Whereas the results in §3 are based on a theorem exhibiting the principal part of a Wronskian of smoothly-varying functions, ( [2] , Th. 9, p. 18), the results in this section are based on claims in ( [2] , Th. 10-(III) and Th. 10-(IV), pp. 19-20) exhibiting the principal part of a Wronskian of rapidly-varying functions under special assumptions. We restate the needed claims correcting some misprints in the original statements.

Lemma 8.1. (Principal parts of Wronskians of rapidly-varying functions). Let the functions ${\varphi }_i\in C^{n-1}\left[T,+\infty \right),\left(i=1,\cdots ,n\right),n\ge 2$, satisfy the following properties:

$\{\begin{array}{l}{\varphi }_i^{\left(k\right)}\left(x\right)\ne 0\forall x\text{large}\text{enough}\text{and}0\le k\le n-1;\hfill \\ \underset{x\to +\infty }{\lim }x{{\varphi }^{\prime }}_i\left(x\right)/{\varphi }_i\left(x\right)=\pm \infty \left(\text{the}\text{sign}\pm \text{depending}\text{on}i\right);\hfill \\ {\varphi }_i^{\left(k\right)}\left(x\right)/{\varphi }_i\left(x\right)~{\left({{\varphi }^{\prime }}_i\left(x\right)/{\varphi }_i\left(x\right)\right)}^k,x\to +\infty ,\left(1\le k\le n-1,1\le i\le n\right);\hfill \end{array}$(8.1)

and this means that each ${\varphi }_i$ is rapidly varying at $+\infty$ and, for $n\ge 3$, it is rapidly varying at $+\infty$ of order $n-2$ in the strong restricted sense of Definition 1.3 in [1] . Then:

(I) Under the further assumption that

$\frac{{{\varphi }^{\prime }}_1\left(x\right)}{{\varphi }_1\left(x\right)}\gg \cdots \gg \frac{{{\varphi }^{\prime }}_n\left(x\right)}{{\varphi }_n\left(x\right)},x\to +\infty ,$(8.2)

there exists a permutation $\left({\varphi }_{i_1},\cdots ,{\varphi }_{i_n}\right)$ forming an asymptotic scale at $+\infty$ and

$\begin{array}{l}W\left({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right)\right)\\ ~{\left(-1\right)}^{n\left(n-1\right)/2}\left(\underset{i=1}{\overset{n}{{\displaystyle \prod }}}{\varphi }_i\left(x\right)\right)\cdot \left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}{\left(\frac{{{\varphi }^{\prime }}_i\left(x\right)}{{\varphi }_i\left(x\right)}\right)}^{n-i}\right),x\to +\infty .\end{array}$(8.3)

(II) Under the further assumption that

${{\varphi }^{\prime }}_i\left(x\right)/{\varphi }_i\left(x\right)=a_i\varphi \left(x\right)+o\left(\varphi \left(x\right)\right),x\to +\infty ,1\le i\le n,$(8.4)

for some fixed function $\varphi$ and arbitrary constants $a_i$ (possibly $a_i=0$ for some $i$), we have:

$\begin{array}{l}W\left({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right)\right)\\ =\left(\underset{i=1}{\overset{n}{{\displaystyle \prod }}}{\varphi }_i\left(x\right)\right)\cdot {\left(\varphi \left(x\right)\right)}^{n\left(n-1\right)/2}\cdot \left[V\left(a_1,\cdots ,a_n\right)+o\left(1\right)\right],x\to +\infty .\end{array}$(8.5)

If $\varphi$ is strictly positive and $a_1,\cdots ,a_n$ are pairwise distinct then condition “ $a_1>\cdots >a_n$” implies that $\left({\varphi }_1,\cdots ,{\varphi }_n\right)$ is an asymptotic scale at $+\infty$ and

$\begin{array}{l}W\left({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right)\right)\\ ~V\left(a_1,\cdots ,a_n\right)\cdot \left(\underset{i=1}{\overset{n}{{\displaystyle \prod }}}{\varphi }_i\left(x\right)\right)\cdot {\left(\varphi \left(x\right)\right)}^{n\left(n-1\right)/2},x\to +\infty ;\end{array}$(8.6)

whereas “ $a_1<\cdots <a_n$” implies that $\left({\varphi }_n,\cdots ,{\varphi }_1\right)$ is an asymptotic scale at $+\infty$ and the foregoing relation is still true.

