On a generalization of Taylor–Maclourin formula for classes of Dzrbashyan functions $C^{*\infty}_{\alpha}$

B. A. Sahakyan ; G. S. Kocharyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 2 (2010), pp. 3-11

Résumé

In the paper for any $\rho\geq1$ and an arbitrary increasing sequence of positive numbers $\{\lambda_j\}^{\infty}_0$, the systems of operators and functions are introduced: $\{L^{n/p}_{\infty}\}^{\infty}_0$, $\{\varphi_n(x)\}^{\infty}_0$, $x\in[0, +\infty)$; $L^{0/p}_{\infty}f\equiv f$, $L^{n/p}_{\infty}f\equiv\displaystyle\prod^{n-1}_{j=0}(D^{1/\rho}_{\infty}+\lambda_j)f$, $n\geq 1$, where $L^{1/p}_{\infty}f\equiv\dfrac{d}{dx}D^{\alpha}_{\infty} f$, $D^{-\alpha}_{\infty} f\equiv\dfrac{1}{\Gamma(\alpha)}\displaystyle\int_x^{\infty}(t-x)^{\alpha-1}f(t)dt$, $D^{n/\rho}_{\infty} f\equiv D^{1/\rho}_{\infty} D^{(n-1)/\rho}_{\infty}f(1-\alpha=1/\rho)$; $\varphi_0(x)=\exp(-\lambda^{\rho}_0x)$, $\varphi_n(x)=\displaystyle\sum_{k=0}^nC_k^{(n)}\exp(-\lambda^{\rho}_nx)$, $C_k^{(n)}=\left(\displaystyle\prod^n_{j=0,(j\neq k)}(\lambda_j-\lambda_k)\right)^{-1}.$ Some properties of these systems are investigated, as well as specific differential equations of fractional order are solved. Finally, for some classes of functions Taylor–McLaurens type formulas are obtained.

Détail

Keywords: Weil operators, Taylor–McLaurens type formulas.