Further studies on ordinary differential equations involving the M-fractional derivative

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In the current paper, the power series based on the $ M $-fractional derivative is formally introduced. More peciesely, the Taylor and Maclaurin expansions are generalized for fractional-order differentiable functions in accordance with the $ M $-fractional derivative. Some new definitions, theorems, and corollaries regarding the power series in the $ M $ sense are presented and formally proved. Several ordinary differential equations (ODEs) involving the $ M $-fractional derivative are solved to examine the validity of the results presented in the current study.