As in Part I our approach starts with a preliminary example.

Example 8.1. For the function

$\varphi \left(x\right):=\text{exp}\left(c_1{\text{e}}^{c_{2}x}\right),c_1\ne 0,c_2>0;R\left(x\right):=c_1{\text{e}}^{c_{2}x},$(8.7)

by grouping and reordering the terms of the sum in (4.5a), we get:

$\{\begin{array}{l}{R}^{\left(k\right)}\left(x\right)=c_1{c}_2^k{\text{e}}^{c_{2}x},\left(k\ge 1\right);\hfill \\ {\psi }_k\left(x\right):=\frac{{\varphi }^{\left(k\right)}\left(x\right)}{\varphi \left(x\right)}={\left(c_1{c}_2\right)}^k\exp \left(k{c}_{2}x\right)+\underset{j=1}{\overset{k-1}{{\displaystyle \sum }}}{a}_j\exp \left(c_{2}jx\right),\hfill \\ \left(\text{with suitable coefficients}{a}_j\right);\hfill \end{array}$(8.8)

whence:

$\{\begin{array}{l}{{\psi }^{\prime }}_k\left(x\right)~{\left(c_1{c}_2\right)}^{k}k{c}_2\exp \left(k{c}_{2}x\right),\hfill \\ {{\psi }^{″}}_k\left(x\right)~{\left(c_1{c}_2\right)}^k{\left(k{c}_2\right)}^2\exp \left(k{c}_{2}x\right),\hfill \\ {{\psi }^{″}}_k\left(x\right)/{{\psi }^{\prime }}_k\left(x\right)~k{c}_2,\hfill \end{array}x\to +\infty ,\left(k\ge 1\right).$(8.9)

For the Wronskian $W\left({{\psi }^{\prime }}_1,\cdots ,{{\psi }^{\prime }}_{n-1}\right)$ in (4.2) this is the situation in Lemma 8.1-(II) with $\varphi \left(x\right)\equiv 1$, and (8.6) yields:

$W\left({{\psi }^{\prime }}_1,\cdots ,{{\psi }^{\prime }}_{n-1}\right)~V\left(c_2,2c_2,\cdots ,\left(n-1\right)c_2\right)\cdot \underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}{{\psi }^{\prime }}_i\left(x\right),x\to +\infty ;$

whence, by ( [2] , formula (61)), the final formula follows:

$\begin{array}{l}{H}_n\left[\exp \left(c_1{\text{e}}^{c_{2}x}\right)\right]~\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right)c_1^{n\left(n-1\right)/2}{c}_2^{n\left(n-1\right)}\\ \times \exp \left(n\left(n-1\right)c_{2}x/2\right)\cdot \exp \left(n{c}_1{\text{e}}^{c_{2}x}\right),x\to +\infty ,\left(n\ge 2\right);\end{array}$(8.10)

with $c_1\ne 0,c_2>0$, a formula to be commented on in Proposition 9.4 below.

(Let the reader keep in mind that a notation $f\in {ℛ}_{\pm \infty }\left(+\infty \right)$ contains the restriction “ $f\left(x\right)>0$ for all $x$ large enough”, whereas a notation $f\in \left\{{ℛ}_{\pm \infty }\left(+\infty \right)\text{of}\text{order}k\ge 1\right\}$ implies that “ $f\left(x\right)$ has the same strict sign for all $x$ large enough”.)

Let us consider a function of type:

$\varphi \left(x\right):=\text{exp}\left(R\left(x\right)\right)\text{where}R\in \left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}2n-2\right\},$(8.11)

where, for the argument’s sake, we restrict our attention to a function $R$ positively divergent as $x\to +\infty$. The algebraic structure of “ $\varphi =\text{exp}\circ R$” implies, by ( [3] , Prop.7.6-(III), p. 827) that

$\varphi \in \left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}2n-2\right\},$(8.12)

whence, by (1.33) the following relations follow:

$\begin{array}{l}{\psi }_k\left(x\right):={\varphi }^{\left(k\right)}\left(x\right)/\varphi \left(x\right)\\ ~{\left({\varphi }^{\prime }\left(x\right)/\varphi \left(x\right)\right)}^k={\left(R^{\prime }\left(x\right)\right)}^k,x\to +\infty ,\left(k\ge 1\right),\end{array}$(8.13)

as in the situation of Theorem 5.1, formula (5.9). The special assumption on $R$ implies that all the derivatives of its are positively divergent as $R$ itself, as noticed in ( [4] , Def. 4.1, p. 807), so that the right-hand side in the representation (4.5a) is a linear combination, with positive coefficients, of functions in the class ${ℛ}_{+\infty }\left(+\infty \right)$ and, by Proposition 2.7, inference in (2.23), the function ${\psi }_k\left(x\right)$ and its derivatives belong to this class, i.e.

