We consider the differential equation associated with a positive parameter $\omega$. By a Sturm-Liouville function, we mean a nontrivial real solution of (1.1). Let $\left\{x_k\left(\omega \right)\right\}$ denote the ascending sequence of the zeros of a Sturm-Liouville function in the interval $[a,b]$. The Sturm comparison theorem (see e.g., [1, p.314] or [3, p.56]) states that the second differences of the sequence $\left\{x_k\left(\omega \right)\right\}$ are all positive if ${\rho }^{\prime }\left(x\right)<0,$ and are all negative if ${\rho }^{\prime }\left(x\right)>0.$ Our main purpose here is to go beyond the second differences and to show that higher consecutive differences of sequences constructed from $\left\{x_k\left(\omega \right)\right\}$ are regular in sign. Lorch and Szego [3] initiated the study of the sign-regularity of higher differences of the sequences associated with Sturm-Liouville functions. In particular, if $c_{\nu k}\left(\alpha \right)$ denotes the kth positive zero of the general Bessel (cylinder) function they proved that, for $\left|\nu \right|>1/2$, and conjectured [3, p.71] on the basis of numerical evidence that, for $\left|\nu \right|<1/2$,

The symbol ${\Delta }^m{a}_k$ means, as usual, the mth (forward) difference of the sequence $\left\{a_k\right\}$:

Note that $C_{\nu }(x;\alpha )$ is a solution of the equation with $q\left(x\right)=1-({\nu }^2-(1/4))x^{-2}.$ Since $q^{\prime }\left(x\right)=2({\nu }^2-(1/4))x^{-3},$ we see that the Sturm comparison theorem gives the results ${\text{(1.2)}}_1$ and ${\text{(1.3)}}_0.$ They also mentioned in [3] that the signs of the first M differences of zeros of a Sturm-Liouville function of (1.4) could be inferred from the signs of $q^{\left(m\right)}\left(x\right)$, $m=1,2,...,M.$ Muldoon [7] made some progress in ${\text{(1.3)}}_m.$ He proved that ${\text{(1.3)}}_m$ holds when $1/3\le \left|\nu \right|<1/2$ ([7, Corollary 4.2]).

Our approach here is based on the ideas and results of [10], where the string equation $y^{\prime \prime }+\lambda \rho \left(x\right)y=0$ with $y\left(0\right)=y\left(1\right)=0$ was considered. Using the eigenvalues and the nodal points, we constructed a sequence of piecewise continuous linear functions which converges to ${\rho }^{-1/2}\left(x\right)$ uniformly on $[0,1]$. We also obtained a formula for derivatives of ${\rho }^{-1/2}\left(x\right)$ in terms of the eigenvalues and the differences of the nodal points.

This paper is organized as follows. In Section 2, we use the zeros $x_k\left(\omega \right)$ of a Sturm-Liouville function as nodes to obtain a difference-derivative theorem (Lemma 2.1). We also give asymptotic estimates for ${\rho }^{-1/2}\left(x_k\left(\omega \right)\right)$ as $\omega \to \infty$ (Lemma 2.3). Then we are able to express the higher differences ${\Delta }^{m+1}{x}_k\left(\omega \right)$ in terms of the derivatives of ${\rho }^{-1/2}\left(x\right)$ at those zeros. Moreover, the expression can be used to determine the regular manner of these differences (Theorems 2.4 and 2.5). Besides, we construct sequences from $x_k\left(\omega \right)$, whose all mth differences have the same sign (Corollary 2.6). The proofs of Lemmas 2.1 and 2.3 rely on a system of interlaced inductions, which will be given in Section 5. In Section 3, we use an approximation process for the zeros of the general Bessel function to prove the conjecture of Lorh and Szego (Theorem 3.1). In Section 4, the zeros of various orthogonal polynomials with higher degrees are shown to share similar sign-regularity (Theorems 4.1 and 4.2).

