A Historical Survey of Fractional Calculus with Respect to Functions (?-fractional Calculus)

Abstract

Derivatives and integrals of one function with respect to another function are well known from basic calculus, using the chain rule and Riemann–Stieltjes integration. The fractional-order versions of these ideas give rise to a theory of fractional calculus with respect to functions, which is often referred to nowadays as ?-fractional calculus. The history of these operators is longer than most researchers realise, as they have been discovered and re-discovered several times through the decades. In this survey article, we trace the full history of fractional calculus with respect to functions, as well as spotlighting some key properties that are underappreciated in the literature yet very powerful in the understanding and study of these operators. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.