1. Introduction

The discovery of new types of orthogonal polynomials, called the exceptional polynomials, has been the most interesting development in the area of exactly solvable models in quantum mechanics in the last two years [1–10]. Unlike the classical orthogonal polynomials, these new polynomials have the remarkable properties that they still form complete sets with respect to some positive-definite measure, although they start with degree polynomials instead of a constant. Four sets of infinite families of such polynomials, namely, the Laguerre type L1, L2, and the Jacobi type J1, J2 polynomials, with , were constructed in [3–5]. These systems were derived by deforming the radial oscillator potential and the Darboux-Pöschl-Teller (DPT) potential in terms of an eigen polynomial of degree (). The lowest () examples, the -Laguerre and -Jacobi polynomials, are equivalent to those introduced in the pioneering work of Gomez-Ullate et al. [1, 2] within the Sturm-Liouville theory. The results in [1, 2] were reformulated in the framework of quantum mechanics and shape-invariant potentials by Quesne et al. [6–8]. By construction, these new orthogonal polynomials satisfy a second order differential equation (the Schrödinger equation) without contradicting Bochner’s theorem [11], since they start at degree instead of the degree zero constant term. Generalization of exceptional orthogonal polynomials to discrete quantum mechanical systems was done in [12, 13].

Later, equivalent but much simpler looking forms of the Laguerre- and Jacobi-type polynomials than those originally presented in [3–5] were given in [9]. These nice forms were derived based on an analysis of the second order differential equations for the polynomials within the framework of the Fuchsian differential equations in the entire complex -plane. They allow us to study in depth some important properties of the polynomials, such as the actions of the forward and backward shift operators on the polynomials, Gram-Schmidt orthonormalization for the algebraic construction of the polynomials, Rodrigues formulas, and the generating functions of these new polynomials.

Recently, these exceptional orthogonal polynomials were generated by means of the Darboux-Crum transformation [14, 15]. Physical models which may involve these new polynomials were considered in [16].

One important aspect related to these new polynomials, which was only briefly mentioned in [9] but has not been investigated in depth so far, is the structure of their zeros. It is the purpose of this paper to look into this. Particularly, we investigate the behaviors of the zeros as the parameters of the polynomials change through numerical analysis.

The plan of this paper is as follows. In Section 2, we briefly review the forms of the exceptional polynomials. Sections 3 and 4 study the behaviors of the extra and the ordinary zeros, respectively, of the exceptional polynomials as one of and increases while the other parameters being kept fixed. Section 5 presents analytical proofs that explain the movements of the extra zeros of the exceptional polynomials as changes at fixed . In Section 6, we consider behaviors of the zeros at large and/or . Section 7 summarizes the paper. In the Appendix, we list a few lower degree exceptional orthogonal polynomials for reference. We do believe these heuristic results exemplify the essential features of the zeros of the exceptional orthogonal polynomials and that they would instigate more rigorous research of this novel and interesting subject.

2. Exceptional Orthogonal Polynomials

Four sets of infinitely many exceptional orthogonal polynomials were derived in [3–5], among them two are deformations of the Laguerre polynomials, and the others are deformations of the Jacobi polynomials. A unified nice form of these polynomials was given in [9], in which these polynomials are expressed as a bilinear form of the original polynomials, the Laguerre or Jacobi polynomials and the deforming polynomials, depending on the set of parameters and their shifts and a nonnegative integer , which is the degree of the deforming polynomials. The two sets of exceptional Laguerre polynomials (, ) are [9] in which , and Here are the classical Laguerre polynomials (A.1). It is interesting to note the following alternative expressions for the L1 and L2 exceptional polynomials: The two sets of exceptional Jacobi polynomials (, ) are [9] in which , , , and where are the classical Jacobi polynomials (A.2). The new exceptional orthogonal polynomials can be viewed as deformations of the classical orthogonal polynomials by the parameter , and the two polynomials and played the role of the deforming polynomials.

The differential equations satisfied by these new polynomials are as follows:

The zeros of orthogonal polynomials have always attracted the interest of researchers. In this paper, we will study the properties of the zeros of these new exceptional polynomials as some of their basic parameters change, mainly by numerical analysis.

In the case of polynomial , it has zeros in the (ordinary) domain, where the weight function is defined, that is, for the L1 and L2 polynomials and for the J1 and J2 polynomials. The behavior of these zeros, which we will call the ordinary zeros, is the same as those of other ordinary orthogonal polynomials [17–20]. We will say more about these zeros in Section 4. Besides these zeros, there are extra zeros outside the ordinary domain. For convenience, we will adopt the following notation for the zeros of the various polynomials involved: We emphasize that for the L1 and L2 Laguerre polynomials, and for the J1 and J2 Jacobi polynomials.

Figures 1–10 depict the distribution of the zeros for some representative parameters of the systems, namely, , and . From these figures, one can deduce certain patterns of the distribution of the zeros as those parameters vary. We will discuss these behaviors below.

