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SandBox[Polynomial Sequences: Matrix]

Edit detail for SandBox[Polynomial Sequences: Matrix] revision 8 of 11

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Editor: rrogers Time: 2018/10/04 15:48:46 GMT+0
Note: Typo-- added Id matrix Id_M(dim) and Frobenius–Euler polynomials of order r - and some rearrangement

changed:
-
--- 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/sh/i2f7lehirme848p/AAA3jUgIkLNshPK88HuOR4MJa?dl=0
--- H-generating-9.pdf
---
---
---)Clear all
--- 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

changed:
---
---
--- 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
---
---
---
--- 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
---

changed:
---
---
--- Upper strict Jordan
--- When used on the left of variables x it multiplies by x
--- With last entry 0
--- Whenc used on the right of coefficient matrix it shifts
--- the columns to the right by one.
---
---
---
--- 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.
---

changed:
---
---
--- H and Pascals matrix
---
---
---
--- 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
---
---
--- Id matrix
--- Exp_Matrix defined below
---
Id_M(dim) ==
  exp_tmp: Matrix(Integer) := new(dim,dim,0)
  exp :=Exp_Matrix(exp_tmp)
  exp
---
---
--- H and Pascals matrix
---

changed:
---
---
---Generate Pascal Matrices with a scalar (x) coefficient
--- exp(x*H)
---
---
---
---Generate Pascal Matrices with a scalar (x) coefficient
--- exp(x*H)
---

changed:
---
---
---H Left handed Psuedo inverse H_PIl
--- H_PIl*H ~ I
--- inverts H except for the last row.
---
---
---
---H Left handed Psuedo inverse H_PIl
--- H_PIl*H ~ I
--- inverts H except for the last row.
---

changed:
---
---
--- 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.
---
---
---
--- 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.
---

changed:
---
---
--- 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
-
--- Log of particular matrix
--- Requires A - I be nilpotent and square; 
--- i.e. A is Unipotent/Lower unittriangular
---
---
---

--- Log of particular matrix
--- Requires A - I be nilpotent and square; 
--- i.e. A is Unipotent/Lower unittriangular
---

changed:
---
--- (A^B) A a matrix, B either integer or matrix
--- A must be Unipotent
---
---
--- (A^B) A a matrix, B either integer or matrix
--- A must be Unipotent
---

changed:
---
--- Find p in P^-1* A* P = J
--- J is really a shift
--- A must be Stricly Lower Triangular 
---
---
--- Find p in P^-1* A* P = J
--- J is really a shift
--- A must be Stricly Lower Triangular 
---

changed:
---
--- Same as ST_H except the equation is P^-1*A *P=H
---
---
--- Same as ST_H except the equation is P^-1*A *P=H
---

changed:
---
---
--- Start of Examples
---
---
---
--- Start of Examples
---

changed:
--- Enough is enough in taylor stream
--- Enough is enough in taylor stream

changed:
--- To change a coefficient array to the corresponding polynomial array
--- To change a coefficient array to the corresponding polynomial array

changed:
---
---
--- Appell Sequences
--- f(t)*exp(x*t)
---
--- Euler Polynmials from WikiPedia
--- 2/(e^t +1) exp(x*t)
---
---
---
--- Appell Sequences
--- f(t)*exp(x*t)
---
--- Euler Polynmials from WikiPedia
--- 2/(e^t +1) exp(x*t)
---

changed:
--- Test
--- Test

changed:
---
--- Bernoulli Polynomials from WikiPedia
--- (t/(e^t -1))* exp(x*t)
---
---
--- 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.
---

changed:
---
--- Bernoulli of order a from Roman
----- (t/(e^t -1))^a * exp(x*t)
---
---
--- Bernoulli of order a from Roman
------ (t/(e^t -1))^a * exp(x*t)
---- This is the same as above exponenitated
---

