 1 & 3 & 1 & 0 \ 0 & 1 & 7 & 6 & 1 (21)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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---
--- 
-
\label{eq22}-(22)
Type: Variable(-)
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--- Scheffer Sequences require Jordan form
---
--- g(t)*exp(x*f(t)) ->g(H)*ST_H(f(H))
--- 
--- Laguerre Polynomials Roman PP 110
--- (1-t)^(-a-1)*exp(x*(t/(t-1)))
Lag_A  := Exp_Matrix_AB(1-H,(-a-1))*ST_H(H*(H-1)^-1)
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Compiling function Exp_Matrix_AB with type (SquareMatrix(5,Integer)
      , Polynomial(Integer)) -> Matrix(Fraction(Polynomial(Fraction(
      Complex(Integer)))))
\label{eq23}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ {a + 1}& - 1 & 0 & 0 & 0 \ {{{a}^{2}}+{3 \ a}+ 2}&{-{2 \ a}- 4}& 1 & 0 & 0 \ {{{a}^{3}}+{6 \ {{a}^{2}}}+{{11}\ a}+ 6}&{-{3 \ {{a}^{2}}}-{{1 5}\ a}-{18}}&{{3 \ a}+ 9}& - 1 & 0 \ {{{a}^{4}}+{{10}\ {{a}^{3}}}+{{35}\ {{a}^{2}}}+{{50}\ a}+{24}}&{-{4 \ {{a}^{3}}}-{{36}\ {{a}^{2}}}-{{104}\ a}-{96}}&{{6 \ {{a}^{2}}}+{{42}\ a}+{72}}&{-{4 \ a}-{16}}& 1 (23)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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--- Checking
Lag_S:=(1-t)^(-a-1)*exp(x*(t/(t-1)));
Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)
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Lag_S_C3:=6*coefficient(Lag_S,3)
\label{eq24}\begin{array}{@{}l} \displaystyle -{{x}^{3}}+{{\left({3 \ a}+ 9 \right)}\ {{x}^{2}}}+{{\left(-{3 \ {{a}^{2}}}-{{15}\ a}-{18}\right)}\ x}+{{a}^{3}}+{6 \ {{a}^{2}}}+ \ \ \displaystyle {{11}\ a}+ 6 (24)
Type: Expression(Integer)
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Lag_P:=Lag_A*XC
\label{eq25}\left[ \begin{array}{c} 1 \ {- x + a + 1} \ {{{x}^{2}}+{{\left(-{2 \ a}- 4 \right)}\ x}+{{a}^{2}}+{3 \ a}+ 2} \ {-{{x}^{3}}+{{\left({3 \ a}+ 9 \right)}\ {{x}^{2}}}+{{\left(-{3 \ {{a}^{2}}}-{{15}\ a}-{18}\right)}\ x}+{{a}^{3}}+{6 \ {{a}^{2}}}+{{11}\ a}+ 6} \ {{{x}^{4}}+{{\left(-{4 \ a}-{16}\right)}\ {{x}^{3}}}+{{\left({6 \ {{a}^{2}}}+{{42}\ a}+{72}\right)}\ {{x}^{2}}}+{{\left(-{4 \ {{a}^{3}}}-{{36}\ {{a}^{2}}}-{{104}\ a}-{96}\right)}\ x}+{{a}^{4}}+{{10}\ {{a}^{3}}}+{{35}\ {{a}^{2}}}+{{50}\ a}+{24}} (25)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
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Lag_P(4,1)-Lag_S_C3
\label{eq26}0(26)
Type: Expression(Complex(Integer))
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--- Note the "series" give "Power Series" to get the
--- Taylor series one has to multiply by factorial(n)
---

--- You might want to comment this out; it has a lot of terms  at the end.