The notation used throughout is standard. A function $\phi \left(x\right)$ is said to be M-monotonic (resp., absolutely M-monotonic) on an interval I if

If ${\text{(1.5)}}_M$ holds for $M=\infty$, then $\phi \left(x\right)$ is said to be completely (resp., absolutely) monotonic on I. A sequence $\left\{a_k\left(\omega \right)\right\}$, depending on a positive parameter $\omega$, is said to be asymptotically M-monotonic (resp., asymptotically absolutely M-monotonic) if for $\omega$ sufficiently large.

In this section we consider the differential equation where $\omega$ is a positive parameter. We shall assume throughout that $\rho \left(x\right)$ is a positive $C^{\infty }$-function on the interval $[a,b]$. The notation $f\left(x\right)$ is reserved for the function ${\rho }^{-1/2}\left(x\right)$. Let $y(x;\omega )$ be a nontrival real solution of (2.1), and let $x_1\left(\omega \right)<x_2\left(\omega \right)<\cdots$ be the zeros of $y(x;\omega )$ in the interval $[a,b]$. For $a\le x<b,$ we denote by $k(x;\omega )$ the smallest positive interge k such that $x\le x_k\left(\omega \right).$ It is known(see e.g., [9, 10]) that

It follows that $\pi {min}_{[a,b]}f\le \omega \Delta x_k\left(\omega \right)\le \pi {max}_{[a,b]}f.$ In particular, we have

Thus, by (2.2) and the continuity of f, we obtain $f\left(x\right)={lim}_{\omega \to \infty }\frac{\omega }{\pi }\Delta x_{k(x;\omega )}\left(\omega \right)$ and, for any fixed l,

Note that (2.4) means that, as $\omega \to \infty$, the sequence $x_k\left(\omega \right)$ behaves as equally distributed.

If $\phi$ is m-times differentiable in $(t,t+md)$ and the lower derivatives of $\phi$ are continuous on $[t,t+md]$, a mean-value theorem [8, p. 52, no. 98] for differences and derivatives states that there exists a $\delta$, such that where ${\Delta }_d\phi \left(t\right)=\phi (t+d)-\phi \left(t\right)$. It is interesting to look for a difference-derivative theorem which can express the differences of a smooth function on the sequence $\left\{x_k\left(\omega \right)\right\}$ in terms of its derivatives at this sequence. The following lemma provides such a result.

Lemma 2.1. Let $x_k=x_k\left(\omega \right)$. If φ is a $C^{\infty }$-function on $[a,b]$, then, for $m=1,2,...,$

Moreover,

where the coefficients $A_{q,k}^{\left(m\right)}$ satisfy the recurrence relation:

To prove Lemma 2.1, we need a more detailed investigation on the behaviour of $x_k\left(\omega \right)$. We use the Prüfer method to achieve our purpose. For each nontrivial solution $y(x;\omega )$ of (2.1), we define the Prüfer angle $\theta (x;\omega )$ as follows:

Then $\theta (x;\omega )$ satisfies the differential equation

If we specify the initial condition for $\theta (x;\omega )$ to be $\theta (a;\omega )={\theta }_a\left(\omega \right)$ with $0\le {\theta }_a\left(\omega \right)<\pi ,$ then, by the standard results (see e.g., [1, p. 315]), we have and $k\pi \le \theta (x;\omega )\le (k+1)\pi ,$$x\in [x_k\left(\omega \right),x_{k+1}\left(\omega \right)].$ Let $x_k=x_k\left(\omega \right).$ Integrating both sides of (2.8) from $x_k$ to $x_{k+1},$ and using (2.9), we find

Taking the Taylor expansion of $(1/f)\left(x\right)$ at $x_k$ and using (2.3), we obtain

The estimate of the second integral in (2.10) is stated as the following lemma. Its proof consists of a reducible system of integrals which will be given in Appendix.

