Let s(n,k) and S(n,k) denote the Stirling numbers of the first (with signs) and second kinds, respectively. Next consider the sequence |s(n+2,n)| which begins: (2,11,35,85,175,...) . Using this to form a difference table yields:

2 11 35 85 175

9 24 50 90

15 26 40

11 14

3 = 1$\cdot$3 = 3!!

In other words, $\Delta^4$|s(n+2,n)| = 3!! . Similarly, we have the sequence |s(n+3,n)| which begins with (6,50,225,735,1960,4536,9450,...) . After taking differences this ultimately gives $\Delta^6$|s(n+3,n)| = 15 = 1$\cdot$3$\cdot$5 = 5!! . In general, we get $\,$ (D$_1$): $\Delta^{2k}$|s(n+k,n)| = (2k-1)!! . $\;$ Curiously, the exact same result holds for Stirling numbers of the second kind, i.e. (D$_2$): $\Delta^{2k}$S(n+k,n) = (2k-1)!! . $\,$ What lies behind this coincidence?

Questions: (1) The properties D$_1$ and D$_2$ above must be well known. Is there a convenient reference for the proofs?

(2) The two kinds of Stirling numbers form an inverse pair of triangular arrays. This can be expressed as a matrix equation A$\cdot$B = I where both A and B are lower triangular matrices given by A = (a$_{i,j}$) = (s(i,j)) and B =(b$_{i,j}$) = (S(i,j)) . $\,$ For any such inverse pair, either one completely determines the other and therefore, in principle, information about one can be transferred to the other. Of course, this may or may not be practical to do depending on the situation. In general, is there any useful way to relate finite differences of one member of an inverse pair to those of its mate?

(3) Are there any other inverse pairs of arrays whose finite differences are related in an especially simple way?


  • $\begingroup$ Write $c(n,k)=|s(n,k)|$. Denote $\sum_n c(n+k,n) x^{n+k}=f_k(x)$. Then $f_0(x)=x/(1-x)$ and from the relation $c(n+k,n)=c(n+k-1,n-1)+(n+k-1)c(n+k-1,n)$ we get $f_k(x)(1-x)=\sum_n (c(n+k,n)-c(n+k-1,n-1))x^{n+k}=x^2 f_{k-1}'(x)$. This should help. $\endgroup$ – Fedor Petrov Jul 11 '18 at 21:07

Let $c(n,k) = |s(n,k)| = (-1)^{n-k} s(n,k)$. It is well known that for fixed $k$, $S(n+k,n)$ and $s(n+k,n)$ are polynomials in $n$ of degree $2k$ with leading coefficient $(2k-1)!!/(2k)!$. If we extend these polynomials to all values of $n$ then $S(-n+k,-n)=c(n,n-k)$, or with appropriate restrictions on $n$ and $k$, $s(k,n)= c(-n,-k)$. Similar results apply to inverse pairs that come from exponential generating functions for powers of compositional inverse generating functions (for Stirling numbers these are $e^x-1$ and $\log(1+x)$). This is a consequence of Lagrange inversion, or more precisely, a form of Lagrange inversion called the Schur-Jabotinsky theorem.

For more on the duality between Stirling numbers of the first and second kinds, see I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory Ser. A 24 (1978), 24–33, and D. E. Knuth, Two notes on notation, American Mathematical Monthly 99 (1992), 403–422.

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