Definition and examples
For nonnegative integers k, the Stirling polynomials Sk(x) are defined by the generating function equation
The first 10 Stirling polynomials are:
Yet another variant of the Stirling polynomials is considered in (see also the subsection on Stirling convolution polynomials below).
Special values include:
S k ( − m ) = ( − 1 ) k ( k + m − 1 k ) S k + m − 1 , m − 1 , where S m , n denotes Stirling numbers of the second kind. Conversely, S n , m = ( − 1 ) n − m ( n m ) S n − m ( − m − 1 ) ; S k ( − 1 ) = δ k , 0 ; S k ( 0 ) = ( − 1 ) k B k , where Bk are Bernoulli numbers under the convention B1 = −1/2; S k ( 1 ) = ( − 1 ) k + 1 ( ( k − 1 ) B k + k B k − 1 ) ; S k ( 2 ) = ( − 1 ) k 2 ( ( k − 1 ) ( k − 2 ) B k + 3 k ( k − 2 ) B k − 1 + 2 k ( k − 1 ) B k − 2 ) ; S k ( k ) = k ! ; S k ( m ) = ( − 1 ) k ( m k ) s m + 1 , m + 1 − k , where s m , n are Stirling numbers of the first kind. They may be recovered by s n , m = ( − 1 ) n − m ( n − 1 n − m ) S n − m ( n − 1 ) .The sequence S k ( x − 1 ) is of binomial type, since S k ( x + y − 1 ) = ∑ i = 0 k ( k i ) S i ( x − 1 ) S k − i ( y − 1 ) . Moreover, this basic recursion holds: S k ( x ) = ( x − k ) S k ( x − 1 ) x + k S k − 1 ( x + 1 ) .
Explicit representations involving Stirling numbers can be deduced with Lagrange's interpolation formula:
Here, L n ( α ) are Laguerre polynomials.
These following relations hold as well:
( k + m k ) S k ( x − m ) = ∑ i = 0 k ( − 1 ) k − i ( k + m i ) S k − i + m , m ⋅ S i ( x ) , where S k , n is the Stirling number of the second kind and
( k − m k ) S k ( x + m ) = ∑ i = 0 k ( k − m i ) s m , m − k + i ⋅ S i ( x ) , where s k , n is the Stirling number of the first kind.
By differentiating the generating function it readily follows that
Closely related to Stirling polynomials are Nørlund polynomials (or generalized Bernoulli polynomials) with generating function
The relation is given by S k ( x ) = B k ( x + 1 ) ( x + 1 ) .
Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article and in the Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as
σ n ( x ) = [ x x − n ] ⋅ 1 x ( x − 1 ) ⋯ ( x − n ) . It follows that these polynomials satisfy the next recurrence relation given by
( x + 1 ) σ n ( x + 1 ) = ( x − n ) σ n ( x ) + x σ n − 1 ( x ) , n ≥ 1. These Stirling "convolution" polynomials may be used to define the Stirling numbers, [ x x − n ] and { x x − n } , for integers n ≥ 0 and arbitrary complex values of x . The next table provides several special cases of these Stirling polynomials for the first few n ≥ 0 .
This variant of the Stirling polynomial sequence has particularly nice ordinary generating functions of the following forms:
( z e z e z − 1 ) x = ∑ n ≥ 0 x σ n ( x ) z n ( 1 z ln 1 1 − z ) x = ∑ n ≥ 0 x σ n ( x + n ) z n . More generally, if S t ( z ) is a power series that satisfies ln ( 1 − z S t ( z ) t − 1 ) = − z S t ( z ) t , we have that
S t ( z ) x = ∑ n ≥ 0 x σ n ( x + t n ) z n . For integers 0 ≤ k ≤ n and r , s ∈ C , these polynomials satisfy the two Stirling convolution formulas given by
( r + s ) σ n ( r + s + t n ) = r s ∑ k = 0 n σ k ( r + t k ) σ n − k ( s + t ( n − k ) ) and
n σ n ( r + s + t n ) = s ∑ k = 0 n k σ k ( r + t k ) σ n − k ( s + t ( n − k ) ) . When n , m ∈ N , we also have that the polynomials, σ n ( m ) , are defined through their relations to the Stirling numbers
{ n m } = ( − 1 ) n − m + 1 n ! ( m − 1 ) ! σ n − m ( − m ) ( when m < 0 ) [ n m ] = n ! ( m − 1 ) ! σ n − m ( n ) ( when m > n ) , and their relations to the Bernoulli numbers given by
σ n ( m ) = ( − 1 ) m + n − 1 m ! ( n − m ) ! ∑ 0 ≤ k < m [ m m − k ] B n − k n − k , n ≥ m > 0 σ n ( m ) = − B n n ⋅ n ! , m = 0. 