Editing Binomial coefficient (section)

== Binomial coefficients as polynomials ==
For any nonnegative integer ''k'', the expression <math display="inline">\binom{t}{k}</math> can be written as a polynomial with denominator {{math|''k''!}}:
: <math>\binom{t}{k} = \frac{t^\underline{k}}{k!} = \frac{t(t-1)(t-2)\cdots(t-k+1)}{k(k-1)(k-2)\cdots2 \cdot 1};</math>
this presents a [[polynomial]] in ''t'' with [[rational number|rational]] coefficients.

As such, it can be evaluated at any real or complex number ''t'' to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in [[Binomial theorem#Newton's generalized binomial theorem|Newton's generalized binomial theorem]].

For each ''k'', the polynomial <math>\tbinom{t}{k}</math> can be characterized as the unique degree ''k'' polynomial {{math|''p''(''t'')}} satisfying {{math|1=''p''(0) = ''p''(1) = ⋯ = ''p''(''k'' − 1) = 0}} and {{math|1=''p''(''k'') = 1}}.

Its coefficients are expressible in terms of [[Stirling numbers of the first kind]]:
: <math>\binom{t}{k} = \sum_{i=0}^k s(k,i)\frac{t^i}{k!}.</math>
The [[derivative]] of <math>\tbinom{t}{k}</math> can be calculated by [[logarithmic differentiation]]:
: <math>\frac{\mathrm{d}}{\mathrm{d}t} \binom{t}{k} = \binom{t}{k} \sum_{i=0}^{k-1} \frac{1}{t-i}.</math>
This can cause a problem when evaluated at integers from <math>0</math> to <math>t-1</math>, but using identities below we can compute the derivative as:
: <math>\frac{\mathrm{d}}{\mathrm{d}t} \binom{t}{k} = \sum_{i=0}^{k-1} \frac{(-1)^{k-i-1}}{k-i} \binom{t}{i}.</math>

=== 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}}

=== Integer-valued polynomials ===
{{main|Integer-valued polynomial}}
Each polynomial <math>\tbinom{t}{k}</math> is [[integer-valued polynomial|integer-valued]]: it has an integer value at all integer inputs <math>t</math>. (One way to prove this is by induction on ''k'' using [[Pascal's identity]].) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, ({{EquationNote|4}}) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring ''R'' of a characteristic 0 field ''K'', a polynomial in ''K''[''t''] takes values in ''R'' at all integers if and only if it is an ''R''-linear combination of binomial coefficient polynomials.

=== Example ===
The integer-valued polynomial {{math|3''t''(3''t'' + 1) / 2}} can be rewritten as
:<math>9\binom{t}{2} + 6 \binom{t}{1} + 0\binom{t}{0}.</math>