Preliminary

Clear[b,m,n,r,s,t,u,v,q]

id4=IdentityMatrix[4];

prod=((-q^(m b))^u q^(p u(u-1)/2)(-q^(n b))^v q^(p v(v-1)/2)(-q^(p-2m b))^r q^(p r(r-1))(-q^(p-2n b))^s q^(p s(s-1)))//PowerExpand;

Equation (1.9)

G1=Exponent[prod,q]

Equation (1.10)

soln=(Solve[((G1)/.{p->0,p/2->0})==b t,v]/.{1/n->(OverBar[n])}//Expand)/.{ n s (OverBar[n])->s};

consv={v->2 s+2 m r (OverBar[n])+t (OverBar[n])-m u (OverBar[n])+ p v}

Equation (1.11)

G2=Expand[G1/.consv];
g=Expand[((G2)-b t)/p]

f=f(r,s,u,v) in Equation (1.12)

f=(r+s+u+v)/.consv

Case 1

A={{-1-((4 (m^2) (-1+n (OverBar[n])))/(3 p)),0,((2 (m^2) (-1+n (OverBar[n])))/(3 p)),-((2 m n)/3)},{((4 m (OverBar[n]))/3)-((4 m n (-1+n (OverBar[n])))/(3 p)),1,-((2 m (OverBar[n]))/3)+((2 m n (-1+n (OverBar[n])))/(3 p)),((2 (m^2))/3)},{((4 (m^2) (-1+n (OverBar[n])))/(3 p)),0,-1-((2 (m^2) (-1+n (OverBar[n])))/(3 p)),((2 m n)/3)},{((4 m (-1+n (OverBar[n])) (p (OverBar[n])-n (-1+n (OverBar[n]))))/(p^2)),0,-((2 m (-1+n (OverBar[n])) (p (OverBar[n])-n (-1+n (OverBar[n]))))/(p^2)),1+((2 (m^2) (-1+n (OverBar[n])))/p)}};
bb={((m (m+n-2 n w (OverBar[n])))/(3 p)),((m (-m+n+2 m w (OverBar[n])))/(3 p)),1-((m (m+n-2 n w (OverBar[n])))/(3 p)),((m (-1+n (OverBar[n])) (-m+n+2 m w (OverBar[n])))/(p^2))};
{A//MatrixForm,bb//MatrixForm}

Equation (1.14)

Factor[Factor[Simplify[Simplify[A.A]/.{((m^2)+(n^2)-p)->0}]/.{((n^2)-p)->-m^2}]/.{(m^4) ->m^2(p-n^2)}]//MatrixForm

Equation (1.15)

Simplify[Simplify[(A+id4).bb]/.{((m^2)+(n^2)-p)->0}/.{((n^2)-p)->-m^2}]//MatrixForm

Equation (1.16)

mat2=A.{r,s,u,v}+bb;
gx=g/.{t->w};
gxprime=gx/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]}/.{t->w};
dif=Simplify[Expand[gx-gxprime]]
Factor[%/.{((m^2)+(n^2)-p)->0}]

Equation (1.17) ,   f(x)-f(x’) is odd

ft=f/.{t->w}

Simplify[ft-(ft/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]})]
%/.{((m^2)+(n^2)-p)->0}

