Let s(n,j) and S(n,j) denote the (unsigned) first and second kind Stirling numbers, respectively. For fixed k, we consider the sequences s(n+k,n) and S(n+k,n) with n = 1,2,3, ...$\,$ . For example, if k=2, then the first few values of S(n+2,n) are (1,7,25,65,140). $\,$ This generates a difference table:
1 7 25 65 140
6 18 40 75
12 22 35
10 13
3
In other words,$\,$ $\Delta^4$S(n+2,n) = 3 =3!! = 1$\cdot$3 $\;$. $\,$ Next, if we try k=3, then S(n+3,n) begins with (1,15,90,350,1050,2646,5880) . Completing the difference table this time gives $\Delta^6$S(n+3,n) = 15 = 5!! = 1$\cdot$3 $\cdot$5 $\;$ . $\;$ In general, it appears that $\Delta^{2k}$S(n+k,n) = (2k - 1)!! $\;$ . $\,$ Curiously, the same result also seems to hold for 1st kind Stirling numbers, i.e. $\Delta^{2k}$s(n+k,n) = (2k-1)!! $\;$ . Both of these statements can be checked for small values of k . $\,$ Are they true in general?
Questions: (1) $\,$ Is it true that $\,$ $\Delta^{2k}$S(n+k,n) = (2k-1)!! $\;$ ?
(2) $\;$ Is it similarly true that $\Delta^{2k}$s(n+k,n) = (2k-1)!! $\;$ ?
(3) The (signed) 1st and 2nd kind Stirling numbers form an inverse pair which can, for instance, be encoded as a matrix equation: (1st Stirling)(2nd Stirling) = (Identity) . Each array completely determines the other and therefore, in principle, any information about one can be transferred to the other. Of course, it may or may not be practical to do so depending on the circumstances.
a) Given an arbitrary inverse pair, is there a nice way to express finite differences of one in terms of differences of the other?
b) Are there any other inverse pairs where an equality of finite differences holds similar to that of the Stirling pair above? $\qquad$ Thanks