changed:
---
--- Hermite Polynomials wiki version
---exp(x*t-(t^2)/2) *exp(x*t)
---
---
----
---- 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)
Frob_Euler:=Exp_Matrix_AB(Frob_Euler_term,r)
---
---
--- Hermite Polynomials wiki version
---exp(x*t-(t^2)/2) *exp(x*t)
---

changed:
---
---

changed:
--- Hermite/Gould Hopper of order 3
--- exp(x*t+y*t^3)
--- Hermite/Gould Hopper of order 3
--- exp(x*t+y*t^3)

changed:
---
---
--- Associated Sequences requires Jordan transform
--- exp(x*f(t)) -> SJ_H(f(H))
---
--- Lower Factorial Exponential
--- Roman PP 57
--- exp(x*Ln(t+1)
---
---
---
--- Associated Sequences requires Jordan transform
--- exp(x*f(t)) -> SJ_H(f(H))
---
--- Lower Factorial Exponential
--- Roman PP 57
--- exp(x*Ln(t+1)
---

changed:
--- Just to verify
--- Just to verify

changed:
---
---  Exponential Polynomials 
--- Roman PP 64
--- Also http://mathworld.wolfram.com/BellPolynomial.html
--- exp(x*(exp(t)-1))
---
---
---  Exponential Polynomials 
--- Roman PP 64
--- Also http://mathworld.wolfram.com/BellPolynomial.html
--- exp(x*(exp(t)-1))
---

changed:
---
--- 
---
--- 

changed:
--- 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)))
--- 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)))

changed:
--- Checking
--- Checking

changed:
--- 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
--- 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

changed:
--- and the Sheffer form
--- We need the Jordan form of Ln_Uni_Matrix((1-H*d)*(1-H)^-1)
--- and the Sheffer form
--- We need the Jordan form of Ln_Uni_Matrix((1-H*d)*(1-H)^-1)

changed:
--- 
--- Let's check
--- 
--- Let's check

added:

fricas

--- 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
FPFI ==> Fraction(Polynomial(Fraction(Integer)))

Type: Void

fricas

---
---
--- 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

fricas

---
---
--- 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

fricas

---
---
--- 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

fricas

---
---
--- 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

fricas

---
---
--- 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

fricas

---
---
---Generate Pascal Matrices with a scalar (x) coefficient
--- exp(x*H)
---
Gen_Pascal(x,dim) ==
        H : Matrix(FPFI):=x*Gen_H(dim)
        Exp_Matrix(x*H)

Type: Void

fricas

---
---
---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

fricas

---
---
--- 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

fricas

---
---

--- 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

fricas

---
--- (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

fricas

---
--- 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

fricas

---
--- 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

fricas

---
---
--- Start of Examples
---
dim:=5

$\label{eq1}5$

(1)

Type: PositiveInteger?

fricas

H:=Gen_H(dim)

fricas

Compiling function Gen_H with type PositiveInteger -> Matrix(
      Fraction(Polynomial(Fraction(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(Integer))))

fricas

GP :=Gen_Pascal(1,dim)

fricas

Compiling function Exp_Matrix with type Matrix(Fraction(Polynomial(
      Fraction(Integer)))) -> Matrix(Fraction(Polynomial(Fraction(
      Integer))))

fricas

Compiling function Gen_Pascal with type (PositiveInteger, 
      PositiveInteger) -> Matrix(Fraction(Polynomial(Fraction(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(Integer))))

fricas

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(Integer))))

fricas

--- Enough is enough in taylor stream

fricas

)set streams calculate 4

t := series 't

$\label{eq5}t$

(5)

Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)

fricas

--- 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))

fricas

---
---
--- Appell Sequences
--- f(t)*exp(x*t)
---
--- Euler Polynmials from WikiPedia
--- 2/(e^t +1) exp(x*t)
---
Euler_A:=2*((GP+1)^-1)

$\label{eq7}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ -{1 \over 2}& 1 & 0 & 0 & 0 \ 0 & - 1 & 1 & 0 & 0 \ {1 \over 4}& 0 & -{3 \over 2}& 1 & 0 \ 0 & 1 & 0 & - 2 & 1$

