An irrational power series over a finite field \(\mathbb{F}_q\) of characteristic \(p\) is called hyperquadratic if it satisfies an algebraic equation of the form \(x=(Ax^r+B)/(Cx^r+D)\), where \(r\) is a power of \(p\) and the coefficients belong to the ring of polynomials \(\mathbb{F}_q[T]\). These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. The following general result is presented in the paper. Theorem. Let \(p\) be a prime number, \(q=p^s\) with an integer \(s\geq1\), \(r=p^t\) with an integer \(t\geq0\). Let \(l\) be an integer with \(l\geq1\). Let \((a_1,\dots,a_l)\in(\mathbb{F}_q[T])^l\) be given with \(\deg a_i>0\) for \(1\leq i\leq l\). Let \(P,Q\) and \(R\) be polynomials in \(\mathbb{F}_q[T]\). We assume that \(PR\neq 0\) and \(\deg Q| T| \), of the following algebraic equation \[ x=(Ax^r+B)/(Cx^r+D), \] where \[ A=Rx_{l,1},\quad B=Px_{l-1,1}-Qx_{l,1},\quad C=Rx_{l-1,2},\quad D=Px_{l-2,2}-Qx_{l-1,2} \] and the sequence of polynomials \(\{x_{i,k}\}_{i\geq-1,k\geq1}\) in \(\mathbb{F}_q[T]\) defined recursively by equalities: \[ x_{-1,k}=0,\quad x_{0,k}=1,\quad x_{i,k}=a_{k+i-1}x_{i-1,k}+x_{i-2,k}. \] The author applies this theorem to describe several families of hyperquadratic expansions having a regular pattern.