The Relation Between Nabla Fractional Differences and Nabla Integer Differences


Jia Baoguo, Lynn Erbe, Christopher Goodrich, Allan Peterson




In this paper we obtain two interrelated results. The first result is the following inequality: extbf{Theorem.} Assume that $f\colon\mathbb N_a\to\mathbb R$ satisfies $\nabla^v_af(t)\geq0$, for each $t\in\mathbb N_{a+1}$, $v>0$, $v\notin\mathbb N_1$, and choose $N\in\mathbb N_1$, such that $N-1<v<N$. Then for each $k\in\mathbb N_{a+N}$, we have \[ abla^{N-1}f(a+k)\geq-um_{i=0}^{N-2}H_{-v+i}(a+k,a+i)abla^if(a+i+1)-um_{k-1}^{i=N}H_{-v+N-2}(a+k,a+i-1)abla^{N-1}f(a+i), \] where \[ H_{-v+N-2}(a+k,a+i-1)=\frac{(k-i+1)^{verline{-v+N+2}}}{ȁmma(-v+N-1)}<0. \] As an application of the above inequality we prove the following result: extbf{Theorem.} Assume that $f\colon\mathbb N_a\to\mathbb R$ satisfies $\nabla^v_af(t)\geq0$, for each $t\in\mathbb N_{a+1}$, where $5<v<6$. Then $\nabla^5f(t)\geq0$, for $t\in\mathbb N_{a+6}$. This demonstrates that, in some sense, the positivity of the $v$-th order fractional difference has a strong connection to the positivity of an integer-order difference of the function $f$.