${\psi }_k^{\left(i\right)}\in {ℛ}_{+\infty }\left(+\infty \right),\left(i\ge 1\right),$(8.14)

We now want to apply Lemma 8.1 to the Wronskian in (4.2) and this procedure requires complete information on the order of rapid variation of the involved functions as well as on the asymptotic behaviors of the functions ${{\psi }^{″}}_k/{{\psi }^{\prime }}_k,$ $1\le k\le n-1$. In this heuristic sketch, we pay no attention to the order of rapid variation and try to guess the mentioned asymptotic behaviors. In the proof of Theorem 5.1 direct calculations were bypassed by the use of a preliminary result reported in §2; in the present situation, by formally differentiating twice the asymptotic relation in (8.13) and using (1.32) in estimating $R^{″}$, we obtain:

$\{\begin{array}{c}{{\psi }^{\prime }}_k\left(x\right)~k{\left(R^{\prime }\left(x\right)\right)}^{k-1}{R}^{″}\left(x\right)~k{\left(R^{\prime }\left(x\right)\right)}^{k-1}{\left(R^{\prime }\left(x\right)\right)}^2{\left(R\left(x\right)\right)}^{-1}\\ ~k{\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-1},x\to +\infty ,\left(k\ge 1\right);\\ {{\psi }^{″}}_k\left(x\right)~k\left(k+1\right){\left(R^{\prime }\left(x\right)\right)}^k{R}^{″}\left(x\right){\left(R\left(x\right)\right)}^{-1}-k{\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-2}\\ =k\left(k+1\right){\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-2}\left[1+o\left(1\right)\right]-k{\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-2}\\ ~k^2{\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-2},x\to +\infty ,\left(k\ge 1\right);\end{array}$(8.15)

whence:

${{\psi }^{″}}_k\left(x\right)/{{\psi }^{\prime }}_k\left(x\right)~k\left(R^{\prime }\left(x\right)/R\left(x\right)\right),x\to +\infty ,\left(k\ge 1\right),$(8.16)

and applying Lemma 8.1 we get a final result. To make rigorous the preceding reasonings notice that formal differentiation of an asymptotic relation is not granted even between two functions having the same type of asymptotic variation, (rapid variation in the present case), as shown in ( [5] , §7.2, Counterexamples 2 and 3, p. 472). Hence, the very first steps in both chains of relations in (8.15) require proofs, whereas the subsequent steps are legitimate.

Theorem 8.2 For a function of type:

$\varphi \left(x\right):=\text{exp}\left(R\left(x\right)\right)\text{where}R\in \left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}2n-2\right\},$(8.17)

the following relation holds true:

$\begin{array}{l}{H}_n\left[\text{exp}\left(R\left(x\right)\right)\right]~\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right)\cdot {\left(R\left(x\right)\right)}^{-n\left(n-1\right)/2}\cdot {\left(R^{\prime }\left(x\right)\right)}^{n\left(n-1\right)}\\ \times \exp \left(nR\left(x\right)\right),x\to +\infty ,\left(n\ge 2\right),\end{array}$(8.18)

trivially checked for $n=2$ using both (4.4) (rewritten at the outset of this paper) and (8.20) below.

After the proof, we compare the two asymptotic relations (5.5) and (8.18).

Proof. Under the assumptions in (8.17) we have $\varphi \left(x\right)\in {ℛ}_{\pm \infty }\left(+\infty \right)$ according as “ $R\left(+\infty \right)=\pm \infty$” by Proposition 1.3-(I). But, using a result about compositions of functions, ( [3] , Prop.7.6-(III), p. 827), we get, more precisely, that: $\varphi \left(x\right)\in \left\{{ℛ}_{\pm \infty }\left(+\infty \right)\text{of}\text{order}2n-2\right\}$.

