GENERALIZED DOBINSKI FORMULAS Peter Bala, June 2014 Let S = (S(n,k))n,k>=1 denote the triangle of Stirling numbers of the second kind, A008277 / 1 | 1 1 S = | 1 3 1 | 1 7 6 1 | ... | The row generating polynomials of S are known as the Bell polynomials and denoted by Bell(n,x) Bell(n,x) = sum {j = 1..n} S(n,j)*x^j. An alternative generating function for the Bell polynomials is given by the formula Bell(n,x) = exp(-x)*sum {k = 1..inf} k^n*x^k/k!. (1) The particular case of this result when x = 1 is due to Dobinski (1877) and we shall refer to (1) as Dobinski's formula. We give two simple generalizations of this result. Let L = (L(n,k))n,k>=1 be an arbitrary lower triangular matrix / L(1,1) | L(1,2) L(2,2) L = | L(1,3) L(3,2) L(3,3) | ... | with row and column indexing starting at 1, so that the row generating polynomials P(n,x) of the triangle L begin P(1,x) = L(1,1)*x, P(2,x) = L(1,2)*x + L(2,2)*x^2 P(3,x) = L(1,3)*x + L(3,2)*x^2 + L(3,3)*x^3 and so on. Proposition A. The row polynomials P(n,x) of the lower triangular array L and the row polynomials Q(n,x) of the lower triangular array L*S are related by the Dobinski-type formula Q(n,x) = exp(-x)*sum {k = 1..inf} P(n,k)*x^k/k!. (2) The original Dobinski formula (1) is the particular case when L equals the infinite identity matrix. Proof. The rhs of (2) is exp(-x)*sum {k = 1..inf} P(n,k)x^k/k! = exp(-x)*sum {k = 1..inf} (sum{i = 1..n} L(n,i)*k^i)*x^k/k! = sum{i = 1..n} L(n,i)*exp(-x)*(sum {k = 1..inf} (k^i)*x^k/k!) = sum {i = 1..n} L(n,i)*Bell(i,x) from (1) = sum {i = 1..n} L(n,i)*(sum {j = 1..i} S(i,j)*x^j) = sum {i = 1..n} L(n,i)*(sum {j = 1..n} S(i,j)*x^j) = sum {j = 1..n} [sum {i = 1..n} L(n,i)*S(i,j)]*x^j = Q(n,x) since the sum in square brackets in the penultimate line is equal to the (n,j)-th entry of the matrix product L*S. End proof. We call the arrays L and L*S a Dobinski pair. For example, the arrays S^k and S^(k+1), for any integer k, will form a Dobinski pair. The case k = 0 is the original Dobinski formula (1). If we take k = -1 then Proposition A gives a Dobinski-type relation between the falling factorials and the monomial polynomials x^n = exp(-x)*sum {k = 1..inf} k*(k-1)*...*(k-n+1)*x^k/k!. (3) Some other examples of Dobinski pairs are listed in the table below. The notation | | indicates the unsigned version of the array is being considered. Pascal denotes Pascal's triangle A007318 but taken with a row and column offset of 1 (so that the row generating polynomials of Pascal are x*(1 + x)^(n-1) for n >= 1). Dobinski pairs = = = = = = = = = = = = = = = = = = = = = = = = = = = L L*S = = = = = = = = = = = = = = = = = = = = = = = = = = = Identity matrix A008277 Pascal A143494 Pascal^2 A143495 Pascal^3 A143496 Pascal^4 A193685 A008275 Identity matrix |A008275| A105278 A039683 A122850 |A039683| A035342 A051141 A004747 (signed version) |A051141| A035469 A051142 A000329 (signed version) |A051142| A049029 A209849 A075497 A079641 (signed version) A080417 A039757 2*Identity matrix A Type B generalization of Dobinski's formula Let StB denote the triangle of type B Stirling numbers of the second kind, A039755 / 1 | 1 1 StB = | 1 4 1 | 1 13 9 1 | ... | where now we take the row and column indexing as starting at 0. Let L = (L(n,k))n,k>=0 be an arbitrary lower triangular matrix / L(0,0) | L(1,0) L(1,1) L = | L(2,0) L(2,1) L(2,2) | ... | with row generating polynomials P(n,x) beginning P(0,x) = L(0,0), P(1,x) = L(1,0) + L(1,1)*x, P(2,x) = L(2,0) + L(2,1)*x + L(2,2)*x^2, etc. The following type B version of Dobinski's formula can be proved in an exactly similar fashion to Proposition A. Proposition B. The row polynomials P(n,x) of the lower triangular array L and the row polynomials R(n,x) of the lower triangular array L*StB are related by the Dobinski-type formula R(n,x) = exp(-x/2)*sum {k = 0..inf} P(n,2*k+1)*(x/2)^k/k!. (4) Examples Using the theory of exponential Riordan arrays it can be shown that the triangle A113278 is of the form L*StB where / 1 | 0 1 L = | 0 -2 1 | 0 8 -6 1 | ... | (compare with A039683) is the triangle of coefficients of the generalized falling factorial polynomials P(n,x) defined as P(0,x) = 1 and P(n,x) = x*(x-2)*(x-4)*...*(x-2*(n-1)) for n >= 1. Thus, by Proposition B, the n-th row polynomial R(n,x) of A113278 is given by the type B Dobinski formula R(n,x) = exp(-x/2)*sum {k = 0..inf} (2*k+1)*(2*k-1)*...*(2*k+1-2*(n-1))*(x/2)^k/k!. As a second example, if we take L = A008275, the triangle of Stirling nubers of the first kind then, again using the theory of exponential Riordan arrays, it is easy to show that L*StB = A122848. Hence, by Proposition B, the n-th row polynomial R(n,x) of A122848 is given by the type B Dobinski formula R(n,x) = exp(-x/2)*sum {k = 0..inf} (2*k+1)*(2*k)*...*(2*k+1-n)*(x/2)^k/k!.