proof the the general L

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=-1x1+0x1 1=1 hence k=0 Let l=2 (-1)/4    y2x2 = (-1)/6   (x0y3+y0x3) (-1)/4    2x-1 = (-1)/6   (-1x3+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