First step. All the terms in the right-hand side in (4.5a) belong to the same class of $R$ but with the pertinent order, namely $\left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}2n-2-k\right\}$, by a result on products of higher-order varying functions, ( [3] , Prop. 7.3-(II), p. 821), because the highest derivative appearing therein is $R^{\left(k\right)}$ which belongs to this class. This would imply that

${\psi }_k\in \left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}2n-2-k\right\}\left(\text{irrespective}\text{of}R\left(+\infty \right)=\pm \infty \right).$

Second step. We shall now have recourse to Proposition 2.8-(I) to prove the following stronger relation:

${\psi }_k\in \left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}2n-3\right\}\left(\text{irrespective}\text{of}R\left(+\infty \right)=\pm \infty \right).$(8.19)

To this end, we need to estimate the growth-order of each term into the summation symbol in (4.5a) and we shall repeatedly use the basic relation (1.33), referred to $R$ and written in the form:

$R^{\left(j\right)}\left(x\right)~{\left(R^{\prime }\left(x\right)\right)}^j\cdot {\left(R\left(x\right)\right)}^{1-j},x\to +\infty ,$(8.20)

for all indexes $1\le i\le 2n-1$. We have:

$\begin{array}{l}\underset{1\le p\le k}{{\displaystyle \prod }}{\left(R^{\left(p\right)}\left(x\right)\right)}^{i_p}~\underset{1\le p\le k}{{\displaystyle \prod }}{\left(R^{\prime }\left(x\right)\right)}^{p{i}_p}\cdot {\left(R\left(x\right)\right)}^{\left(1-p\right)i_p}\\ ={\left(R^{\prime }\left(x\right)\right)}^{i_1+2i_2+\cdots +k{i}_k}{\left(R\left(x\right)\right)}^{\left(1-1\right)i_1+\left(1-2\right)i_2+\cdots +\left(1-k\right)i_k}\\ ={\left(R^{\prime }\left(x\right)\right)}^k{\left(R\left(x\right)\right)}^{i_1+i_2+\cdots +i_k-k}\stackrel{\text{by}\left(\text{4}\text{.5b}\right)}{=}O\left({\left(R^{\prime }\left(x\right)\right)}^k{\left(R\left(x\right)\right)}^{-1}\right)\\ =o\left({\left(R^{\prime }\left(x\right)\right)}^k\right),\end{array}$(8.21)

and it follows that the first isolated term in (4.5a) dominates (as growth-order) each term into the summation symbol. Hence, no matter how the terms into the summation symbol in (4.5a) are ordered, Proposition 2.8-(I) can be applied inferring (8.19).

Third step. We now proceed to finding out the asymptotic behaviors of ${{\psi }^{\prime }}_k,{{\psi }^{″}}_k$. We rewrite (4.5a) as

$\{\begin{array}{l}{\psi }_k\left(x\right)={\left(R^{\prime }\left(x\right)\right)}^k+{\displaystyle {\sum }_{\cdots }^{\cdots }{a}_{i_1,\cdots ,i_k}{P}_k\left(x\right)}\\ P_k\left(x\right):={\displaystyle {\prod }_{1\le p\le k}{\left(R^{\left(p\right)}\left(x\right)\right)}^{i_p}},\end{array}$(8.22)

though using an imprecise notation as $P_k$ depends on the $k$-tuple $\left(i_1,\cdots ,i_k\right)$, and get:

$\begin{array}{c}{P^{\prime }}_k\left(x\right)=\underset{j=1}{\overset{k}{{\displaystyle \sum }}}\left[i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{i_j-1}{R}^{\left(j+1\right)}\left(x\right)\cdot \underset{1\le p\le k}{\overset{p\ne j}{{\displaystyle \prod }}}{\left(R^{\left(p\right)}\left(x\right)\right)}^{i_p}\right]\\ =\underset{j=1}{\overset{k}{{\displaystyle \sum }}}\left[i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-1}{R}^{\left(j+1\right)}\left(x\right)\cdot \underset{1\le p\le k}{{\displaystyle \prod }}{\left(R^{\left(p\right)}\left(x\right)\right)}^{i_p}\right]\\ \equiv \underset{1\le j\le k}{{\displaystyle \sum }}\left[i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-1}{R}^{\left(j+1\right)}\left(x\right)\cdot P_k\left(x\right)\right].\end{array}$(8.23)

For the factors in the last sum we repeatedly use (8.20) so getting:

$\begin{array}{l}{P}_k\left(x\right)~\underset{1\le p\le k}{{\displaystyle \prod }}{\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^{p{i}_p}{\left(R\left(x\right)\right)}^{i_p}\\ \equiv {\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^{i_1+2i_2+\cdots +k{i}_k}R{\left(x\right)}^{i_1+\cdots +i_k}\\ \equiv {\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^{k}R{\left(x\right)}^{i_1+\cdots +i_k}\\ \stackrel{\text{by}\left(\text{4}\text{.5b}\right)}{=}O\left({\left(R^{\prime }\left(x\right)\right)}^k{\left(R\left(x\right)\right)}^{-1}\right);\end{array}$(8.24)

and

$\begin{array}{l}{\left(R^{\left(j\right)}\left(x\right)\right)}^{-1}{R}^{\left(j+1\right)}\left(x\right)\cdot P_k\left(x\right)\\ ~{\left(R^{\prime }\left(x\right)\right)}^{-j}\cdot {\left(R\left(x\right)\right)}^{j-1}{\left(R^{\prime }\left(x\right)\right)}^{j+1}{\left(R\left(x\right)\right)}^{-j}\cdot P_k\left(x\right)\\ =R^{\prime }\left(x\right){\left(R\left(x\right)\right)}^{-1}\cdot O\left({\left(R^{\prime }\left(x\right)\right)}^k{\left(R\left(x\right)\right)}^{-1}\right)\\ =O\left({\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-2}\right);\end{array}$(8.25)

both estimates depending solely on $k$ and not on the many indexes. Replacing (8.25) into (8.23) gives:

$\{\begin{array}{l}{P^{\prime }}_k\left(x\right)=O\left({\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-2}\right);\\ {\left({\displaystyle {\sum }_{\cdots }^{\cdots }{a}_{i_1,\cdots ,i_k}{P}_k\left(x\right)}\right)}^{\prime }=O\left({\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-2}\right);\end{array}$(8.26)

and from (8.22) the estimate for ${{\psi }^{\prime }}_k$ follows:

$\begin{array}{l}{{\psi }^{\prime }}_k\left(x\right)=k{\left(R^{\prime }\left(x\right)\right)}^{k-1}\cdot R^{″}\left(x\right)+O\left({\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-2}\right)\\ \stackrel{\text{by}\left(8.20\right)}{=}k{\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-1}+O\left({\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-2}\right)\\ ~k{\left(R^{\prime }\left(x\right)\right)}^{k+1}{\left(R\left(x\right)\right)}^{-1},x\to +\infty ,\left(k\ge 1\right);\end{array}$(8.27)

the last relation depending on the fact that $\left|R\left(x\right)\right|=+\infty \left(1\right)$, $x\to +\infty$. As concerns ${{\psi }^{″}}_k$ we have from (8.23):

$\begin{array}{c}{P^{″}}_k\left(x\right)=\underset{1\le j\le k}{{\displaystyle \sum }}\{\left[-i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-2}{\left(R^{\left(j+1\right)}\left(x\right)\right)}^2+i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-1}{R}^{\left(j+2\right)}\left(x\right)\right]\cdot P_k\left(x\right)\\ +i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-1}{R}^{\left(j+1\right)}\left(x\right){P^{\prime }}_k\left(x\right)\},\end{array}$(8.28)

wherein each single factor ${\left(R^{\left(m\right)}\left(x\right)\right)}^h$ belongs to one of the classes ${ℛ}_{\pm \infty }\left(+\infty \right)$ with the appropriate order. For the quantity in square brackets we have the following estimate:

$\begin{array}{l}\left[-i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-2}{\left(R^{\left(j+1\right)}\left(x\right)\right)}^2+i_j{\left(R^{\left(j\right)}\left(x\right)\right)}^{-1}{R}^{\left(j+2\right)}\left(x\right)\right]\\ \stackrel{\text{by}\left(8.20\right)}{=}-i_j{\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^2\left[1+o\left(1\right)\right]+i_j{\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^2\left[1+o\left(1\right)\right]\\ =o\left({\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^2\right),\left(\text{an estimate independent of}j\right);\end{array}$(8.29)

whereas $P_k\left(x\right)$ is estimated in (8.24), ${P^{\prime }}_k\left(x\right)$ is estimated in (8.26) and

