Solving Linear Equation of a Scattering Problem

There is an equation I need to solve like this;

Ln+an*Mn=∑Nnm*cm On+an*Pn=∑Qnm*cm summation for m's -infinity to infinity

Ln,Mn,Nnm,On,Pn are known.an,cm are unknown. I need to get an for calculation. If anyone can help me , I'll appreciate.

Here is my code:

 clear all
format long
tic
N_cut=20;
eps0=(10^-9)/(36*pi);
mu0=4*pi*10^-7;
epsr1=1.;
epsr2=2.7;
mur1=1.;
mur2=1.;
eps1=epsr1*eps0;
eps2=epsr2*eps0;
mu1=mur1*mu0;
mu2=mur2*mu0;
freq=900*10^6;
omeg=2*pi*freq;
sigma2=3e-7;
k0=omeg*sqrt(eps0*mu0);
k1=sqrt(omeg*omeg*eps2*mu2+1i*omeg*sigma2*mu2);
lambda=2*pi/k0;
a0=4.25e-2;
a1=3e-3;
d=1e-2;
phi_prime=pi/4;
nu0=sqrt(mu0/eps0);
nu1=sqrt((1i*omeg*mu0)/(sigma2+1i*omeg*eps2));
M_phi=180;
phibegin=0;
phiend=2*pi;
deltaphi=(phiend-phibegin)/M_phi;
phig=phibegin:deltaphi:phiend;
R_obs=4.2555e-2;

for n=1:N_cut L(n)=((-1j)^n)*besselj(n,k0*a0); M(n)=((-1j)^n)*besselh(n,2,k0*a0); bessel_der_1(n)=0.5*(besselj(n-1,k0*a1)-besselj(n+1,k0*a1)); hankel_der_1(n)=0.5*(besselh(n-1,2,k0*a1)-besselh(n+1,2,k0*a1)); bessel_der_2(n)=0.5*(besselj(n-1,k1*a1)-besselj(n+1,k1*a1)); hankel_der_2(n)=0.5*(besselh(n-1,2,k1*a1)-besselh(n+1,2,k1*a1)); O(n)=(1/nu0)*((-1j)^n)*bessel_der_1(n); P(n)=(1/nu0)*((-1j)^n)*hankel_der_1(n); for m=1:N_cut K(m)=(-besselh(m,2,k1*a1))/(besselj(m,k1*a1)); N(n,m)=((-1j)^m)*besselj(n-m,k1*d)*besselj(n,k1*a1)*exp(-1j*(n-m)*phi_prime)*K(m)+((-1j)^m)*besselj(n-m,k1*d)*besselh(n,2,k1*a1)*exp(-1j*(n-m)*phi_prime); Q(n,m)=((1/nu1)*((-1j)^m)*besselj(n-m,k1*d)*bessel_der_2(n)*exp(-1j*(n-m)*phi_prime)*K(m)+(1/nu1)*((-1j)^m)*besselj(n-m,k1*d)*hankel_der_2(n)*exp(-1j*(n-m)*phi_prime));

    end
end

for mg=1:M_phi+1 for n=1:N_cut Es(mg,n)=((-1j)^n)*(a(n)*besselh(n,2,k0*R_obs))*exp(-1j*n*phig(mg)); end end F_Es=sum(Es,2); figure plot(rad2deg(phig),abs(F_Es),'r') grid on