SphericalBesselFunction

To express the spherical Bessel functions in terms of Lommel polynomials:

1. First, recall that spherical Bessel functions are related to the regular Bessel functions:

$$j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+\frac{1}{2}}(x)$$

where $j_n(x)$ is the spherical Bessel function of order n, and $J_\nu(x)$ is the regular Bessel function of order $\nu$.

2. Now, the Bessel functions can be expressed in terms of Lommel polynomials. The relationship is:

$$J_\nu(x) = (\frac{x}{2})^\nu [R_{\nu,0}(\frac{1}{x^2}) - R_{\nu+1,1}(\frac{1}{x^2})]$$

where $R_{\mu,\nu}(z)$ are the Lommel polynomials.

3. Substituting this into the spherical Bessel function expression:

$$j_n(x) = \sqrt{\frac{\pi}{2x}} (\frac{x}{2})^{n+\frac{1}{2}} [R_{n+\frac{1}{2},0}(\frac{1}{x^2}) - R_{n+\frac{3}{2},1}(\frac{1}{x^2})]$$

4. Simplifying:

$$j_n(x) = \sqrt{\frac{\pi}{2}} (\frac{1}{2})^{n+\frac{1}{2}} x^n [R_{n+\frac{1}{2},0}(\frac{1}{x^2}) - R_{n+\frac{3}{2},1}(\frac{1}{x^2})]$$

This is the expression for spherical Bessel functions in terms of Lommel polynomials.