Case 2

A={{1+((4 (m^2) (-1+n (OverBar[n])))/(3 p)),0,-((2 (m^2) (-1+n (OverBar[n])))/(3 p)),((2 m n)/3)},{-((4 m (OverBar[n]))/3)+((4 m n (-1+n (OverBar[n])))/(3 p)),-1,((2 m (OverBar[n]))/3)-((2 m n (-1+n (OverBar[n])))/(3 p)),-((2 (m^2))/3)},{-((4 (m^2) (-1+n (OverBar[n])))/(3 p)),0,1+((2 (m^2) (-1+n (OverBar[n])))/(3 p)),-((2 m n)/3)},{-((4 m (-1+n (OverBar[n])) (p (OverBar[n])-n (-1+n (OverBar[n]))))/(p^2)),0,((2 m (-1+n (OverBar[n])) (p (OverBar[n])-n (-1+n (OverBar[n]))))/(p^2)),-1-((2 (m^2) (-1+n (OverBar[n])))/p)}};
Clear[b]
bb={-((m (-6 b+m+n-2 n t (OverBar[n])))/(3 p)),-((-(m^2)-6 b n+m n+2 (m^2) t (OverBar[n]))/(3 p)),((m (-6 b+m+n-2 n t (OverBar[n])))/(3 p)),-(((n+(m^2) (OverBar[n])) (6 b-m-n+2 n t (OverBar[n])))/(p^2))}/.{t->w-3b};
{A//MatrixForm,bb//MatrixForm}

Equation (1.14)

Factor[Factor[Simplify[Simplify[A.A]/.{((m^2)+(n^2)-p)->0}]/.{((n^2)-p)->-m^2}]/.{(m^4) ->m^2(p-n^2)}]//MatrixForm

Equation (1.15)

(Simplify[Simplify[(A+id4).bb]]/.{((m^2)+(n^2)-p)->0})//MatrixForm

Equation (1.16)

mat2=A.{r,s,u,v}+bb;
pas2=pas/.{t->w-3b};
pas22=pas2/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]}/.{t->w-3b};
dif=Simplify[Expand[pas1-pas11]]
%/.{((m^2)+(n^2)-p)->0}

Equation (1.17) ,   f(x)-f(x’) is odd

ft=f/.{t->w-3b}

Simplify[ft-(ft/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]})]
Simplify[%//.{((m^2)+(n^2)-p)->0,((m^2)+(n^2))->p}]

Case 3

A={{((4 m n+p+2 m (m-n) (m+n) (OverBar[n]))/(3 p)),0,-(((m-n) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(3 p)),(1/3) (m-n) (m+n)},{-((2 (m-n) (m+n+(m^2) (OverBar[n])-m n (OverBar[n])))/(3 p)),-1,(((m-n) (m+n+(m^2) (OverBar[n])-m n (OverBar[n])))/(3 p)),-(1/3) (((m-n))^2)},{-((2 (m-n) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(3 p)),0,((2 m n+2 p+m (m-n) (m+n) (OverBar[n]))/(3 p)),-(1/3) (m-n) (m+n)},{-((2 (m+n+(m^2) (OverBar[n])-m n (OverBar[n])) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(p^2)),0,(((m+n+(m^2) (OverBar[n])-m n (OverBar[n])) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(p^2)),-((m (2 n+(m^2) (OverBar[n])-(n^2) (OverBar[n])))/p)}};
bb=Simplify[{(((m-n) (3 b-m+m w (1-3 b (OverBar[m])) (OverBar[n])+n w (1-3 b (OverBar[m])) (OverBar[n])))/(3 p)),((3 b m+3 b n-m n+(n^2)-(m^2) w (1-3 b (OverBar[m])) (OverBar[n])+2 m n w (1-3 b (OverBar[m])) (OverBar[n])-(n^2) w (1-3 b (OverBar[m])) (OverBar[n]))/(3 p)),-(((m-n) (3 b-m+m w (1-3 b (OverBar[m])) (OverBar[n])+n w (1-3 b (OverBar[m])) (OverBar[n])))/(3 p)),-(((m+n+(m^2) (OverBar[n])-m n (OverBar[n])) (3 b-m+m w (1-3 b (OverBar[m])) (OverBar[n])+n w (1-3 b (OverBar[m])) (OverBar[n])))/(p^2))}];
{A//MatrixForm,bb//MatrixForm}

Equation (1.14)

(FullSimplify[Factor[Factor[Simplify[Simplify[A.A]/.{((m^2)+(n^2)-p)->0}]/.{((n^2)-p)->-m^2}]/.{(m^4) ->m^2(p-n^2)}/.{((m^2)+(n^2))->p}/.{n^4->n^2(p-m^2),n^3->n(p-m^2)}]]/.{((m^2)+(n^2))->p})//MatrixForm

