N=n-1;
k=(n-1)/2;

x=n-p-1;

% q=2;
for j=0:(n-1)/2
    s_RHS(j+1)=(-1)^j*nCk(n-p,j)*T_mn(p,(n-1)/2-j);
    % The Kravchuk polynomial has following alternative expressions:
    s_LHS_1(j+1) = (-1)^j * nCk(x,j)* nCk((n-1)-(n-p-1),((n-1)/2)-j);
    s_LHS_2(j+1)=(-2)^j * nCk((n-1)-j,((n-1)/2)-j)* nCk((n-p-1),j);
    s_LHS_3(j+1)=(-1)^j* 2^(((n-1)/2)-j)* nCk(((n-1)/2)+j,j)* nCk(p,((n-1)/2)-j) ;
end
[s_RHS' s_LHS_1' s_LHS_2' s_LHS_3']
[  sum([s_RHS' s_LHS_1' s_LHS_2' s_LHS_3'])]