Mingarelli identity

Consider the n solutions of the following (uncoupled) system of second order linear differential equations over the t–interval [a, b]:

${\displaystyle (p_{i}(t)x_{i}^{\prime })^{\prime }+q_{i}(t)x_{i}=0,\,\,\,\,\,\,\,\,\,\,x_{i}(a)=1,\,\,x_{i}^{\prime }(a)=R_{i}}$ where ${\displaystyle i=1,2,\ldots ,n}$.

Let ${\displaystyle \Delta }$ denote the forward difference operator, i.e.

${\displaystyle \Delta x_{i}=x_{i+1}-x_{i}.}$

The second order difference operator is found by iterating the first order operator as in

${\displaystyle \Delta ^{2}(x_{i})=\Delta (\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i},}$,

with a similar definition for the higher iterates. Leaving out the independent variable t for convenience, and assuming the x_i(t) ≠ 0 on (a, b], there holds the identity,^[2]

${\displaystyle {\begin{aligned}x_{n-1}^{2}\Delta ^{n-1}(p_{1}r_{1})]_{a}^{b}=&\int _{a}^{b}(x_{n-1}^{\prime })^{2}\Delta ^{n-1}(p_{1})-\int _{a}^{b}x_{n-1}^{2}\Delta ^{n-1}(q_{1})\\&-\sum _{k=0}^{n-1}C(n-1,k)(-1)^{n-k-1}\int _{a}^{b}p_{k+1}W^{2}(x_{k+1},x_{n-1})/x_{k+1}^{2},\end{aligned}}}$

where

• ${\displaystyle r_{i}=x_{i}^{\prime }/x_{i}}$ is the logarithmic derivative,
• ${\displaystyle W(x_{i},x_{j})=x_{i}^{\prime }x_{j}-x_{i}x_{j}^{\prime }}$, is the Wronskian determinant,
• ${\displaystyle C(n-1,k)}$ are binomial coefficients.

When n = 2 this equality reduces to the Picone identity.