Good morning,
my question is about proving the linear independence of sequences. In the theory of linear difference equations, one needs a fact, that for all distinct $\lambda_1,\ldots,\lambda_m \in \mathbb{C}$ and all $k_1,\ldots,k_m \in \mathbb{N}$, the functions (sequences) $n^{0} \lambda_1^n, n^1 \lambda_1^n, \ldots, n^{k_1 - 1} \lambda_1^n,$ $n^{0} \lambda_2^n, n^1 \lambda_2^n, \ldots, n^{k_2 - 1} \lambda_2^n,$ $\ldots,$ $n^{0} \lambda_m^n, n^1 \lambda_m^n, \ldots, n^{k_m - 1} \lambda_m^n$ are linearly independent. Although this fact is extensively used in literature, I haven't found any satisfactory proof and my attempts to prove this by myself weren't successful as well.
I know how to prove that $\lambda_1^n,\lambda_2^n,\ldots,\lambda_m^n$ are linearly independent for distinct $\lambda_1,\ldots,\lambda_m \in \mathbb{C}$ (using Vandermonde determinant). Moreover, I know how to prove that $n^0 \lambda_i^n, n^1 \lambda_i^n, \ldots, n^{k_i - 1} \lambda_i^n$ are linearly independent (if one supposes that they are dependent, one gets an equation that says that a polynomial with at least one non-zero coefficient is zero for all $n \in \mathbb{N}$, i.e., a contradiction). But I don't know how to combine these results. Simply the fact that both sets of sequences are linearly independent does not prove that their union is linearly independent. Any ideas would be helpful.
Thank you in advance.