Equation (1.15)

(Simplify[Simplify[(A+id4).bb]]/.{((m^2)+(n^2)-p)->0})//MatrixForm

Equation (1.16)

mat2=A.{r,s,u,v}+bb;
pas3=pas/.{t->(1-3 b (OverBar[m]))w};
pas33=pas3/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]}/.{t->(1-3 b (OverBar[m]))w};
dif=Simplify[Expand[pas3-pas33]]

Simplify[%/.{((m^2)+(n^2)-p)->0}]

Equation (1.17) ,   f(x)-f(x’) is odd

ft=f/.{t->(1-3 b (OverBar[m]))w}

FullSimplify[ft-(ft/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]})]
Simplify[%/.{((m^2)+(n^2)-p)->0,((m^2)+(n^2))->p}/.{-m^2->-p+n^2}]

Case 4

A={{((4 m n-p+2 m (m-n) (m+n) (OverBar[n]))/(3 p)),0,-(((m+n) (m+n+(m^2) (OverBar[n])-m n (OverBar[n])))/(3 p)),(1/3) (m-n) (m+n)},{((2 (m+n) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(3 p)),1,-(((m+n) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(3 p)),(1/3) (((m+n))^2)},{-((2 (m+n) (m+n+(m^2) (OverBar[n])-m n (OverBar[n])))/(3 p)),0,((2 m n-2 p+m (m-n) (m+n) (OverBar[n]))/(3 p)),-(1/3) (m-n) (m+n)},{-((2 (m+n+(m^2) (OverBar[n])-m n (OverBar[n])) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(p^2)),0,(((m+n+(m^2) (OverBar[n])-m n (OverBar[n])) (-m+n+(m^2) (OverBar[n])+m n (OverBar[n])))/(p^2)),-((m (2 n+(m^2) (OverBar[n])-(n^2) (OverBar[n])))/p)}};
bb={((3 b (m-n)+n (m+n)+(-(m^2)+(n^2)) w (-1+3 b (OverBar[m])) (OverBar[n]))/(3 p)),-(((m+n) (-3 b+m+(m+n) w (-1+3 b (OverBar[m])) (OverBar[n])))/(3 p)),((3 b (-m+n)-n (m+n)+3 p+(m-n) (m+n) w (-1+3 b (OverBar[m])) (OverBar[n]))/(3 p)),(((m+n+m (m-n) (OverBar[n])) (-3 b+m+(m+n) w (-1+3 b (OverBar[m])) (OverBar[n])))/(p^2))};
{A//MatrixForm,bb//MatrixForm}

Equation (1.14)

(FullSimplify[Factor[Factor[Simplify[Simplify[A.A]/.{((m^2)+(n^2)-p)->0}]/.{((n^2)-p)->-m^2}]/.{(m^4) ->m^2(p-n^2)}/.{((m^2)+(n^2))->p}/.{n^4->n^2(p-m^2),n^3->n(p-m^2)}]]/.{((m^2)+(n^2))->p})//MatrixForm

Equation (1.15)

(Simplify[Simplify[(A+id4).bb]]/.{((m^2)+(n^2)-p)->0})//MatrixForm

Equation (1.16)

mat2=A.{r,s,u,v}+bb;
pas4=pas/.{t->(1-3 b (OverBar[m]))w};
pas44=pas4/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]}/.{t->(1-3 b (OverBar[m]))w};
dif=Simplify[Expand[pas3-pas33]]

Simplify[%/.{((m^2)+(n^2)-p)->0}]

Equation (1.17) ,   f(x)-f(x’) is odd

ft=f/.{t->(1-3 b (OverBar[m]))w}

Simplify[ft-(ft/.{r->mat2[[1]],s->mat2[[2]],u->mat2[[3]],v->mat2[[4]]})]
Simplify[Simplify[%/.{((m^2)+(n^2)-p)->0,(m^2)+(n^2)->p}]/.{p->m^2+n^2}]

Various