(7)

Type: SquareMatrix?(5,Fraction(Integer))

fricas

--- Test
Euler_A*XC

$\label{eq8}\left[ \begin{array}{c} 1 \ {x -{1 \over 2}} \ {{{x}^{2}}- x} \ {{{x}^{3}}-{{3 \over 2}\ {{x}^{2}}}+{1 \over 4}} \ {{{x}^{4}}-{2 \ {{x}^{3}}}+ x}$

(8)

Type: Matrix(Polynomial(Fraction(Integer)))

fricas

---
--- 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

fricas

Compiling function Gen_H_PIl with type PositiveInteger -> Matrix(
      Fraction(Polynomial(Fraction(Integer))))

fricas

Compiling function Gen_RI with type (PositiveInteger, 
      PositiveInteger) -> Matrix(Fraction(Polynomial(Fraction(Integer))
      ))

$\label{eq9}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ -{1 \over 2}& 1 & 0 & 0 & 0 \ {1 \over 6}& - 1 & 1 & 0 & 0 \ 0 &{1 \over 2}& -{3 \over 2}& 1 & 0 \ -{1 \over{30}}& 0 & 1 & - 2 & 1$

(9)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

---
--- 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)

fricas

Compiling function Ln_Uni_Matrix with type Matrix(Fraction(
      Polynomial(Fraction(Integer)))) -> Matrix(Fraction(Polynomial(
      Fraction(Integer))))

fricas

Compiling function Exp_Matrix_AB with type (Matrix(Fraction(
      Polynomial(Fraction(Integer)))), Polynomial(Integer)) -> Matrix(
      Fraction(Polynomial(Fraction(Integer))))

$\label{eq10}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ {{1 \over 2}\ a}& 1 & 0 & 0 & 0 \ {{{1 \over 4}\ {{a}^{2}}}+{{1 \over{12}}\ a}}& a & 1 & 0 & 0 \ {{{1 \over 8}\ {{a}^{3}}}+{{1 \over 8}\ {{a}^{2}}}}&{{{3 \over 4}\ {{a}^{2}}}+{{1 \over 4}\ a}}&{{3 \over 2}\ a}& 1 & 0 \ {{{1 \over{16}}\ {{a}^{4}}}+{{1 \over 8}\ {{a}^{3}}}+{{1 \over{4 8}}\ {{a}^{2}}}-{{1 \over{120}}\ a}}&{{{1 \over 2}\ {{a}^{3}}}+{{1 \over 2}\ {{a}^{2}}}}&{{{3 \over 2}\ {{a}^{2}}}+{{1 \over 2}\ a}}&{2 \ a}& 1$

(10)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

---
----
---- 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)

fricas

Compiling function Exp_Matrix with type Matrix(Integer) -> Matrix(
      Fraction(Polynomial(Fraction(Integer))))

fricas

Compiling function Id_M with type PositiveInteger -> Matrix(Fraction
      (Polynomial(Fraction(Integer))))

$\label{eq11}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ {1 \over{u - 1}}& 1 & 0 & 0 & 0 \ {{u + 1}\over{{{u}^{2}}-{2 \ u}+ 1}}&{2 \over{u - 1}}& 1 & 0 & 0 \ {{{{u}^{2}}+{4 \ u}+ 1}\over{{{u}^{3}}-{3 \ {{u}^{2}}}+{3 \ u}- 1}}&{{{3 \ u}+ 3}\over{{{u}^{2}}-{2 \ u}+ 1}}&{3 \over{u - 1}}& 1 & 0 \ {{{{u}^{3}}+{{11}\ {{u}^{2}}}+{{11}\ u}+ 1}\over{{{u}^{4}}-{4 \ {{u}^{3}}}+{6 \ {{u}^{2}}}-{4 \ u}+ 1}}&{{{4 \ {{u}^{2}}}+{{16}\ u}+ 4}\over{{{u}^{3}}-{3 \ {{u}^{2}}}+{3 \ u}- 1}}&{{{6 \ u}+ 6}\over{{{u}^{2}}-{2 \ u}+ 1}}&{4 \over{u - 1}}& 1$

