Addition formula for generalized Laguerre polynomials

Question

For the Hermite polynomials, there is the following addition formula

Is there a similar formula for the generalized Laguerre Polynomials, in particular for $a=b=1/2$.

I.e. what is $L^m_k(0.5x + 0.5 y)$ in terms of $L^m_k(x), L^m_k(y)$?

Answer 1

According to wikipedia, the addition formula for the generalized Laguerre polynomials is

$$L_{n}^{\alpha +\beta +1}(x+y)=\sum _{i=0}^{n}L_{i}^{\alpha}(x)L_{n-i}^{\beta}(y)$$

So, letting $\alpha=m-1$ and $\beta=0$, we have $$L_{k}^{m}(0.5x+0.5y)=\sum _{i=0}^{k}L_{i}^{m-1}(0.5x)L_{k-i}(0.5y)\tag1$$

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Added :

I need the Laguerres to be evaluated at x and y, not 0.5x and so on

The following may help.

Using one of the multiplication theorems

$$e^{(1-t)z}L_{n}^{\alpha}(zt)=\sum _{k=0}^{\infty }{\frac {(1-t)^{k}z^{k}}{k!}}L_{n}^{\alpha +k}(z)$$

we can write

$$\begin{align}L_{i}^{m-1}(0.5x)&=e^{-x/2}\sum_{s=0}^{\infty}\frac{(x/2)^s}{s!}L_{i}^{m-1+s}(x) \\\\L_{k-i}(0.5y)&=e^{-y/2}\sum_{t=0}^{\infty}\frac{(y/2)^t}{t!}L_{k-i}^{t}(y)\end{align}$$ So, it follows from $(1)$ that $$L_{k}^{m}(0.5x+0.5y)=e^{-(x+y)/2}\sum_{i=0}^{k}\bigg(\sum_{s=0}^{\infty}\frac{(x/2)^s}{s!}L_{i}^{m-1+s}(x)\bigg)\bigg(\sum_{t=0}^{\infty}\frac{(y/2)^t}{t!}L_{k-i}^{t}(y)\bigg)$$