If hn is a polynomial in n of degree m, prove that the...

If $h_{n}$ is a polynomial in $n$ of degree $m$, prove that the constants $c_{0}, c_{1}, \ldots, c_{m}$ such that $$h_{n}=c_{0}\left(\begin{array}{l}n \\ 0\end{array}\right)+c_{1}\left(\begin{array}{l}n \\1 \end{array}\right)+\cdots+c_{m}\left(\begin{array}{c}n \\m\end{array}\right)$$ are uniquely determined. (Cf. Theorem 8.2.2.)

If $h_{n}$ is a polynomial in $n$ of degree $m$, prove that the constants $c_{0}, c_{1}, \ldots, c_{m}$ such that
$$h_{n}=c_{0}\left(\begin{array}{l}n \\
0\end{array}\right)+c_{1}\left(\begin{array}{l}n \\1
\end{array}\right)+\cdots+c_{m}\left(\begin{array}{c}n \\m\end{array}\right)$$
are uniquely determined. (Cf. Theorem 8.2.2.)

Key Concepts

Uniqueness of Polynomial Representation

When expressing a polynomial of degree m as a linear combination of any basis – including the binomial coefficient basis – the coefficients in the representation are uniquely determined provided the basis is linearly independent. This uniqueness is a consequence of a fundamental theorem in linear algebra: every vector in a finite-dimensional vector space has a unique representation with respect to any basis of that space, which in this context translates to the unique determination of the constants c0, c1, …, cm in the expansion.

Binomial Coefficient Basis

The binomial coefficients (n choose k) for k = 0, 1, ..., m form a set of polynomials in n that are often used as an alternative basis for the space of polynomials of degree at most m. Unlike the standard monomial basis, these polynomials have combinatorial significance and satisfy useful recursive relations, which can simplify many proofs and computations in combinatorial analysis.

Polynomial Function and Degree

A polynomial of degree m is a function that can be expressed as a sum of terms, each a constant multiplied by a power of the variable, with the highest power being m. This concept is fundamental in algebra and analysis, as it provides a structured way to study and approximate functions using finite sums, ensuring that the behavior of the function is controlled and predictable by its leading term.

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