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and the Stirling number of the second kind is given by the generating function to be ( e t − 1 ) m = m !

The Stirling number of the first kind is given by ( x ) n = x ( x − 1 ) ⋯ ( x − n + 1 ) = ∑ l = 0 n S 1 ( n, l ) x l. (10).

The recurrence relation for the Stirling numbers of the first kind is given by s_{1}(n+1, k)=s_{1}(n,k-1 -ns_{1}(n,k-1 -ns

For (rin 0,1)), Legendre's complete elliptic integral [27] of the second kind is given by mathcal{E}(r)= int_{0}^{pi/2}sqrt{1-r^{2}sin ^{2}(t)},dt.

A relation between the λ-Bernoulli polynomials (mathfrak{B}_{n}(x;lambda)), the Apostol-Daehee polynomials (mathfrak{D}_{n}(x;lambda)) and the Stirling numbers of the second kind is given by the following theorem.

The Stirling number of the first kind is given by the generating function, (x)_{m} =sum_{l=0}^{m} S_{1}(m,l x^{l} quad (mgeq0) (6) and the Stirling number of the second kind is defined by the generating function, bigl(e^{t}-1bigr)^{m} =m!sum _{l=m}^{infty}S_{2}(l,m frac{t^{l}}{l!} quad (mgeq0) (mbox{see [7, 8, 15, 17]}).

and the Stirling numbers of the second kind are given by ( e t − 1 ) n = n !

Unsigned Stirling numbers of the first kind are given by x n ̲ = x ( x + 1 ) ⋯ ( x + n − 1 ) = ∑ l = 0 n | S 1 ( n, l ) | x l. (1.5).

One of the fruitful generalizations of this kind was given by Branciari in 2000 [1] who replaced the triangular inequality by a more general one, which was later usually called a rectangular or a quadrilateral inequality.

and the higher-order Cauchy polynomials of the second kind are given by ( t ( 1 + t ) log ( 1 + t ) ) α ( 1 + t ) x = ∑ n = 0 ∞ C ˆ n ( x ) t n n ! ( α ∈ Z ≥ 0 ).

Recall that the Korobov numbers (K_{n}(lambda)) of the first kind are given by (sum_{ngeq0}K_{n}(lambda)frac{t^{n}}{n!}=frac{lambda t}{(1+t)^{lambda}-1} ^{lambda}-1}).