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The Stirling number of the first kind is defined by (2.7).
Let the q-Bessel function of the second kind be defined in terms of (9)–(11).
The Stirling number of the second kind is defined by the generating function to be ( e t − 1 ) m = m !
As is well known, the Stirling numbers S ( l, n ) of the second kind are defined by the generating function to be ( e t − 1 ) n = n !
The Daehee numbers of the first kind are defined by frac{log(1+t)}{t}=sum_{n=0}^{infty}D_{n} frac{t^{n}}{n!} (cf. [7]).
The familiar q-Stirling numbers S ( n, k ) of the second kind are defined by ( e q ( t ) − 1 ) k [ k ] q !
The Stirling number of the first kind is defined by ( x ) n = ∑ l = 0 n S 1 ( n, l ) x l ( n ≥ 0 ) ( see [16] ).
The Stirling number of the second kind is defined by x n = ∑ l = 0 n S 2 ( n, l ) ( x ) l ( n ≧ 0 ).
The Stirling number of the first kind is defined by ( x ) n = x ( x − 1 ) ⋯ ( x − n + 1 ) = ∑ k = 0 n S 1 ( n, k ) x k. (9).
Stirling numbers of the first kind are defined by bigl(log(1+t bigr)^{n} = n! sum _{m=n}^{infty}S_{1} (m,n) {t^{m} over m!}, (7) and the Stirling numbers of the second kind are defined by bigl(e^{t}-1bigr)^{n}= n! sum _{l=n}^{infty}S_{2} (n,l) frac{t^{l}}{l!} quad (n ge0).
Still by analogy with classical statistic for scalar real random variables defined in ({mathbb {R}^), the second characteristic function (CF) of the second kind is defined as the natural logarithm of the first CF of the second kind.
