I am studying Legendre equation but not in rigorous mannar.
From Legendre's polynoimal's power series property, combined with it's orthonormal property. It is enough to convince me $P_{n}$(x) forms a complete set.
However, things start to get confusing when some article e.g. wikipedia state that $P_{n}$(x) form a complete set from Sturm-Liouvile theory.
Isn't it that legendre equation has TWO solutions, i.e. the converging kind ($P_{n}(x)$ and the diverging kind ($Q_{n}(x)$)? Sturm-Liouvile theory should state that ALL eigenfunctions combined to form a complete set. Then how does $P_{n}$(x) alone form a complete set already?
It would be nice if someone can explain it in a non-rigorous manner or point out my misconception. Appreciate your help.
I've changed this answer. Hopefully this answer will better address your concerns.
Start with a classical Sturm-Liouville problem on $[0,\pi]$: $$ -y''=\lambda y, \;\; y(0)=0,\;y(\pi)=0. $$ This problem has solutions $$ \sin(nx),\;\;\; n=1,2,3,\cdots. $$ These functions form a complete orthogonal set for $L^2[0,\pi]$. Similary, the problem $$ -y''=\lambda y,\;\;\; y'(0)=y'(\pi)=0 $$ has solutions $$ 1,\cos(x),\cos(2x),\cos(3x),\cdots. $$ These functions form a complete orthogonal set for $L^2[0,\pi]$, just as the $\sin(nx)$ functions form a complete orthogonal set for $L^2[0,\pi]$. You don't need to include both sets in order to obtain a complete set on $[0,\pi]$. In fact, you can use the $\sin(nx)$ functions to write any $\cos(mx)$ function in the $L^2[0,\pi]$ sense: $$ \cos(mx) = \sum_{n=1}^{\infty}\frac{\int_0^\pi\cos(mx')\sin(nx')dx'}{\int_0^\pi\sin^2(nx')dx'}\sin(nx). $$ Similarly, you can use the $\cos(mx)$ functions to write any $\sin(nx)$ function $$ \sin(nx) = \sum_{m=0}^{\infty}\frac{\int_0^\pi\sin(nx')\cos(mx')dx'}{\int_0^\pi\cos(mx')^2dx'}\cos(mx). $$ Even though the $\cos(mx)$ functions are mutually orthogonal on $[0,\pi]$ and the $\sin(nx)$ functions are mutually orthogonaal on $[0,\pi]$, the $\sin$ and $\cos$ functions are not mutually orthogonal on $[0,\pi]$; if they were mutually orthgonal, the above expansions would not work.
You cannot mix the orthogonal functions for one Sturm-Liouville problem on $[0,\pi]$ with those from another Sturm-Liouville problem on $[0,\pi]$ and hope to have another orthogonal set because either set can be used to expand the functions in the other set.
There are multiple complete orthogonal sets of eigenfunctions of $-y''=\lambda y$, as many as there are endpoint conditions of the form $$ Ay(0)+By'(0)=0,\;\; Cy(\pi)+Dy'(\pi)=0, $$ where not both $A,B$ are $0$ and not both $C,D$ are $0$. The solutions of one problem can be expanded in the eigenfunctions of the other problem and vice-versa.
There are some singular Sturm-Liouville problems on $[a,b]$ where you don't have two possible endpoint conditions at $a$ and/or at $b$. This has to do with the so called limit-point, limit-circle endpoint classifications of the Sturm-Liouville problem. The limit-point/limit-circle classification has to do with the eigenfunctions of the differential operator that are in the appropriate weighted $L^2_w[a,b]$ space.