Lemma 2.2. Let $x_k=x_k\left(\omega \right)$. Then, for $m=2,3,...,$ we have

where the functions $F_r$ depend on $f={\rho }^{-1/2},$ and

Note that the first two functions $F_r$ appeared in ${\left(2.12\right)}_m$ are of the forms

For $m=2,3,...,$ using the estimates (2.11), ${\left(2.12\right)}_m$ and ${\left(2.13\right)}_m$ and multiplying (2.10) by $f\left(x_k\right)/\pi ,$ we find the estimate for $f\left(x_k\right):$ where the functions $g_r=f{(1/f)}^{\left(r\right)},$$r=0,1,2,...,m.$ Note that $g_0=1.$ Moreover, if we apply the mth order difference operator to ${\left(2.15\right)}_m$, then we can find the estimates for differences of the function $f\left(x\right)$ at those zeros. Indeed, we have

Lemma 2.3. Let $f\left(x\right)$ and $x_k=x_k\left(\omega \right)$ as above. Then, for $m=1,2,3,...,$ we have

Moreover,

The proofs of Lemmas 2.1 and 2.3 will be given in Section 5.

Now, if we apply Lemma 2.1 to the function $f\left(x\right)$, then, by ${\left(2.17\right)}_m,$ we have the estimate for the higher differences of $x_k=x_k\left(\omega \right):$

Moreover, using (2.3) and ${\text{(2.7)}}_m$, iterating ${\text{(2.18)}}_m$ for m from 1 to M, and then taking $\omega$ sufficiently large, we can ensure the monotonicity of the sequence $\{\Delta x_k\left(\omega \right)\}$ by f.

Theorem 2.4. Let $x_k=x_k\left(\omega \right)$ and $f\left(x\right)={\rho }^{-1/2}\left(x\right)$ be as those mentioned above. If $f\left(x\right)$ is M-monotonic on the interval $[a,b]$, then the sequence $\{\Delta x_k\left(\omega \right)\}$ is asymptotically M-monotonic.

Proof. Since it suffices to show that as $\omega \to \infty$, to conclude that

We prove ${\text{(2.20)}}_M$ by induction on M. When $M=1$, ${\text{(2.20)}}_1$ reduces to $A_{1,k}^{\left(1\right)}\ge 0,$ which is true because $A_{1,k}^{\left(1\right)}=\Delta x_k,$ by ${\text{(2.7)}}_1.$ Now, suppose that ${\text{(2.20)}}_N,$$1\le N<M,$ is true. By ${\text{(2.18)}}_N,$ we have which is nonnegative as $\omega \to \infty ,$ by induction hypothesis, (2.19) and ${\text{(2.16)}}_{N+1}.$ Thus, by ${\text{(2.7)}}_{N+1},$${(-1)}^N{A}_{1,k}^{(N+1)}={(-1)}^N{\Delta }^{N+1}{x}_k\ge 0,$ and , for $q=1,2,...,N+1,$ by induction hypothesis again. This prove ${\text{(2.20)}}_{N+1}$ and thus the theorem. □

Note that, if the factors ${(-1)}^m$ are deleted from the assumptions (2.19), then, by making the obvious changes in the above proof, the conclusion (2.21) remains valid, provided they are amended by eliminating the factors ${(-1)}^m.$ Thus we have

Theorem 2.5. Let $x_k=x_k\left(\omega \right)$ and $f\left(x\right)={\rho }^{-1/2}\left(x\right)$ be as those mentioned above. If $f\left(x\right)$ is absolutely M-monotonic on the interval $[a,b]$, then the sequence $\{\Delta x_k\left(\omega \right)\}$ is asymptotically absolutely M-monotonic.

As consequences of Lemma 2.1, Theorems 2.4 and 2.5, we can use the zeros of a solution of (2.1) to construct sequences whose all mth differences have the same sign.

Corollary 2.6. (a) Let $f\left(x\right)$ be M-monotonic on $[a,b].$ If $\phi \left(x\right)$ is also M-monotonic on $[a,b],$ then the sequence $\left\{\phi \left(x_k\right)\right\}$ is asymptotically M-monotonic.