$\begin{array}{l}\left(R^{\left(j+1\right)}\left(x\right)/R^{\left(j\right)}\left(x\right)\right){P^{\prime }}_k\left(x\right)\stackrel{\text{by}\left(8.20\right)}{~}\left(R^{\prime }\left(x\right)/R\left(x\right)\right){P^{\prime }}_k\left(x\right)\\ =O\left({\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-3}\right).\end{array}$(8.30)

Collecting together all these estimates we get:

$\{\begin{array}{l}{P^{″}}_k\left(x\right)=o\left({\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-3}\right)+O\left({\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-3}\right)\\ =O\left({\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-3}\right);\\ {\displaystyle \underset{\cdots }{\overset{\cdots }{\sum }}{a}_{i_1,\cdots ,i_k}{P^{″}}_k\left(x\right)}=O\left({\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-3}\right);\end{array}$(8.31)

noticing once again that the estimates are independent of the indexes $i_1,\cdots ,i_k$, so that we may infer:

$\begin{array}{l}{{\psi }^{″}}_k\left(x\right)\stackrel{\text{by}\left(8.22\right)}{\equiv }{\left(k{\left(R^{\prime }\left(x\right)\right)}^{k-1}{R}^{″}\left(x\right)\right)}^{\prime }+\underset{\cdots }{\overset{\cdots }{{\displaystyle \sum }}}{a}_{i_1,\cdots ,i_k}{P^{″}}_k\left(x\right)\\ =k\left(k-1\right){\left(R^{\prime }\left(x\right)\right)}^{k-2}{\left(R^{″}\left(x\right)\right)}^2+k{\left(R^{\prime }\left(x\right)\right)}^{k-1}{R}^{‴}\left(x\right)\\ +O\left({\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-3}\right)\\ \stackrel{\text{by}\left(8.20\right)}{~}{k}^2{\left(R^{\prime }\left(x\right)\right)}^{k+2}{\left(R\left(x\right)\right)}^{-2}.\end{array}$(8.32)

Summing up, both the asymptotic relations for ${{\psi }^{\prime }}_k,{{\psi }^{″}}_k$ in (8.15) have been proved so that (8.16) holds true and we may legitimately apply Lemma 8.1-(II), namely relation (8.6), to the Wronskian in (4.2) so obtaining:

$\begin{array}{l}W\left({{\psi }^{\prime }}_1\left(x\right),\cdots ,{{\psi }^{\prime }}_{n-1}\left(x\right)\right)\\ ~V\left(1,\cdots ,n-1\right)\cdot \left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}{{\psi }^{\prime }}_i\left(x\right)\right)\cdot {\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^{\left(n-1\right)\left(n-2\right)/2}\\ \stackrel{\text{by}\left(8.27\right)}{~}\left(\underset{i=0}{\overset{n-2}{{\displaystyle \prod }}}i!\right)\left(n-1\right)!\cdot {\left(R\left(x\right)\right)}^{-\left(n-1\right)-\left[\left(n-1\right)\left(n-2\right)\right]/2}\\ \times {\left(R^{\prime }\left(x\right)\right)}^{2+3+\cdots +n+\left[\left(n-1\right)\left(n-2\right)/2\right]}\\ ~\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right)\cdot {\left(R\left(x\right)\right)}^{-n\left(n-1\right)/2}\cdot {\left(R^{\prime }\left(x\right)\right)}^{n\left(n-1\right)},x\to +\infty ,\left(n\ge 2\right);\end{array}$(8.33)

whence, by (4.2), one gets (8.18). $\square$

A comparison between Theorem 5.1 and Theorem 8.2.

Comparing (5.5) and (8.18) we shall point out two meaningful facts.

1. Apart from the constants appearing in the two asymptotic relations, the structure of the comparison functions is the same because in (5.5) we have “ $R^{\prime }\left(x\right)/R\left(x\right)~\gamma x^{-1}$” so that the principal part of the comparison function in (5.5) is

$\begin{array}{l}{x}^{-n\left(n-1\right)}\cdot {\left(R\left(x\right)\right)}^{n\left(n-1\right)/2}\cdot \text{exp}\left(nR\left(x\right)\right)\\ ~{\gamma }^{-n\left(n-1\right)}\cdot {\left(R^{\prime }\left(x\right)/R\left(x\right)\right)}^{n\left(n-1\right)}\cdot {\left(R\left(x\right)\right)}^{n\left(n-1\right)/2}\cdot \text{exp}\left(nR\left(x\right)\right)\\ \equiv {\gamma }^{-n\left(n-1\right)}\cdot {\left(R\left(x\right)\right)}^{-n\left(n-1\right)/2}\cdot {\left(R^{\prime }\left(x\right)\right)}^{n\left(n-1\right)}\cdot \text{exp}\left(nR\left(x\right)\right),\end{array}$(8.34)

coinciding with the comparison function in (8.18), a numerical constant apart.

