Editing Binomial coefficient (section)

=== Binomial coefficients as a basis for the space of polynomials ===
Over any [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic 0]] (that is, any field that contains the [[rational number]]s), each polynomial ''p''(''t'') of degree at most ''d'' is uniquely expressible as a linear combination <math display="inline">\sum_{k=0}^d a_k \binom{t}{k}</math> of binomial coefficients, because the binomial coefficients consist of one polynomial of each degree. The coefficient ''a''<sub>''k''</sub> is the [[finite difference|''k''th difference]] of the sequence ''p''(0), ''p''(1), ..., ''p''(''k''). Explicitly,<ref>This can be seen as a discrete analog of [[Taylor's theorem]]. It is closely related to [[Newton's polynomial]]. Alternating sums of this form may be expressed as the [[Nörlund–Rice integral]].</ref>
{{NumBlk2|:|<math>a_k = \sum_{i=0}^k (-1)^{k-i} \binom{k}{i} p(i).</math>|4}}