In this paper, we introduce a parametric type of Cauchy polynomials and study their characteristic combinatorial properties. particular, show some determinant expressions.

We use a fixed point method to prove the Cauchy–Rassias stability of homomorphisms associated to the Pexiderized Cauchy–Jensen type functional equation r f ( x+ y r ) + sg ( x− y s ) = 2h(x), r,s ∈ R\{0}

Let H be an infinite--dimensional Hilbert space and K(H) be the set of all compact operators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate of higher derivation and higher Jordan derivation on K(H) associated with the following cauchy-Jencen type functional equation 2f(frac{T+S}{2}+R)=f(T)+f(S)+2f(R) for all T,S,Rin K(H).

Let H be an innite dimensional Hilbert space and K(H) be the set of all compactoperators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate ofhigher derivation and higher Jordan derivation on K(H) associated with the following Cauchy-Jensentype functional equation 2f((T + S)/2+ R) = f(T ) + f(S) + 2f(R) for all T, S, R are in K(...

We prove that the obstacle problem for a non-uniformly parabolic operator of Kolmogorov type, with Cauchy (or Cauchy-Dirichlet) boundary conditions, has a unique strong solution u. We also show that u is a solution in the viscosity sense.