Prove that the following formula holds for the k th-order...

Prove that the following formula holds for the $k$ th-order differences of a sequence $h_{0}, h_{1}, \ldots, h_{n}, \ldots:$
$$\Delta^{k} h_{n}=\sum_{j=0}^{k}(-1)^{k-j}\left(\begin{array}{l}k \\j\end{array}\right) h_{n+j}$$

Key Concepts

Recurrence Relations

Recurrence relations describe sequences where each term is defined as a function of preceding terms. The kth-order differences often emerge in the process of transforming or solving these recurrence relations, helping to simplify or solve for the behavior of the sequence. Understanding the interplay between recurrence relations and finite differences is essential in discrete mathematics and its applications.

Finite Differences

Finite differences are the discrete analogs of derivatives in calculus. They measure the change between successive values in a sequence. The kth-order difference represents the repeated application of the difference operator, analogous to taking higher-order derivatives, and is central to analyzing discrete functions and sequences, particularly in the study of difference equations and numerical methods.

Binomial Coefficients

Binomial coefficients arise naturally in the expansion of expressions and play a key role in combinatorics. In the context of finite differences, they weight the terms in the formula for the kth-order difference, reflecting how each shifted term in the sequence contributes to the overall difference. Their properties and symmetries facilitate the derivation and proof of many identities, including those involving finite differences.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It is particularly useful for proving formulas involving sequences or recursively defined operations, such as finite differences. By demonstrating the base case and proving that if the formula holds for a particular order it also holds for the next, one can validate the formula for all natural numbers.

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