Context
I am studying self-adjoint eigenfunction problems using [1]. I am working through Example 1 on page 54 in [1].
Example 1 (page 54 in [1])
Suppose we have the linear differential operator $$ L[y] = \sum_{s = 1}^2\frac{d^s}{dx^s}\left(r_s(x)\frac{d^sy}{dx^s}\right).$$ Because of the form of the operator, I know that $L[\cdot]$ is formally self adjoint.
Let, $u(x)$ be a solution to the system $$L[u] = 0,$$ and let $v(x)$ be a solution to the adjoint system $$L^*[v] = 0.$$
I want to determine the bilinear concomitant, $J(u,v)$.
Question 1
What is the definition of a bilinear concomitant?
I have found no answer in wikipedia.org, encyclopediaofmath.org, or CRC Math Encyclopedia. A related question [2], seems to indicate that a bilinear concomitant is also known as a conjunct. A search of wikipedia, encyclopediaofmath, and CRC again come up empty handed for conjunct. A related question [4], has a pdf link to [5]. In [5], the term bilinear concomitant is used several times, but I have seen no definition for it.
Question 2
In [1], Zwillinger states that
"$J(u,v)$ is called the bilinear concomitant and is defined by \begin{equation} J(v,w) = \sum_{m=1}^{n} \,\, \sum_{j+k=m-1} (-1)^k \left(\frac{d^k}{dt^k}\left(p_m\,u\right)\right)\left(\frac{d^jv}{dt^j} \right) ." \end{equation}
This has the feeling of some type of determinant of a Hessian, which is what appeared in the entry for comitant in [3].
I am not even sure how to interpret Zwillinger's definition, I think it may be either \begin{equation} J(v,w) = \sum_{m=1}^{n} \,\, \sum_{k=0}^{m-1} (-1)^k \left(\frac{d^{(k)}}{dt^{(k)}}\left(p_m\,u\right)\right)\left(\frac{d^{(m-1-k)}v}{dt^{(m-1-k)}} \right), \end{equation} or \begin{equation} J(v,w) = \sum_{m=1}^{n} \,\, \sum_{k=1}^{m-1} (-1)^k \left(\frac{d^{(k)}}{dt^{(k)}}\left(p_m\,u\right)\right)\left(\frac{d^{(m-1-k)}v}{dt^{(m-1-k)}} \right) \end{equation}
Which, if any, of these two interpretations is the correct reading of Zwillinger's definition for bilinear concomitant?
References
[1] Zwillinger, "Handbook on Differential Equations," Academic Press, Inc. 1989, 1st Edition.
[3] https://encyclopediaofmath.org/wiki/Comitant
[4] When do boundary conditions specify a unique solution to ODEs?
[5] http://people.cs.uchicago.edu/~lebovitz/Eodesbook/bv.pdf