(11)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

Frob_Euler:=Exp_Matrix_AB(Frob_Euler_term,r)

fricas

Compiling function Exp_Matrix_AB with type (Matrix(Fraction(
      Polynomial(Fraction(Integer)))), Variable(r)) -> Matrix(Fraction(
      Polynomial(Fraction(Integer))))

$\label{eq12}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ {r \over{u - 1}}& 1 & 0 & 0 & 0 \ {{{r \ u}+{{r}^{2}}}\over{{{u}^{2}}-{2 \ u}+ 1}}&{{2 \ r}\over{u - 1}}& 1 & 0 & 0 \ {{{r \ {{u}^{2}}}+{{\left({3 \ {{r}^{2}}}+ r \right)}\ u}+{{r}^{3}}}\over{{{u}^{3}}-{3 \ {{u}^{2}}}+{3 \ u}- 1}}&{{{3 \ r \ u}+{3 \ {{r}^{2}}}}\over{{{u}^{2}}-{2 \ u}+ 1}}&{{3 \ r}\over{u - 1}}& 1 & 0 \ {{{r \ {{u}^{3}}}+{{\left({7 \ {{r}^{2}}}+{4 \ r}\right)}\ {{u}^{2}}}+{{\left({6 \ {{r}^{3}}}+{4 \ {{r}^{2}}}+ r \right)}\ u}+{{r}^{4}}}\over{{{u}^{4}}-{4 \ {{u}^{3}}}+{6 \ {{u}^{2}}}-{4 \ u}+ 1}}&{{{4 \ r \ {{u}^{2}}}+{{\left({{12}\ {{r}^{2}}}+{4 \ r}\right)}\ u}+{4 \ {{r}^{3}}}}\over{{{u}^{3}}-{3 \ {{u}^{2}}}+{3 \ u}- 1}}&{{{6 \ r \ u}+{6 \ {{r}^{2}}}}\over{{{u}^{2}}-{2 \ u}+ 1}}&{{4 \ r}\over{u - 1}}& 1$

(12)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

---
---
--- Hermite Polynomials wiki version
---exp(x*t-(t^2)/2) *exp(x*t)
---
Herm_Wiki_A := Exp_Matrix(-H^2/2)

$\label{eq13}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ - 1 & 0 & 1 & 0 & 0 \ 0 & - 3 & 0 & 1 & 0 \ 3 & 0 & - 6 & 0 & 1$

(13)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

---
Herm_Roman_A := Exp_Matrix(-v*H^2/2)

$\label{eq14}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ - v & 0 & 1 & 0 & 0 \ 0 & -{3 \ v}& 0 & 1 & 0 \ {3 \ {{v}^{2}}}& 0 & -{6 \ v}& 0 & 1$

(14)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

--- Hermite/Gould Hopper of order 3
--- exp(x*t+y*t^3)
HGH_A := Exp_Matrix(y*H^3)

$\label{eq15}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 \ {6 \ y}& 0 & 0 & 1 & 0 \ 0 &{{24}\ y}& 0 & 0 & 1$

(15)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

---
---
--- 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))

fricas

Compiling function Ln_Uni_Matrix with type SquareMatrix(5,Integer)
       -> Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

Compiling function ST_H with type Matrix(Fraction(Polynomial(
      Fraction(Integer)))) -> Matrix(Fraction(Polynomial(Fraction(
      Integer))))

$\label{eq16}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & - 1 & 1 & 0 & 0 \ 0 & 2 & - 3 & 1 & 0 \ 0 & - 6 &{11}& - 6 & 1$

(16)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

--- Just to verify
x:='x;

