Further solutions to the recurrence relation for the Laguerre polynomials

Question

The Laguerre polynomials are the polynomials that satisfy the linear homogoneous recurrence relation $$L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}, \quad k \geq 1,$$ with initial conditions $L_0(x) = 1$ and $L_1(x) = 1-x$. It is known that $$L_n(x) =\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k$$ for all $n \geq 0$. Can one give an explicit formula for another polynomial solution of the same recurrence relation that is not just a polynomial multiple of the Laguerre polynomials?