The sequence spaces \(\ell (p)\) and \( m (p)\) \((= \ell_\infty (p))\), where \(p = (p_k)\) is a sequence of real numbers \(0 < p_k \leq 1\), were introduced by S. Simons [Proc. Lond. Math. Soc., III. Ser. 15, 422-436 (1965; Zbl 0128.33805)]. In 1966, I. J. Maddox introduced the notions of \([A,p]\)-strongly summable and \([A,p]\)-strongly bounded sequences, where \(A = (a_{nk})\) denotes an infinite complex matrix and \(p = (p_k)\) a sequence of strictly positive reals. Through these notions he defined the sequence spaces \([A,p]\) and \([A, p]_\infty\) of \([A,p]\)-strongly summable and \([A,p]\)-strongly bounded sequences, respectively. Various choices of the matrix \(A\) lead to the definition of such sequences spaces as \( \ell (p)\), \(\ell_\infty (p)\), \(c(p)\), \(c_0 (p)\), etc.
Since then extensive work has been done by I. J. Maddox and others with respect to the algebraic and topological structure of these spaces, the characterisation of their various duals, the characterisation of matrix transformation classes between such spaces, etc. In 1981 H. Kizmaz [Can. Math. Bull. 24, 169-176 (1981; Zbl 0454.46010)] introduced the sequence spaces \[ \alpha (\Delta) = \bigl\{ x = (x_k) \mid \Delta x = (\Delta x_k) = (x_k - x_{k + 1}) \in \alpha \bigr\} \] where \(\alpha\) is one of the symbols \(\ell_\infty, c, c_0\) of bounded, convergent and null sequences, respectively. Extending Kizmaz’s definitions in a Maddox way, the present authors define the sequence spaces \[ \text{S} \alpha (p) = \bigl\{ x = (x_k) \mid \Delta x = (\Delta x_k) = (x_k - x_{k + 1}) \in \alpha (p) \bigr\} \] where \(p = (p_k)\) is a sequence of strictly positive reals, and, using rather known techniques, obtain the first and second algebraic duals of \(\text{S} \ell_\infty (p)\) and characterise the matrix transformation class \((\text{S} \ell_\infty (p), \ell)\).