Type: Variable(x)

fricas

$\label{eq17}1$

(17)

Type: PositiveInteger?

fricas

$\label{eq18}x$

(18)

Type: Variable(x)

fricas

x*(x-1)

$\label{eq19}{{x}^{2}}- x$

(19)

Type: Polynomial(Integer)

fricas

x*(x-1)*(x-2)

$\label{eq20}{{x}^{3}}-{3 \ {{x}^{2}}}+{2 \ x}$

(20)

Type: Polynomial(Integer)

fricas

x*(x-1)*(x-2)*(x-3)

$\label{eq21}{{x}^{4}}-{6 \ {{x}^{3}}}+{{11}\ {{x}^{2}}}-{6 \ x}$

(21)

Type: Polynomial(Integer)

fricas

---
---  Exponential Polynomials 
--- Roman PP 64
--- Also http://mathworld.wolfram.com/BellPolynomial.html
--- exp(x*(exp(t)-1))
---
Exp_A := ST_H(GP-1)

fricas

Compiling function ST_H with type SquareMatrix(5,Integer) -> Matrix(
      Fraction(Polynomial(Fraction(Integer))))

$\label{eq22}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & 1 & 1 & 0 & 0 \ 0 & 1 & 3 & 1 & 0 \ 0 & 1 & 7 & 6 & 1$

(22)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

---
--- 
-

$\label{eq23}-$

(23)

Type: Variable(-)

fricas

--- 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)

fricas

Compiling function Exp_Matrix_AB with type (SquareMatrix(5,Integer)
      , Polynomial(Integer)) -> Matrix(Fraction(Polynomial(Fraction(
      Integer))))

$\label{eq24}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ {a + 1}& - 1 & 0 & 0 & 0 \ {{{a}^{2}}+{3 \ a}+ 2}&{-{2 \ a}- 4}& 1 & 0 & 0 \ {{{a}^{3}}+{6 \ {{a}^{2}}}+{{11}\ a}+ 6}&{-{3 \ {{a}^{2}}}-{{1 5}\ a}-{18}}&{{3 \ a}+ 9}& - 1 & 0 \ {{{a}^{4}}+{{10}\ {{a}^{3}}}+{{35}\ {{a}^{2}}}+{{50}\ a}+{24}}&{-{4 \ {{a}^{3}}}-{{36}\ {{a}^{2}}}-{{104}\ a}-{96}}&{{6 \ {{a}^{2}}}+{{42}\ a}+{72}}&{-{4 \ a}-{16}}& 1$

(24)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

--- Checking
Lag_S:=(1-t)^(-a-1)*exp(x*(t/(t-1)));

Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)

fricas

Lag_S_C3:=6*coefficient(Lag_S,3)

$\label{eq25}\begin{array}{@{}l} \displaystyle -{{x}^{3}}+{{\left({3 \ a}+ 9 \right)}\ {{x}^{2}}}+{{\left(-{3 \ {{a}^{2}}}-{{15}\ a}-{18}\right)}\ x}+{{a}^{3}}+{6 \ {{a}^{2}}}+ \ \ \displaystyle {{11}\ a}+ 6$

(25)

Type: Expression(Integer)

fricas

Lag_P:=Lag_A*XC

$\label{eq26}\left[ \begin{array}{c} 1 \ {- x + a + 1} \ {{{x}^{2}}+{{\left(-{2 \ a}- 4 \right)}\ x}+{{a}^{2}}+{3 \ a}+ 2} \ {-{{x}^{3}}+{{\left({3 \ a}+ 9 \right)}\ {{x}^{2}}}+{{\left(-{3 \ {{a}^{2}}}-{{15}\ a}-{18}\right)}\ x}+{{a}^{3}}+{6 \ {{a}^{2}}}+{{11}\ a}+ 6} \ {{{x}^{4}}+{{\left(-{4 \ a}-{16}\right)}\ {{x}^{3}}}+{{\left({6 \ {{a}^{2}}}+{{42}\ a}+{72}\right)}\ {{x}^{2}}}+{{\left(-{4 \ {{a}^{3}}}-{{36}\ {{a}^{2}}}-{{104}\ a}-{96}\right)}\ x}+{{a}^{4}}+{{10}\ {{a}^{3}}}+{{35}\ {{a}^{2}}}+{{50}\ a}+{24}}$

