[F(x):=,S(k,x):=]

M(k,x,h):=INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)/(2*k)!

[V(y):=,U(y):=]

INT(V(y)*DIF(U(y),y),y,x,x+h)=LIM(V(y)*U(y),y,x+h,0)-LIM(V(y)*U(y),y,x,0)-INT~
(U(y)*DIF(V(y),y),y,x,x+h)

[V(y):=S(2*k,(y-x)/h),U(y):=DIF(F(y),y,2*k-1)]

;Simp(#4)
INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)=-INT(DIF(F(y),y,2*k-1)*LIM(DIF(S(~
2*k,@),@),@,(y-x)/h,0),y,x,x+h)/h+LIM(S(2*k,(y-x)/h)*DIF(F(y),y,2*k-1),y,x+h,~
0)-LIM(S(2*k,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,0)

[U(y)=DIF(F(y),y,2*k-1),U(y):=]

;Simp(User)
INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)=-INT(U(y)*LIM(DIF(S(2*k,@),@),@,(~
y-x)/h,0),y,x,x+h)/h+U(x+h)*S(2*k,1)-U(x)*S(2*k,0)

[S(k,x):=k*SUM(p^(k-1),p,1,x-1),k:epsilonInteger (1, inf)]

;User=Simp(User)
[S(2*k,1),S(2*k,0)]=[0,0]

[S(k,x):=,B(k):=]

;Simp(User)
INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)=-INT(U(y)*LIM(DIF(S(2*k,@),@),@,(~
y-x)/h,0),y,x,x+h)/h

LIM(DIF(S(2*k,@),@),@,(y-x)/h,0)=LIM(DIF(S(2*k,t),t),t,(y-x)/h,0)

[DIF(S(k,x),x)=k*(S(k-1,x)+B(k-1)),k>2]

DIF(S(2*k,t),t)=(2*k)*(S(2*k-1,t)+B(2*k-1))

;Sub(#13')
LIM(DIF(S(2*k,@),@),@,(y-x)/h,0)=LIM((2*k)*(S(2*k-1,t)-B(2*k-1)),t,(y-x)/h,0)

B(k):=IF(k=0,1,IF(k=1,-1/2,-k*ZETA(1-k)))

;Simp(#16')
LIM(DIF(S(2*k,@),@),@,(y-x)/h,0)=2*k*S(2*k-1,(y-x)/h)

;Simp(User)
INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)=-2*k*INT(U(y)*S(2*k-1,(y-x)/h),y,~
x,x+h)/h

U(y):=DIF(F(y),y,2*k-1)

;Simp(#19)
INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)=-2*k*INT(S(2*k-1,(y-x)/h)*DIF(F(y~
),y,2*k-1),y,x,x+h)/h

;Simp(User')
M(k,x,h):=-INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)/(h*(2*k-1)!)