Figure 1: L1: distributions of the zeros (), (○), and (■) for the L1 Laguerre polynomials, with and . The three diagrams correspond to , , and , respectively. The ordinary zeros lie in .

Figure 2: L2: distributions of the zeros (), (○), and (■) for the L2 Laguerre polynomials, with and . The three diagrams correspond to , , and , respectively. The ordinary zeros lie in .

Figure 3: J2: distributions of the zeros (), (○), and (■) for the J2 Jacobi polynomials, with , and . The three diagrams correspond to , , and , respectively. The ordinary zeros lie in .

Figure 4: L1: distributions of the zeros (), (○), and (■) for the L1 Laguerre polynomials, with and . The three diagrams correspond to , , and , respectively.

Figure 5: L1: distributions of the zeros (), (○), and (■) for the L1 Laguerre polynomials, with and . The three diagrams correspond to , , and , respectively.

Figure 6: L2: distributions of the zeros (), (○), and (■) for the L2 Laguerre polynomials, with and . The four diagrams correspond to , , , and , respectively.

Figure 7: L2: distributions of the zeros (), (○), and (■) for the L2 Laguerre polynomials, with and . The four diagrams correspond to , , , and , respectively.

Figure 8: J2: distributions of the zeros (), (○), and (■) for the J2 Jacobi polynomials, with , , and . The four diagrams correspond to , , , and , respectively.

Figure 9: J2: distributions of the zeros (), (○), and (■) for the J2 Jacobi polynomials, with , , and . The four diagrams correspond to , , , and , respectively.

Figure 10: J2: distributions of the zeros (), (○), and (■) for the J2 Jacobi polynomials, with , , and . The two diagrams correspond to and , respectively.

3. Extra Zeros of Polynomials

Here we discuss the locations of the extra zeros of the exceptional orthogonal polynomials, which lie in different positions for the different types of polynomials. From our numerical analysis, we can summarize the trend as follows.

The extra zeros of L1 polynomials are on the negative real line . The L2 polynomials have one real negative zero which lies to the left of the remaining pairs of complex conjugate roots. The L2 polynomials have pairs of complex conjugate roots.

The situations for the Jacobi polynomials are a bit more complicated. The J1 polynomials have one real negative root which lies to the left of the remaining pairs of complex conjugate roots with negative real parts. The J1 polynomials have pairs of complex conjugate roots with negative real parts. The J2 polynomials have one real positive root which lies to the right of the remaining pairs of complex conjugate roots with positive real parts. The J2 polynomials have pairs of complex conjugate roots with positive real parts.

One notes that the J1 and J2 polynomials are the mirror images of each other, in the sense and , as exemplified by the relation [3–5, 9]. So the behaviors of the zeros of J1 Jacobi polynomials can be obtained from those of the J2 type accordingly. As such, for clarity of presentation, we will only discuss the behaviors of the zeros of the J2 Jacobi polynomials in this paper.

3.1. Behaviors as Increases at Fixed

In all cases, we have This implies that the zeros of coincide with those of , namely, .

At fixed , all the move from at to as . This can be seen from Figures 1–3 and in Tables 1–7. We will provide heuristic arguments for this result in Section 5.

Table 1: List of the zeros , , and for the L1 Laguerre polynomials with , , and (). It can be seen that when , . As increases, approaches to .

Table 2: Same as Table 1 for L1 Laguerre polynomials with and .

Table 3: Same as Table 1 but for L2 Laguerre polynomials with and .

Table 4: Same as Table 3 for L2 Laguerre polynomials with and .

Table 5: Same as Table 1 but for J2 Jacobi polynomials with , and .

Table 6: Same as Table 5 for J2 Jacobi polynomials with , and .

Table 7: Same as Table 5 for J2 Jacobi polynomials with , and .

3.2. Behaviors as Increases at Fixed

The discussions in the last subsection show that by proper numbering are sandwiched between and . Thus to know how behave as increases at fixed , we only need to study how the zeros and flow as increases.

3.2.1. L1 Laguerre

As changes to , the zeros of and decrease (move to the left), and a new set of zeros appear from the right. for .

We show these patterns for some representative parameters in Figures 4 and 5.

3.2.2. L2 Laguerre

For , there is one real root each for and , with .

For , the above two roots bifurcate into two complex roots, with , .

Generally, for even , there are complex zeros with All are sandwiched between and . As an even changes to which is odd, all zeros move to the right with the real and the absolute value of the imaginary parts increased, and a new real zero appears to the left of all the complex zeros on the negative real axis. As increases further, the complex zeros move as described before, and the zero on the negative real axis bifurcates into two complex zeros, giving an even number of complex zeros. These patterns continue as increases.

Figures 6 and 7 show these behaviors for some selected parameters. For large , these zeros distribute in a horse-shoe pattern.

3.2.3. J2 Jacobi

For , there is one real root each for and , with .

For , the above two roots bifurcate into two complex roots, with , .

