Summary: We generalize the string functions \({\mathcal C}_{n,r}(\tau)\) associated with the coset \(\widehat{\text{sl}}(2)_k/u(1)\) to higher string functions \({\mathcal A}_{n,r}(\tau)\) and \({\mathcal B}_{n,r}(\tau)\) associated with the coset \(W(k)/u(1)\) of the \(W\)-algebra of the logarithmically extended \(\widehat{\text{sl}}(2)_k\) conformal field model with positive integer \(k\). The higher string functions occur in decomposing \(W(k)\) characters with respect to level-\(k\) theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered “logarithmic parafermionic characters,” are given by \({\mathcal A}_{n,r}(\tau)\), \({\mathcal B}_{n,r}(\tau)\), \({\mathcal C}_{n,r}(\tau)\), and by the triplet \({\mathcal W}(p)\)-algebra characters of the \((p=k+2,1)\) logarithmic model. We study the properties of \({\mathcal A}_{n,r}\) and \({\mathcal B}_{n,r}\), which nontrivially generalize those of the classic string functions \({\mathcal C}_{n,r}\), and evaluate the modular group representation generated from \({\mathcal A}_{n,r}(\tau)\) and \({\mathcal B}_{n,r}(\tau)\); its structure inherits some features of modular transformations of the higher-level Appell functions and the associated transcendental function \(\Phi\).