2. We now show that the constant in (8.18) is the limit as $\gamma \to +\infty$ of the constant appearing in (5.5). In fact, using the comparison function in (8.18) we see that, by (8.34), the numerical constant in (5.5) is

$\begin{array}{l}{C}_{\gamma }:=V_{\gamma }\cdot \left(n-1\right)!{\gamma }^{n\left(n-1\right)/2}\cdot {\left(\gamma -1\right)}^{n-1}\cdot {\gamma }^{-n\left(n-1\right)}\\ ~V_{\gamma }\cdot \left(n-1\right)!{\gamma }^{\left(2-n\right)\left(n-1\right)/2},\gamma \to +\infty ,\end{array}$(8.35)

where $V_{\gamma }:=V\left(\gamma -2,2\gamma -3,\cdots ,\left(n-1\right)\gamma -n\right)\cdot \left(n-1\right)$. Now, from the explicit expression of a general Vandermondian,

$\begin{array}{l}V\left(c_1,\cdots ,c_n\right)=\left(c_2-c_1\right)\left[\left(c_3-c_2\right)\left(c_3-c_1\right)\right]\left[\left(c_4-c_3\right)\left(c_4-c_2\right)\left(c_4-c_1\right)\right]\\ \cdots \left[\left(c_n-c_{n-1}\right)\left(c_n-c_{n-2}\right)\cdots \left(c_n-c_1\right)\right],\end{array}$

containing $n\left(n-1\right)/2$ factors, one gets:

$\begin{array}{l}V\left(\gamma -2,2\gamma -3,\cdots ,\left(n-1\right)\gamma -n\right)\\ ~1!2!3!\cdots \left(n-2\right)!\cdot {\gamma }^{\left(n-1\right)\left(n-2\right)/2},\gamma \to +\infty ,\end{array}$(8.36)

which, replaced into (8.35), gives:

$C_{\gamma }~\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right),\gamma \to +\infty ,$(8.37)

which is the numerical constant in (8.18).

This result parallels a formal link between higher-order regular variation (which is variation of a certain index $\gamma \in ℝ$) and higher-order rapid variation (which is variation of index $\pm \infty$) as we are going to highlight. If $f\in \left\{{ℛ}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}$, $n\ge 1$, then we have the relations in (1.23), here rewritten:

$\begin{array}{c}{f}^{\left(k\right)}\left(x\right)/f\left(x\right)=\alpha \left(\alpha -1\right)\cdots \left(\alpha -k+1\right)x^{-k}+o\left(x^{-k}\right)\\ \equiv {\alpha }^{\underset{\_}{k}}{x}^{-k}+o\left(x^{-k}\right),x\to +\infty ,\left(1\le k\le n\right),\end{array}$(8.38)

for each fixed $\alpha \in ℝ$; and this is a characterization of the class $\left\{{ℛ}_{\alpha }\left(+\infty \right)\text{of}\text{order}n\right\}$ if ${\alpha }^{\underset{\_}{k}}\ne 0$: see ( [4] , Proposition 3.1-(II), p. 799) for a more precise statement. As, by the definition in (1.17),

$x^{-k}~{\alpha }^{-k}{\left(f^{\prime }\left(x\right)/f\left(x\right)\right)}^k,x\to +\infty ,\left(1\le k\le n\right),$

(8.38) may be rewritten in the form:

$f^{\left(k\right)}\left(x\right)/f\left(x\right)~{\alpha }^{\underset{\_}{k}}{\alpha }^{-k}{\left(f^{\prime }\left(x\right)/f\left(x\right)\right)}^k,x\to +\infty ,\left(1\le k\le n\right),$(8.39)

for each fixed $\alpha$ such that ${\alpha }^{\underset{\_}{k}}\ne 0$. Obviously, ${\lim }_{\alpha \to \pm \infty }{\alpha }^{\underset{\_}{k}}{\alpha }^{-k}=1$ and, as $\alpha \to \pm \infty$, one formally gets from (8.39) the relations

$f^{\left(k\right)}\left(x\right)/f\left(x\right)~{\left(f^{\prime }\left(x\right)/f\left(x\right)\right)}^k,x\to +\infty ,\left(1\le k\le n\right),$(8.40)

which, completed by the same relation for $k=n+1$, characterize the functions $f\in \left\{{ℛ}_{+\infty }\left(+\infty \right)\text{of}\text{order}n\right\}\cup \left\{{ℛ}_{-\infty }\left(+\infty \right)\text{of}\text{order}n\right\}$: see (1.33).

