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In mathematics, the Associated Legendre polynomials is usually involved to solve ordinary differential systems using spectral theory in polar spherical coordinates. In geophysics, one often encountered infinite Associated Legendre series in the form $$S(i,j)=\sum_{n=j}^\infty {\varepsilon}^n n^i P_n^j(\cos \theta)$$ and $$D(i,j)=\sum_{n=j}^\infty \frac{{\varepsilon}^n}{n+i} P_n^j(\cos \theta)$$ with $$i$$ an arbitrary integer and $$j$$ a positive integer, when modeling the static deformations or the stress distributions caused by the earthquake, the ice load and the lunar-sonar tide.

1. Derivation of the recurrence relations
It is easy to know that

$$S(i+1,j)=\varepsilon \frac{d S(i,j)}{d \varepsilon}$$,

and make use of the relations Eq. (5) of this paper

$$S(0,j)=\frac{(2j)!}{j! 2^j} \frac{(\sin \theta)^j \varepsilon^j}{s^{2j+1}}$$, with $$s=(1-2\varepsilon \cos \theta+{\varepsilon}^2)^{1/2}$$.

Here the definition of $$D(i,j)$$ only has a slight difference of $$S_{m,n}$$ of this paper, and it can be got from this article.

2. Some explicit results
Using the recurrence formula in above section, it