Proof. Since $f\left(x\right)$ is M-monotonic on $[a,b],$ we see from the proof of Theorem 2.4 that ${\text{(2.20)}}_M$ holds. On the other hand, the M-monotonicity of $\phi \left(x\right)$ on $[a,b]$ means that

It now follows from ${\text{(2.6)}}_m,$ ${\text{(2.20)}}_M,$ (2.22) and ${\text{(2.5)}}_m$ that for all k and $m=0,1,2,...,M,$ as $\omega \to \infty .$ The proof of (b) is similar to that of part (a). □

Note that, by the definition of the function $f\left(x\right)={\rho }^{-1/2}\left(x\right),$ the conclusion of Theorem 2.4 (resp., Theorem 2.5) can be inferred directly from the assumptions on $\rho \left(x\right).$ In fact, ${(-1)}^m{\rho }^{(m+1)}\left(x\right)\ge 0$ (resp., ${\rho }^{(m+1)}\left(x\right)\le 0$) on $[a,b],$ for $m=0,1,2,...,M-1,$ imply ${(-1)}^m{f}^{\left(m\right)}\left(x\right)\ge 0$ (resp., $f^{\left(m\right)}\left(x\right)\ge 0$) on $[a,b],$ for $m=1,2,...,M.$ To examine the assertions, we can proceed by induction on $M.$ For $M=1,$ by the facts $f\left(x\right)={\rho }^{-1/2}\left(x\right)$ and $f^{\prime }\left(x\right)=(-1/2){\rho }^{-3/2}\left(x\right){\rho }^{\prime }\left(x\right)$, the assertion is valid. For higher derivatives of $f\left(x\right),$ a general term of $f^{\left(m\right)}\left(x\right)$ would appear as with exponentials ${\alpha }_0$ a negative half-integer and ${\alpha }_1,{\alpha }_2,...,{\alpha }_m,$ all nonnegative integers. The induction is carried through by differentiating $S_m.$ We have and under the conditions ${(-1)}^m{\rho }^{(m+1)}\left(x\right)\ge 0$ (resp., ${\rho }^{(m+1)}\left(x\right)\le 0$) and the negative ${\alpha }_0,$ each term in the last sum has opposite sign (resp., the same sign) as $S_m.$ Thus, $f^{\left(m\right)}\left(x\right)$ and $f^{(m+1)}\left(x\right)$ have alternating signs (resp., the same sign), and then the inductions are complete. Hence we obtain

Corollary 2.7. Let $x_k=x_k\left(\omega \right)$ be as above. (a) If ${\rho }^{\prime }\left(x\right)$ is $(M-1)$-monotonic on $[a,b],$ then the sequence $\{\Delta x_k\left(\omega \right)\}$ is asymptotically M-monotonic.

Although Corollary 2.7(a) is a partial result included in [4, Theorem 3.3], the techniques employ in this section are independent of the methods in the series of papers [4, 5, 7] and the results of Hartman [2, Theorems ${\text{18.1}}_n$ and ${\text{20.1}}_n$.] It also gives the connection of the quantities between the differences of the zeros and the coefficient function $\rho \left(x\right).$ However, it might have some numerical interest.

One can find similar results concerned with the critical points of a Sturm-Liouville function of (2.1). In fact, by letting $x_k^{\prime }\left(\omega \right)$ denote the kth critical point of a solution $y(x;\omega )$ of (2.1) in the interval $[a,b]$ and noting the definition of the Prüfer angle the procedures employed in this section are all valid. Thus if we replace $\left\{x_k\left(\omega \right)\right\}$ in Theorems 2.4 and 2.5, and Corollaries 2.6 and 2.7 by $\left\{x_k^{\prime }\left(\omega \right)\right\},$ the conclusions in these Theorems and Corollaries still hold.

Let $c_{\nu k}\left(\alpha \right)$ be the kth positive zero of the general Bessel (cylinder) function where $J_{\nu }\left(x\right)$ and $Y_{\nu }\left(x\right)$ denote the Bessel functions with order $\nu$ of the first and second kind, respectively. The main results in this section are stated as follows:

Theorem 3.1. (a) For $\left|\nu \right|<1/2,$ we have

(b) For $0<\left|\nu \right|<1/2,$ we have

The Airy functions (see e.g., [11, p.18]) satisfy the differential equation $y^{\prime \prime }+\frac{x}{3}y=0.$ We consider a broader class of functions, including the Airy functions, which satisfy the differential equation (see e.g., [12, p. 97(9)]) where $0<\gamma <\infty .$ These functions are closely related to Bessel functions. Indeed, is a nontrivial real solution of (3.1). Note that, for each $\omega >0$, the kth positive zeros ${\xi }_k\left(\omega \right)$ of $z(x;\omega )$ satisfies the identities

Moreover, for each $\omega >0$, and $m=0,1,2,...,$ we have and

The identities (3.2) and (3.3) are really the key for us to study the regularity behaviour of the Bessel zeros.

