Let \(\theta\) and \(\rho\) be real numbers with \(0\leq \theta, \rho <1\) and \(\theta\) irrational. For a complex number \(z\) with \(|z|<1\), set \[ h_{\theta,\rho}(z)=\sum_{k\geq 1} \lfloor k\theta+\rho \rfloor z^k. \] This function is a generalization of the Hecke function \(h_{\theta}(z)\). Similarly, the function \[ F_{\theta,\rho}(z_1,z_2) =\sum_{k_1 \geq 1} \sum_{k_2=1}^{\lfloor k_1\theta+\rho \rfloor} z_1^{k_1} z_2^{k_2} \] generalizes the Mahler function \(F_{\theta}(z_1,z_2)\). The following theorem is proved in the article.
Let \(\theta\) and \(\rho\) be real numbers with \(0 \leq \theta, \rho \leq 1\) and \(\theta\) irrational. Let \(\alpha,\beta\) be nonzero complex algebraic numbers such that \(|\beta \alpha^\theta|<1\) and \(\beta \neq 1\). Then the complex numbers \(\xi_{\mathbf{s}_{\theta,\rho}}(\beta,\alpha)\) and \(\xi_{\mathbf{s'}_{\theta,\rho}}(\beta,\alpha)\) are transcendental. In particular, if \(|\beta|<1\), then the complex numbers \[ h_{\theta,\rho}(\beta), \; F_{\theta,\rho}(\beta,\alpha) \] are transcendental, where \[ \xi_{\mathbf{s}_{\theta,\rho}}(\beta,\alpha)=\sum_{n \geq 1} \big(\lceil n\theta+\rho\rceil -\lceil (n-1)\theta +\rho\rceil \big) \beta^n \alpha^{\lfloor n\theta+\rho\rfloor}, \]
\[ \xi_{\mathbf{s'}_{\theta,\rho}}(\beta,\alpha)=\sum_{n \geq 1} \big(\lfloor n\theta+\rho\rfloor -\lfloor (n-1)\theta +\rho\rfloor \big) \beta^n \alpha^{\lfloor n\theta+\rho\rfloor} \] Let \(a\) and \(b\) be positive integers with \(b \geq 2\). The continued fraction expansion of the real number of the following form \[ (b-1)\xi_{\mathbf{s}_{\theta,\rho}}(1/b,1/a) \quad \text{or} \quad (b-1) \xi_{\mathbf{s'}_{\theta,\rho}}(1/b,1/a), \] is obtained in the paper.