Here I have the integral $$\int_0^{\infty}x^{\ell-1}K_m(x)K_n(\frac bx)\,dx$$ which is the integral of the multiplication of two modified bessel function of the second kind. I find that this integral is expressed as as a series of the form $$\sum_{v=0}^{\infty}c_vb^{\rho+v}$$ in the reference, i.e., "An Infinite Integral Involving a Product of two Modified Bessel functions of the Second Kind" by T. M. Macrobert. which is used to express the integral as generalized hypergeometric function. I do not know why the integral can be expressed as this series. Can you give me some advice?
2026-03-27 13:39:53.1774618793
How to express the integral as a series?
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