In this paper, a generalized multivariate fractional Taylor's and Cauchy's mean value theorem of the kind
$$f(x,y) = \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}f({x_{0,}}{y_0})}}{{\Gamma (j\alpha + 1)}}} + R_n^\alpha (\xi,\eta),\qquad\frac{{f(x,y) - \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}f({x_{0,}}{y_0})}}{{\Gamma (j\alpha + 1)}}} }}{{g(x,y) - \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}g({x_{0,}}{y_0})}}{{\Gamma (j\alpha + 1)}}} }} = \frac{{R_n^\alpha (\xi ,\eta )}}{{T_n^\alpha (\xi ,\eta )}},$$
where $0<\alpha \le 1$, is established. Such expression is precisely the classical Taylor's and Cauchy's mean value theorem in the particular case $\alpha=1$. In addition, detailed expressions for $R_n^\alpha (\xi,\eta)$ and $T_n^\alpha (\xi,\eta)$ involving the sequential Caputo fractional derivative are also given.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v52n1.19.04}, url = {https://global-sci.com/article/87714/on-multivariate-fractional-taylors-and-cauchy-mean-value-theorem} }