Example 8.2. For the function

$\varphi \left(x\right):=\text{exp}\left(c_1{\text{e}}^{c_2{x}^{\alpha }}\right),c_1\ne 0,c_2>0,\alpha >0,$(8.41)

we have the relation:

$\begin{array}{l}{H}_n\left[\text{exp}\left(c_1{\text{e}}^{c_2{x}^{\alpha }}\right)\right]~\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right)\cdot {\left(c_1\right)}^{n\left(n-1\right)/2}{\left(c_2\alpha \right)}^{n\left(n-1\right)}\cdot x^{\left(\alpha -1\right)n\left(n-1\right)}\\ \times \exp \left(n\left(n-1\right)c_2{x}^{\alpha }/2\right)\cdot \exp \left(n{c}_1{\text{e}}^{c_2{x}^{\alpha }}\right),x\to +\infty ,\left(n\ge 2\right),\end{array}$(8.42)

generalizing relation (8.10).

Example 8.3. For the function

$\varphi \left(x\right):=\exp \left(c_3{x}^{\gamma }\right)\cdot \exp \left(c_1{\text{e}}^{c_2{x}^{\alpha }}\right),c_1,c_3\ne 0;c_2>0,\alpha >0,\gamma \in ℝ,$(8.43)

we shall show the relation:

$\begin{array}{l}{H}_n\left[\exp \left(c_3{x}^{\gamma }\right)\cdot \exp \left(c_1{\text{e}}^{c_2{x}^{\alpha }}\right)\right]\\ ~\exp \left(n{c}_3{x}^{\gamma }\right)\cdot H_n\left[\exp \left(c_1{\text{e}}^{c_2{x}^{\alpha }}\right)\right],x\to +\infty ,\left(n\ge 2\right),\end{array}$(8.44)

where the principal part of the last Hankelian is specified in (8.42). In fact, putting

$R\left(x\right):=c_1{\text{e}}^{c_2{x}^{\alpha }},R_{\gamma }\left(x\right):=c_3{x}^{\gamma },$(8.45a)

we have:

$\{\begin{array}{l}R\left(x\right)+R_{\gamma }\left(x\right)~R\left(x\right),x\to +\infty ;\hfill \\ R^{\prime }\left(x\right)+{R^{\prime }}_{\gamma }\left(x\right)~R^{\prime }\left(x\right),x\to +\infty ;\hfill \end{array}$(8.45b)

and from (8.18):

$\begin{array}{l}{H}_n\left[\varphi \left(x\right)\right]\equiv H_n\left[\text{exp}\left(R\left(x\right)+R_{\gamma }\left(x\right)\right)\right]\\ ~\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right)\cdot {\left(R\left(x\right)+R_{\gamma }\left(x\right)\right)}^{-n\left(n-1\right)/2}\cdot {\left(R^{\prime }\left(x\right)+{R^{\prime }}_{\gamma }\left(x\right)\right)}^{n\left(n-1\right)}\\ \times \exp \left(nR\left(x\right)+n{R}_{\gamma }\left(x\right)\right)\\ \stackrel{\text{by}\left(8.45\text{b}\right)}{~}\left(\underset{i=1}{\overset{n-1}{{\displaystyle \prod }}}i!\right)\cdot {\left(R\left(x\right)\right)}^{-n\left(n-1\right)/2}\cdot {\left(R^{\prime }\left(x\right)\right)}^{n\left(n-1\right)}\\ \times \exp \left(nR\left(x\right)\right)\cdot \exp \left(n{R}_{\gamma }\left(x\right)\right)\\ =\text{exp}\left(n{R}_{\gamma }\left(x\right)\right)\cdot H_n\left[\text{exp}\left(R\left(x\right)\right)\right].\end{array}$(8.46)