The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation
\({[{{p_1}(z)f({z + c})}]^2} + {\left[{{p_2}(z)f(z) + {p_3}(z)f^{\prime}(z)}\right]^2} = p(z),\)
or
\({[{{p_1}(z)f(z)}]^2} + {\left[{{p_2}(z)f^{\prime}(z) + {p_3}(z)f(z+c)}\right]^2} = p(z)\)
or
\({\left[{{p_1}(z)f^{\prime}(z)}\right]^2} + {[{{p_2}(z)f(z+c) + {p_3}(z)f(z)}]^2} = p(z),\)
where c is a nonzero complex number, p1, p2 and p3 are polynomials in \(\mathbb{C}\) satisfying p1p3 ≢ 0, and p is a nonzero irreducible polynomial in \(\mathbb{C}\) .
