Given the kernel representation and Bessel polynomial eigenfunctions:
[
K(x, t) = \sum{i,j} c{ij} \phi_i(x) \psi_j(t)
]
The integral equation using the orthogonality of Bessel polynomials (y_n(x)), which are also the eigenfunctions:
[
\int_a^b y_m(x) \left( \int_a^b K(x, t) y_n(t) \, dt \right) w(x) \, dx = \lambda_n \int_a^b y_m(x) y_n(x) w(x) \, dx
]
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Given the kernel representation and Bessel polynomial eigenfunctions: [ K(x, t) = \sum{i,j} c{ij} \phi_i(x) \psi_j(t) ]
The integral equation using the orthogonality of Bessel polynomials (y_n(x)), which are also the eigenfunctions: [ \int_a^b y_m(x) \left( \int_a^b K(x, t) y_n(t) \, dt \right) w(x) \, dx = \lambda_n \int_a^b y_m(x) y_n(x) w(x) \, dx ]
Substitute $K(x, t)$ and apply Fubini's theorem: [ \int_a^b K(x, t) yn(t) \, dt = \sum{i,j} c_{ij} \phi_i(x) \int_a^b \psi_j(t) y_n(t) \, dt ]
Define: [ b_{jn} = \int_a^b \psi_j(t) y_n(t) \, dt ]
Resulting in: [ \int_a^b K(x, t) yn(t) \, dt = \sum{i,j} c_{ij} \phii(x) b{jn} ]
Plug into the outer integral equation: [ \int_a