On lacunary d-statistical convergence of order ?

The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph puplished in Warsaw in 1935. The consept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and later reintroduced by Schoenberg [4]. Over the years and under different names statistical convergence was discussed in the theory of Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Bhardwaj et al. [5], Bilalov and Nazarova [6], C¸ akallı et al. ([7], [8],[9]), Caserta et al. [10], C¸ ınar et al. ([11],[12]), Connor [13], Et et al. ([14],[15]), Fridy [16], Fridy and Orhan [17], Isık et al. ([18],[19],[20]), Kuc¸¨ ukaslan ¨ et al. ([21],[22]), Mursaleen [23], Salat [24], Savas¸ [25], S¸ engul [26] and many others. The order of statistical con- ¨ vergence of a sequence of numbers was given by Gadjiev and Orhan [27] after then statistical convergence of order ? was studied by C¸ olak [28]. By a lacunary sequence we mean an increasing integer sequence ? = (kr) of non-negative integers such that k0 = 0 and hr = (kr ? kr?1) ? ? as r ? ?. The intervals determined by ? will be denoted by Ir = (kr?1, kr] and the ratio kr kr?1 will be abbreviated by qr , and q1 = k1 for convenience. In recent years, lacunary sequences have been studied in ([7],[8],[9],[29],[30],[31],[32], [17],[33],[34])