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Symmetric Quantum Calculus

We generalize the Hahn variational calculus by studying problems of the calculus of variations with higher-order derivatives. The symmetric quantum calculus is studied, namely the $α,β$-symmetric, the $q$-symmetric, and the Hahn symmetric quantum calculus. We introduce the symmetric quantum variational calculus and an Euler-Lagrange type equation for the $q$-symmetric and Hahn's symmetric quantum calculus is proved. We define a symmetric derivative on time scales and derive some of its properties. Finally, we introduce and study the diamond integral, which is a refined version of the diamond-$α$ integral on time scales.