(26)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

Lag_P(4,1)-Lag_S_C3

$\label{eq27}0$

(27)

Type: Expression(Integer)

fricas

--- 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

$\label{eq28}t$

(28)

Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)

fricas

Meix_Series_R :=((1-t*d)*((1-t)^-1))^x*(1-t)^(-b)

$\label{eq29}\begin{array}{@{}l} \displaystyle 1 +{{\left({{\left(- d + 1 \right)}\ x}+ b \right)}\ t}+ \ \ \displaystyle {{{{{\left({{d}^{2}}-{2 \ d}+ 1 \right)}\ {{x}^{2}}}+{{\left(-{{d}^{2}}-{2 \ b \ d}+{2 \ b}+ 1 \right)}\ x}+{{b}^{2}}+ b}\over 2}\ {{t}^{2}}}+ \ \ \displaystyle {{{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1 \right)}\ {{x}^{3}}}+ \ \ \displaystyle {{\left({3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3 \right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+ \ \ \displaystyle {3 \ {{b}^{2}}}+{6 \ b}+ 2$

(29)

Type: UnivariatePuiseuxSeries?(Expression(Integer),t,0)

fricas

--- 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(Integer))) -> Matrix(Fraction(Polynomial(Fraction(
      Integer))))

$\label{eq30}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ 0 &{- d + 1}& 0 & 0 & 0 \ 0 &{-{{d}^{2}}+ 1}&{{{d}^{2}}-{2 \ d}+ 1}& 0 & 0 \ 0 &{-{2 \ {{d}^{3}}}+ 2}&{{3 \ {{d}^{3}}}-{3 \ {{d}^{2}}}-{3 \ d}+ 3}&{-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1}& 0 \ 0 &{-{6 \ {{d}^{4}}}+ 6}&{{{11}\ {{d}^{4}}}-{8 \ {{d}^{3}}}-{6 \ {{d}^{2}}}-{8 \ d}+{11}}&{-{6 \ {{d}^{4}}}+{{12}\ {{d}^{3}}}-{{12}\ d}+ 6}&{{{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1}$

(30)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

Meix_A := Exp_Matrix_AB(1-H,(-b))*Meix_A_fJ

$\label{eq31}\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \ b &{- d + 1}& 0 & 0 & 0 \ {{{b}^{2}}+ b}&{-{{d}^{2}}-{2 \ b \ d}+{2 \ b}+ 1}&{{{d}^{2}}-{2 \ d}+ 1}& 0 & 0 \ {{{b}^{3}}+{3 \ {{b}^{2}}}+{2 \ b}}&{-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2}&{{3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3}&{-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1}& 0 \ {{{b}^{4}}+{6 \ {{b}^{3}}}+{{11}\ {{b}^{2}}}+{6 \ b}}&{-{6 \ {{d}^{4}}}-{8 \ b \ {{d}^{3}}}+{{\left(-{6 \ {{b}^{2}}}-{6 \ b}\right)}\ {{d}^{2}}}+{{\left(-{4 \ {{b}^{3}}}-{{12}\ {{b}^{2}}}-{8 \ b}\right)}\ d}+{4 \ {{b}^{3}}}+{{18}\ {{b}^{2}}}+{{22}\ b}+ 6}&{{{11}\ {{d}^{4}}}+{{\left({{12}\ b}- 8 \right)}\ {{d}^{3}}}+{{\left({6 \ {{b}^{2}}}-{6 \ b}- 6 \right)}\ {{d}^{2}}}+{{\left(-{{12}\ {{b}^{2}}}-{{24}\ b}- 8 \right)}\ d}+{6 \ {{b}^{2}}}+{{18}\ b}+{11}}&{-{6 \ {{d}^{4}}}+{{\left(-{4 \ b}+{12}\right)}\ {{d}^{3}}}+{{1 2}\ b \ {{d}^{2}}}+{{\left(-{{12}\ b}-{12}\right)}\ d}+{4 \ b}+ 6}&{{{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1}$

