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I don't have a general expression as a function of $n$, but there is one in terms of harmonic numbers $H_k$ as a function of $k\in\mathbb{R}^+$ for given $n$; for example $$I_{2,k}=\tfrac{1}{2}\Gamm …

Integration of Equation (34) in MathWorld gives the integral $I_{nm}$ as a sum $$I_{nm}=\sum _{q=0}^m \frac{2^{-q}}{q+1} \binom{-m-1}{q} \binom{m}{q} \, _3F_2\left(-n,n+1,q+1;1,q+2;\tfrac{1}{2}\right …

It helps to rewrite the expression from Gradshteyn, $$Q_\nu^0(z)=\frac{ \Gamma \left(\frac{1}{2}\right) \Gamma (\nu+1)\, _2F_1\left(\frac{\nu}{2}+1,\frac{\nu}{2}+\frac{1}{2};\nu+\frac{3}{2};\frac{1}{z …

The OP asks for a group theoretic derivation that is also elementary. I have not found one which combines these two properties (unless one considers the rotation operator as "elementary"). Considered …

As described in Analytic Combinatorics by Flajolet and Sedgewick, page 4, the pole $t_0$ of the generating function $F(t)$ of smallest absolute value governs the exponential asymptotics $P_n\sim (1/t_ …

In view the Rodrigues formula for the associated Legendre polynomials, one finds $$a_{nm}=\lim_{\vartheta\rightarrow 0} \vartheta^{-m}P^m_n(\cos\vartheta)= (-1)^m\frac{1}{2^n n!} \lim_{x\rightarrow 1} …

This is a particular implementation of a more general method, described in John Boyd's Why Eigenvalues Are Roots: A Derivation of the One-Dimensional Companion Matrix for General Orthogonal Polynomial …

Not a derivation (yet), but at least a reduction to a more familiar form: $$\int_{-1}^{1}\sqrt{\frac{1-x}{2}} P_n(x) \text{d}{x}=4\int_0^1 z^2\,P_n(1-2z^2)\,dz$$ which is a special case, $\mu=3/2$, …

This is theorem 2.1 in On the convergence rates of Legendre approximation (2012) [yes, with a proof in English]