Generally, for even , there are complex zeros with As changes to which is odd, all zeros move toward the -axis, with the real parts decreased, and a new real zero appears to the right of all the complex zeros on the real -axis. As increases further, the complex zeros move as described before, and the zero on the real axis bifurcates into two complex zeros, giving an even number of complex zeros. The absolute value of the imaginary part of the complex zeros may increase initially, but eventually decrease as increases. This pattern continues as increases.

Figures 8 and 9 show these behaviors for some selected parameters. For large , these zeros distribute in a horse-shoe pattern.

4. Ordinary Zeros of Polynomials

In the case of polynomials , it has zeros in the (ordinary) domain, where the weight function is defined, that is, for the L1 and L2 polynomials and for the J1 and J2 polynomials. The behavior of these zeros is the same as that of other ordinary orthogonal polynomials. See, for example, [18] and/or [19, Section 5.4].

4.1. Behaviors as Increases at Fixed

This is guaranteed by the oscillation theorem of the one-dimensional quantum mechanics, since are obtained as the polynomial parts of the eigenfunctions of a shape invariant quantum mechanical problem. Explicitly, as changes to , all zeros of decrease, and a new zero appears from the right. Thus the zeros of and the zeros of interlace with each other: each zero of is surrounded by two zeros of .

Figures 1–3 show these behaviors for selected parameters.

4.2. Behaviors as Increases at Fixed

From Figures 4–7, one sees that for L1 and L2 Laguerre polynomials (whose zeros are positive in the ordinary domains), all the zeros shift to the right as increases.

For J2 Jacobi polynomials, the positive (negative) zeros shift left (right) as increases, that is, they move toward the origin . This is illustrated in Figures 8 and 9.

4.3. Additional Observation for the L1 Case

Using the well-known derivative relation and we get Hence, when , the L1 Laguerre is an even function of , and its zeros are symmetric with respect to .

5. Asymptotic Behavior of as

Here we provide intuitive arguments for the above asymptotic behavior. As mentioned before, for , we have , as . We will show that as , . This amounts to showing that in this limit, dominates over .

5.1. L1 and L2 Cases

We will make use of the above derivative relation (4.1) and (Perron) Theorem of [17], namely, which gives the asymptotic form of for large . For the L1 and L2 cases, we have and , respectively.

One finds

For large with fixed , dominates over , and thus the zeros of are determined by those of as .

5.2. J2 Jacobi

For the asymptotic form of for large , we will make use of Theorem of [17]: and

One finds

Again, for large with fixed , dominates over , and thus the zeros of are determined by those of as .

6. Behaviors at Large and/or

6.1. L1 Laguerre

As increases, we have that all increase. That is, all the zeros move away from the -axis. This can be seen from Figures 4 and 5.

In fact, for large , we have . Hence For , approaches Thus the extra () and the ordinary () zeros of are given by the zeros of and , respectively.

6.2. L2 Laguerre

As increases, we have decreased, , increased, and increased. This is easily seen from Figures 6 and 7. That is, the zeros , and hence all are moving leftwards and away from the -axis, while the ordinary zeros are moving towards the right.

In fact, for large , we have . Hence Using , , and of [9], we arrive at For , approaches Thus the extra () and the ordinary () zeros of are given by the zeros of and , respectively.

6.3. J2 Jacobi

As increases, we have , , , increased, as is evident from Figures 8 and 9. The extra zeros , being in between these zeros, follow the same pattern. That is, the zeros , , and hence all are moving away from the and -axes. The ordinary zeros will have their norm decrease in general as increases. Thus these zeros move towards the -axis.

In fact, for large and , we have (, ) as Using and of [9], we arrive at For and , approaches Thus the extra () and the ordinary () zeros of are given by the zeros of and , respectively.

6.3.1. Additional Observation:

For , all zeros, that is, , , , , gather around . This can be understood as follows. From the series expansion of the Jacobi polynomials, (A.2) one sees that, for , the absolute value of is large near and small at . Hence, in this limit, the zeros of distribute very near . We show this in Figure 10 for certain parameters.

7. Summary

The discovery of new types of orthogonal polynomials, called the exceptional Laguerre and Jacobi polynomials, has aroused great interest in the last two years. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree , and yet they form a complete set with respect to some positive-definite measure. Many essential properties have been studied in [9].

In this paper, we have considered the distributions of the zeros of these new polynomials as some parameters of the Hamiltonians change based on numerical analysis. The polynomial has zeros in the ordinary domain where the weight function is defined, that is, for the L1 and L2 polynomials and for the J1 and J2 polynomials. The behavior of these ordinary zeros are the same as those of other ordinary orthogonal polynomials. In addition to these zeros, there are extra zeros outside the ordinary domain.

For the ordinary zeros, their distribution as increases at a fixed follows the patterns of the zeros of the ordinary classical orthogonal polynomials: they are governed by the oscillation theorem, and the zeros of interlace with the zeros of . On the other hand, when increases at a fixed , the type L1 and L2 Laguerre polynomials will have all their zeros shifted to the right. For the J1 and the J2 Jacobi polynomials, both the positive and negative zeros move toward the origin as increases.