(31)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

--- 
--- Let's check
Meix_C3:=6*coefficient(Meix_Series_R,3)

$\label{eq32}\begin{array}{@{}l} \displaystyle {{\left(-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1 \right)}\ {{x}^{3}}}+ \ \ \displaystyle {{\left({3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3 \right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left(-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2 \right)}\ x}+{{b}^{3}}+ \ \ \displaystyle {3 \ {{b}^{2}}}+{2 \ b}$

(32)

Type: Expression(Integer)

fricas

Meix_P:=Meix_A*XC

$\label{eq33}\left[ \begin{array}{c} 1 \ {{{\left(- d + 1 \right)}\ x}+ b} \ {{{\left({{d}^{2}}-{2 \ d}+ 1 \right)}\ {{x}^{2}}}+{{\left(-{{d}^{2}}-{2 \ b \ d}+{2 \ b}+ 1 \right)}\ x}+{{b}^{2}}+ b} \ {{{\left(-{{d}^{3}}+{3 \ {{d}^{2}}}-{3 \ d}+ 1 \right)}\ {{x}^{3}}}+{{\left({3 \ {{d}^{3}}}+{{\left({3 \ b}- 3 \right)}\ {{d}^{2}}}+{{\left(-{6 \ b}- 3 \right)}\ d}+{3 \ b}+ 3 \right)}\ {{x}^{2}}}+{{\left(-{2 \ {{d}^{3}}}-{3 \ b \ {{d}^{2}}}+{{\left(-{3 \ {{b}^{2}}}-{3 \ b}\right)}\ d}+{3 \ {{b}^{2}}}+{6 \ b}+ 2 \right)}\ x}+{{b}^{3}}+{3 \ {{b}^{2}}}+{2 \ b}} \ {{{\left({{d}^{4}}-{4 \ {{d}^{3}}}+{6 \ {{d}^{2}}}-{4 \ d}+ 1 \right)}\ {{x}^{4}}}+{{\left(-{6 \ {{d}^{4}}}+{{\left(-{4 \ b}+{12}\right)}\ {{d}^{3}}}+{{12}\ b \ {{d}^{2}}}+{{\left(-{{1 2}\ b}-{12}\right)}\ d}+{4 \ b}+ 6 \right)}\ {{x}^{3}}}+{{\left({{1 1}\ {{d}^{4}}}+{{\left({{12}\ b}- 8 \right)}\ {{d}^{3}}}+{{\left({6 \ {{b}^{2}}}-{6 \ b}- 6 \right)}\ {{d}^{2}}}+{{\left(-{{12}\ {{b}^{2}}}-{{24}\ b}- 8 \right)}\ d}+{6 \ {{b}^{2}}}+{{18}\ b}+{11}\right)}\ {{x}^{2}}}+{{\left(-{6 \ {{d}^{4}}}-{8 \ b \ {{d}^{3}}}+{{\left(-{6 \ {{b}^{2}}}-{6 \ b}\right)}\ {{d}^{2}}}+{{\left(-{4 \ {{b}^{3}}}-{{12}\ {{b}^{2}}}-{8 \ b}\right)}\ d}+{4 \ {{b}^{3}}}+{{18}\ {{b}^{2}}}+{{22}\ b}+ 6 \right)}\ x}+{{b}^{4}}+{6 \ {{b}^{3}}}+{{11}\ {{b}^{2}}}+{6 \ b}}$

(33)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

Meix_P(4,1)-Meix_C3

$\label{eq34}0$

(34)

Type: Expression(Integer)