We have the following recurrence relations for the generalized Laguerre polynomials $L_n^{k}$(.) (Recurrence relations) $$L_n^k(x)= L_n^{k+1}(x) -L_{n-1}^{k+1}(x); \quad \forall k \in \mathbb N.$$ Is there a formula for $$L_n(x) - L_{n-1}(x) =?$$ where $L_n(x)$ are the Laguerre polynomials.
Note that, in terms of the Laguerre polynomials $L_n(x)$, we have $$ L_n^0(x)= L_n(x).$$
The connection to be made is with the kernel associated with Laguerre polynomials. More precisely, consider the following function called the Christoffel-Darboux formula:
$$K_n(x,y)=a \dfrac{L_{n}(x)L_{n+1}(y)-L_{n+1}(x)L_n(y)}{x-y}$$
that can be found in a more general form (see formula (5.1.11) p. 101) in the essential (easily "downloadable") book by G. Szegö called "Orthogonal polynomials" or in this wikipedia article.
(for a certain constant $a$ depending on $n$).
Indeed, taking $y=0$ (a case considered on page 102 of Szegö's book), as $L_n(0)=1$, this formula boils down to:
$$xK_n(x,0)=a(L_{n}(x)-L_{n+1}(x))$$
i.e.,
$$L_{n+1}(x)-L_{n}(x)=-\dfrac{1}{a}xK_n(x,0)$$
Remark: Christoffel-Darboux/kernel formulas exist for all families of orthogonal polynomials.