To prove Theorem 3.1, we consider the family of differential equations: on the interval $[0,b].$ Let $y_a(x;\omega )$ be a nontrivial real solution of (3.4) and let the sequence $\left\{x_k(\omega ;a)\right\}$ be the zeros of $y_a(x;\omega )$ with the ascending order in $[0,b].$ Following Theorem 2.4 with $f\left(x\right)={(x+a)}^{-\gamma /2}$ and Corollary 2.6(a) with the function $\phi \left(x\right)={(x+a)}^{-1/2\nu },$ we have and as $\omega \to \infty .$ If we specify the initial conditions for the solution $y_a(x;\omega )$ of (3.4) to be then it is easy to verify that $y_a(x;\omega )=z(x+a;\omega )$ for $x\in [0,b],$ and hence, for each $k,$$x_k(\omega ;a)+a$ converges to ${\xi }_k$ as $a\to 0^{+}.$ Thus, for each $\omega >0,$ by (3.2) and (3.3), we have and

Recalling (2.10) and ${\text{(2.12)}}_{m+3}$ with the function $\rho \left(x\right)={(x+a)}^{\gamma }$ and denoting $x_k=x_k(\omega ;a),$ we have

Note that $\nu =1/(\gamma +2)$ and $f\left(x\right)={(x+a)}^{(2\nu -1)/2\nu }.$ By (2.14), we have

Thus, (3.9) becomes

If we apply the difference operator ${\Delta }^{m+1}$ to (3.10), by ${\text{(2.5)}}_{m+2}$ and ${\text{(2.13)}}_{m+3}$, then we can find

Moreover, multiplying (3.11) by ${(-1)}^m{\omega }^{m+3},$ we have

By (3.12), (3.6), ${\text{(2.5)}}_{m+2}$ and $0<\nu <1/2,$ we have

Now, for each $a>0,$ if we choose $\omega =\omega \left(a\right)$ sufficiently large such that (3.13) and (3.5) hold, then, by (3.7) and (3.8), we have and

Secondly, according to $Y_{\nu }\left(x\right)=(J_{\nu }\left(x\right)cos\pi \nu -J_{-\nu }\left(x\right))/sin\pi \nu$ (see e.g., [12, p. 64]), it is easily to verify that $C_{-\nu }(x;\alpha )=C_{\nu }(x;\alpha +\pi \nu ),$ and hence

Thus, for $0<\left|\nu \right|<1/2,$ (3.14) holds and (3.15) holds in the modified form

Thirdly, for $\nu =0,$ any positive zero $c_{\nu k}\left(\alpha \right)$ of $C_{\nu }(x;\alpha )$ is definable as a continuously increasing function of the real variable $\nu$ (see e.g., [12, p. 508]), so that, by an approximating process, (3.14) hold for all $\left|\nu \right|<1/2.$

Finally, since neither $\left\{{\Delta }^2{c}_{\nu k}\left(\alpha \right)\right\}$ nor $\{\Delta c_{\nu k}^{2\left|\nu \right|}\left(\alpha \right)\}$ are constant sequences, the results of Lorch, Szego and Muldoon for completely monotonic sequences ([3, p. 72] or [6, Theorem 2]) guarantee the strict inequalities of (3.14) and (3.16). This completes the proof of Theorem 3.1.

Some important classical orthogonal polynomials are related to Sturm-Liouville functions such as Hermite and Jacobi polynomials. In [3, p. 73], Lorch, Szego and their coworkers conjectured, on the basis of numerical evidence, that the $\theta$-zeros of the Legendre polynomials, the special cases of Jacobi polynomials, and the positive zeros of the Hermite polynomials form the sequences whose mth differences have the constant sign. In this section, we shall apply the results in Section 2 and 3 to obtain some partial answers for these conjectures.