proof the the general L

Paper details:
for the question , please I need a full proof up to the general L-1 not only L=1 and m=0

Question : For ` 1, prove that K = 0 if
K =
X`??1
m=0
(??1)m+1
m!(2` ?? m)!
”
(2` ?? m)Xm Y2`??m??1 + mXm??1 Y2`??m
#
+
X`??1
m=0
(??1)m+1
m!(2` ?? m)!
”
(2` ?? m) Ym X2`??m??1 + mYm??1 X2`??m
#
+ (??1)`+1 1
(`!)2Y`X`
1

Proof of the General L
(Name)
(Institution)
K= ?_(m=0)^(l-1)¦(-1^(m+1))/(m!(2l-m)) [(2l-m)(xmy2l-m-1+ymx2l-m-1)+(mxm-1y2l-m+mym-1x2l-m)] + (-1)l+1 1/((l!)^2)ylxl

Relation recursion formula

(-1)l+1 1/((l!)^2)ylxl + K =  [(2l-m)(xmy2l-m-1+ymx2l-m-1)+(mxm-1y2l-m+mym-1x2l-m)]

Taking l=1 such that m=0K=0
Let l= 1, m=0
1/1    y1x1=- (x0y1+y0x1)
1x-1=-1×1+0x1
1=1 hence k=0
Let l=2
(-1)/4    y2x2 = (-1)/6   (x0y3+y0x3)
(-1)/4    2x-1 = (-1)/6   (-1×3+0x+1)
1/2    =1/2
Hence k=0
L=3, m=0
1/36    x2y2 = 6/720   x0y5 + y0x5
1/36    x2y2 = 1/120   x0y5 + y0x5
K= ½ x2 (x0y2+y0x2) + ¼ y1x1
Taking integrals of both sides
?_0^(l-1)¦?¼ y1x1 ?= ?_0^(l-1)(x0y2+y0x2)
= 1/4 x y12x1/4 = x0y22/2 + x02 y2/2 + y02x2/2 + x22y0

proof the the general L

Paper details:
for the question , please I need a full proof up to the general L-1 not only L=1 and m=0

Question : For ` 1, prove that K = 0 if
K =
X`??1
m=0
(??1)m+1
m!(2` ?? m)!
”
(2` ?? m)Xm Y2`??m??1 + mXm??1 Y2`??m
#
+
X`??1
m=0
(??1)m+1
m!(2` ?? m)!
”
(2` ?? m) Ym X2`??m??1 + mYm??1 X2`??m
#
+ (??1)`+1 1
(`!)2Y`X`
1

Proof of the General L
(Name)
(Institution)
K= ?_(m=0)^(l-1)¦(-1^(m+1))/(m!(2l-m)) [(2l-m)(xmy2l-m-1+ymx2l-m-1)+(mxm-1y2l-m+mym-1x2l-m)] + (-1)l+1 1/((l!)^2)ylxl

Relation recursion formula

(-1)l+1 1/((l!)^2)ylxl + K =  [(2l-m)(xmy2l-m-1+ymx2l-m-1)+(mxm-1y2l-m+mym-1x2l-m)]

Taking l=1 such that m=0K=0
Let l= 1, m=0
1/1    y1x1=- (x0y1+y0x1)
1x-1=-1×1+0x1
1=1 hence k=0
Let l=2
(-1)/4    y2x2 = (-1)/6   (x0y3+y0x3)
(-1)/4    2x-1 = (-1)/6   (-1×3+0x+1)
1/2    =1/2
Hence k=0
L=3, m=0
1/36    x2y2 = 6/720   x0y5 + y0x5
1/36    x2y2 = 1/120   x0y5 + y0x5
K= ½ x2 (x0y2+y0x2) + ¼ y1x1
Taking integrals of both sides
?_0^(l-1)¦?¼ y1x1 ?= ?_0^(l-1)(x0y2+y0x2)
= 1/4 x y12x1/4 = x0y22/2 + x02 y2/2 + y02x2/2 + x22y0