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Let 
$[a_1(x),a_2(x),a_3(x),\ldots ]$ be the continued fraction expansion of an irrational number 
$x\in [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all 
$x\in [0,1)$, 
$$ \begin{align*} \liminf_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n} = 0\quad \text{and}\quad \limsup_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n}=1. \end{align*} $$
We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.
We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and 
$\Delta _c^n f$ omit zero in the whole complex plane, then either 
$f(z)=\exp (h_1(z)+C_1 z)$, where 
$h_1$ is an entire function of period c and 
$\exp (C_1 c)\neq 1$, or 
$f(z)=\exp (h_2(z)+C_2 z)$, where 
$h_2$ is an entire function of period 
$2c$ and 
$C_2$ satisfies 
$$ \begin{align*} \bigg(\frac{1+\exp(C_2c)}{1-\exp(C_2 c)}\bigg)^{2n}=1. \end{align*} $$