--- Meixner Polynomials
--- Acuctually this is interesting since it is
--- Listed as Scheffer (which is Exponential)
--- but I think Ordinary OGF is a better choice
--- Anyhow here is a slight reformat from Roman
t := series 't
\label{eq27}t(27)
Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)
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Meix_Series_R :=((1-t*d)*((1-t)^-1))^x*(1-t)^(-b)
\label{eq28}\begin{array}{@{}l} \displaystyle 1 +{{\left({{\left(- d + 1 \right)}\ x}+ b \right)}\ t}+ \ \ \displaystyle {{\frac{{{\left({{d}^{2}}-{2 \ d}+ 1 \right)}\ {{x}^{2}}}+{{\left(-{{d}^{2}}-{2 \ b \ d}+{2 \ b}+ 1 \right)}\ x}+{{b}^{2}}+ b}{2}}\ {{t}^{2}}}+ \ \ \displaystyle {{\frac{{{\left(-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1 \right)}\ {{x}^{3}}}+{{\left({3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3 \right)}\ {{x}^{2}}}+{{\left(-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2 \right)}\ x}+{{b}^{3}}+{3 \ {{b}^{2}}}+{2 \ b}}{6}}\ {{t}^{3}}}+ \ \ \displaystyle {{\frac{{{\left({{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1 \right)}\ {{x}^{4}}}+{{\left(-{6 \ {{d}^{4}}}+{{\left(-{4 \ b}+{12}\right)}\ {{d}^{3}}}+{{12}\ b \ {{d}^{2}}}+{{\left(-{{12}\ b}-{12}\right)}\ d}+{4 \ b}+ 6 \right)}\ {{x}^{3}}}+{{\left({{11}\ {{d}^{4}}}+{{\left({{12}\ b}- 8 \right)}\ {{d}^{3}}}+{{\left({6 \ {{b}^{2}}}-{6 \ b}- 6 \right)}\ {{d}^{2}}}+{{\left(-{{12}\ {{b}^{2}}}-{{24}\ b}- 8 \right)}\ d}+{6 \ {{b}^{2}}}+{{18}\ b}+{11}\right)}\ {{x}^{2}}}+{{\left(-{6 \ {{d}^{4}}}-{8 \ b \ {{d}^{3}}}+{{\left(-{6 \ {{b}^{2}}}-{6 \ b}\right)}\ {{d}^{2}}}+{{\left(-{4 \ {{b}^{3}}}-{{12}\ {{b}^{2}}}-{8 \ b}\right)}\ d}+{4 \ {{b}^{3}}}+{{18}\ {{b}^{2}}}+{{22}\ b}+ 6 \right)}\ x}+{{b}^{4}}+{6 \ {{b}^{3}}}+{{11}\ {{b}^{2}}}+{6 \ b}}{24}}\ {{t}^{4}}}+ \ \ \displaystyle {O \left({{t}^{5}}\right)} (28)
Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)
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--- and the Sheffer form
--- We need the Jordan form of Ln_Uni_Matrix((1-H*d)*(1-H)^-1)
Meix_A_fJ := ST_H(Ln_Uni_Matrix((1-H*d)*(1-H)^-1))
fricas
Compiling function Ln_Uni_Matrix with type SquareMatrix(5,Polynomial
      (Fraction(Complex(Integer)))) -> Matrix(Fraction(Polynomial(
      Fraction(Complex(Integer)))))
\label{eq29}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 &{- d + 1}& 0 & 0 & 0 \ 0 &{-{{d}^{2}}+ 1}&{{{d}^{2}}-{2 \ d}+ 1}& 0 & 0 \ 0 &{-{2 \ {{d}^{3}}}+ 2}&{{3 \ {{d}^{3}}}-{3 \ {{d}^{2}}}-{3 \ d}+ 3}&{-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1}& 0 \ 0 &{-{6 \ {{d}^{4}}}+ 6}&{{{11}\ {{d}^{4}}}-{8 \ {{d}^{3}}}-{6 \ {{d}^{2}}}-{8 \ d}+{11}}&{-{6 \ {{d}^{4}}}+{{12}\ {{d}^{3}}}-{{12}\ d}+ 6}&{{{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1} (29)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
fricas
Meix_A := Exp_Matrix_AB(1-H,(-b))*Meix_A_fJ
