In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A= [α , β; γ , δ] we find out the appropriate cosine operator function C(t)= [a(t), b(t); c(t), d(t)], (t \in R) in a real vector space R2 as the solutions of the Cauchy problem C''(t)=AC(t), C(0)=I, C'(0)=0.

We obtain a formula of decomposition for Φ(A) = A R n S(f (x))ϕ(x) dx + R n ϕ(x) dx using the method of stationary phase. Here (S(t)) t∈R is once integrated, exponentially bounded group of operators in a Banach space X, with generator A, which satisfies the condition: For every x ∈ X there exists δ = δ(x) > 0 such that S(t)x t 1/2+δ → 0 as t → 0. The function ϕ(x) is infinitely differentiable, defined on R n , with values in X, with a compact support supp ϕ, the function f (x) is infinitely differentiable, defined on R n , with values in R, and f (x) on supp ϕ has exactly one nondegenerate stationary point x 0 .

In this paper, we consider the nonlinear superposition operator F in lp spaces of sequences (1 ≤ p ≤ ∞), generated by the function f(s,u)=a(s) + arctan u or f(s,u) = a(s) - arctan u. We find out the Rhodius spectra σR(F) and the Neuberger spectra σN(F) of these operators and finally the radii of these spectra. The superposition operator generated by the function f(s,u) = a(s) ∓ arccot u appears to be a special case of above mentioned operator.

In this paper we give a representation of Stirling numbers of the second kind, we obtain explicit formulas for some cases of Stirling numbers of the second kind and illustrate a method for founding other such formulas. 2010 Mathematics Subject Classification. 11B73, 05A10.