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The expression for the orbital perturbation due to J2 ($\mathbf{a_{J2}} = [a_r \ a_s \ a_w]^T$) as well as its derived effect on the argument of perigee and the right ascension of the ascending node are well known and available in most books.

Regarding higher-order terms related to the zonal harmonics or oblateness ($J_3, J_4, ...$) and multipole perturbations ($J_{22}, J_{3m}, J_{4m}, ...$), usually the general equation of the potential $V(r,\delta,\lambda)$ is provided, since it is an infinite series anyway.

Is there any reliable source where the derived perturbations $\mathbf{a_{J_{nm}}} = [a_r \ a_s \ a_w]^T$ for the cases that consider up to J3, J4, etc. are presented? I am looking into preparing a propagator that numerically integrates the osculating orbital elements affected by these perturbations, and I wonder if these expressions are already available somewhere.

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In Vallado's Fundamentals of Astrodynamics and Applications, Section 8.6.1: Gravity Field of a Central Body, he derives the equations to calculate perturbations due to aspherical bodies. In Section 8.7.1: Application: Simplified Acceleration Model, he walks through an example calculating the acceleration due to J2 through J6 (as well as from atmospheric drag) and provides the explicit equations. In Appendix D.1: Gravitational Coefficients, he provides values of some of the coefficients from the EGM-08 model.

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I just got this announcement from the JPL mail list ssd-announce@list.jpl.nasa.gov:

During the week of April 12, the Horizons ephemeris system will be updated to replace the DE430/431 planetary ephemeris, used since 2013, with the new DE440/441 solution and sixteen most massive small-body perturbers.

The new DE440/441 general-purpose planetary solution includes seven additional years of ground and space-based astrometric data, data calibrations, and dynamical model improvements, most significantly involving Jupiter, Saturn, Pluto, and the Kuiper Belt.

For details, see "The JPL Planetary and Lunar Ephemerides DE440 and DE441", R.S. Park, et al., The Astronomical Journal, 161:105 (15pp), 2021 March.

That paper is The JPL Planetary and Lunar Ephemerides DE440 and DE441 but it may be paywalled.

See

3.2. Point-mass Interaction with Extended Bodies

Equation 28 and all the accompanying material together forms a self-consistent basis. It is too long and I am too error prone for it to be advisable for me to reproduce everything here.

The pair of Development Ephemerides that these are replacing is discussed in The Planetary and Lunar Ephemerides DE430 and DE431 which does not appear to be paywalled and looks to be very similar (which is not a surprise).

III.B. Point Mass Interaction with Extended Bodies

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