Transcendence of certain infinite products

We prove the transcendence results for the infinite product , where Ek(x), Fk(x) are polynomials, α is an algebraic number, and r⩾2 is an integer. As applications, we give necessary and sufficient conditions for transcendence of and , where Fn and Ln are Fibonacci numbers and Lucas numbers respectively, and {ak}k⩾0 is a sequence of algebraic numbers with log‖ak‖=o(rk).