Stirling numbers of the first kind
Simple identities

Using the Kronecker delta one has,

and

Also

and

Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials. Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences. Generalizations of the Stirling numbers of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the Stirling convolution polynomials.

These identities may be derived by enumerating permutations directly.For example, a permutation of n elements with n − 3 cycles must have one of the following forms:

The three types may be enumerated as follows:

Sum the three contributions to obtain

Note that all the combinatorial proofs above use either binomials or multinomials of n {\displaystyle n}.

Therefore if p {\displaystyle p} is prime, then:

p | {\displaystyle p\ |\left} for 1 < k < p {\displaystyle 1<k<p}.
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