\label{eq30}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ b &{- d + 1}& 0 & 0 & 0 \ {{{b}^{2}}+ b}&{-{{d}^{2}}-{2 \ b \ d}+{2 \ b}+ 1}&{{{d}^{2}}-{2 \ d}+ 1}& 0 & 0 \ {{{b}^{3}}+{3 \ {{b}^{2}}}+{2 \ b}}&{-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2}&{{3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3}&{-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1}& 0 \ {{{b}^{4}}+{6 \ {{b}^{3}}}+{{11}\ {{b}^{2}}}+{6 \ b}}&{-{6 \ {{d}^{4}}}-{8 \ b \ {{d}^{3}}}+{{\left(-{6 \ {{b}^{2}}}-{6 \ b}\right)}\ {{d}^{2}}}+{{\left(-{4 \ {{b}^{3}}}-{{12}\ {{b}^{2}}}-{8 \ b}\right)}\ d}+{4 \ {{b}^{3}}}+{{18}\ {{b}^{2}}}+{{22}\ b}+ 6}&{{{11}\ {{d}^{4}}}+{{\left({{12}\ b}- 8 \right)}\ {{d}^{3}}}+{{\left({6 \ {{b}^{2}}}-{6 \ b}- 6 \right)}\ {{d}^{2}}}+{{\left(-{{12}\ {{b}^{2}}}-{{24}\ b}- 8 \right)}\ d}+{6 \ {{b}^{2}}}+{{18}\ b}+{11}}&{-{6 \ {{d}^{4}}}+{{\left(-{4 \ b}+{12}\right)}\ {{d}^{3}}}+{{1 2}\ b \ {{d}^{2}}}+{{\left(-{{12}\ b}-{12}\right)}\ d}+{4 \ b}+ 6}&{{{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1} (30)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
fricas
--- 
--- Let's check
Meix_C3:=6*coefficient(Meix_Series_R,3)
\label{eq31}\begin{array}{@{}l} \displaystyle {{\left(-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1 \right)}\ {{x}^{3}}}+ \ \ \displaystyle {{\left({3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3 \right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left(-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2 \right)}\ x}+{{b}^{3}}+ \ \ \displaystyle {3 \ {{b}^{2}}}+{2 \ b} (31)
Type: Expression(Integer)
fricas
Meix_P:=Meix_A*XC
\label{eq32}\left[ \begin{array}{c} 1 \ {{{\left(- d + 1 \right)}\ x}+ b} \ {{{\left({{d}^{2}}-{2 \ d}+ 1 \right)}\ {{x}^{2}}}+{{\left(-{{d}^{2}}-{2 \ b \ d}+{2 \ b}+ 1 \right)}\ x}+{{b}^{2}}+ b} \ {{{\left(-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1 \right)}\ {{x}^{3}}}+{{\left({3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3 \right)}\ {{x}^{2}}}+{{\left(-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2 \right)}\ x}+{{b}^{3}}+{3 \ {{b}^{2}}}+{2 \ b}} \ {{{\left({{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1 \right)}\ {{x}^{4}}}+{{\left(-{6 \ {{d}^{4}}}+{{\left(-{4 \ b}+{12}\right)}\ {{d}^{3}}}+{{12}\ b \ {{d}^{2}}}+{{\left(-{{1 2}\ b}-{12}\right)}\ d}+{4 \ b}+ 6 \right)}\ {{x}^{3}}}+{{\left({{1 1}\ {{d}^{4}}}+{{\left({{12}\ b}- 8 \right)}\ {{d}^{3}}}+{{\left({6 \ {{b}^{2}}}-{6 \ b}- 6 \right)}\ {{d}^{2}}}+{{\left(-{{12}\ {{b}^{2}}}-{{24}\ b}- 8 \right)}\ d}+{6 \ {{b}^{2}}}+{{18}\ b}+{11}\right)}\ {{x}^{2}}}+{{\left(-{6 \ {{d}^{4}}}-{8 \ b \ {{d}^{3}}}+{{\left(-{6 \ {{b}^{2}}}-{6 \ b}\right)}\ {{d}^{2}}}+{{\left(-{4 \ {{b}^{3}}}-{{12}\ {{b}^{2}}}-{8 \ b}\right)}\ d}+{4 \ {{b}^{3}}}+{{18}\ {{b}^{2}}}+{{22}\ b}+ 6 \right)}\ x}+{{b}^{4}}+{6 \ {{b}^{3}}}+{{11}\ {{b}^{2}}}+{6 \ b}} (32)
Type: Matrix(Fraction(Polynomial(Fraction(Complex(Integer)))))
fricas
Meix_P(4,1)-Meix_C3
\label{eq33}0(33)
Type: Expression(Complex(Integer))



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