About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Physics
Volume 2012 (2012), Article ID 920475, 27 pages
http://dx.doi.org/10.5402/2012/920475
Research Article

Zeros of the Exceptional Laguerre and Jacobi Polynomials

1Department of Physics, Tamkang University, Tamsui 251, Taiwan
2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Received 12 April 2012; Accepted 4 July 2012

Academic Editors: G. Goldin and R. Schiappa

Copyright © 2012 Choon-Lin Ho and Ryu Sasaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional 𝑋 Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree =1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

1. Introduction

The discovery of new types of orthogonal polynomials, called the exceptional 𝑋 polynomials, has been the most interesting development in the area of exactly solvable models in quantum mechanics in the last two years [110]. 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 𝑋 polynomials, with =1,2,, were constructed in [35]. 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 (=1,2,). The lowest (=1) examples, the 𝑋1-Laguerre and 𝑋1-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. [68]. 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 >0 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 𝑋 polynomials than those originally presented in [35] were given in [9]. These nice forms were derived based on an analysis of the second order differential equations for the 𝑋 polynomials within the framework of the Fuchsian differential equations in the entire complex 𝑥-plane. They allow us to study in depth some important properties of the 𝑋 polynomials, such as the actions of the forward and backward shift operators on the 𝑋 polynomials, Gram-Schmidt orthonormalization for the algebraic construction of the 𝑋 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 𝑛 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 𝑛 changes at fixed . In Section 6, we consider behaviors of the zeros at large 𝑔 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 [35], 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 𝝀 and their shifts 𝜹 and a nonnegative integer , which is the degree of the deforming polynomials. The two sets of exceptional Laguerre polynomials (=1,2,, 𝑛=0,1,2,) are [9] 𝑃,𝑛(𝜂;𝝀)def=𝜉(𝜂;𝝀+𝜹)𝑃𝑛(𝜂;𝑔+1)𝜉(𝜂;𝝀)𝜕𝜂𝑃𝑛1(𝜂;𝑔+1):L1𝑛+𝑔+211𝑔+2𝜉(𝜂;𝝀+𝜹)𝑃𝑛(𝜂;𝑔++1)+𝜂𝜉(𝜂;𝝀)𝜕𝜂𝑃𝑛(𝜂;𝑔++1):L2,(2.1) in which 𝝀def=𝑔>0, 𝜹def=1 and 𝑃𝑛(𝜂;𝑔)def=𝐿𝑛(𝑔(1/2))(𝜂),𝜉(𝜂;𝑔)def=𝐿(𝑔+(3/2))𝐿(𝜂):L1(𝑔(1/2))(𝜂).:L2.(2.2) Here 𝐿𝑛(𝛼)(𝜂) are the classical Laguerre polynomials (A.1). It is interesting to note the following alternative expressions for the L1 and L2 exceptional polynomials: 𝑃,𝑛(𝜂;𝝀)def=𝐿(𝑔+(1/2))(𝜂)𝐿𝑛(𝑔+(1/2))(𝜂)𝐿(𝑔+(1/2))1(𝜂)𝐿(𝑔+(1/2))𝑛1((𝜂):L1𝑛+𝑔+(1/2))11𝑔+2𝐿(𝑔(3/2))(𝜂)𝐿𝑛(𝑔++(3/2))(𝜂)+(+1)𝐿(𝑔(3/2))+1(𝜂)𝐿(𝑔++(3/2))𝑛1(𝜂):L2.(2.3) The two sets of exceptional Jacobi polynomials (=1,2,, 𝑛=0,1,2,) are [9] 𝑃,𝑛(𝜂;𝝀)def=1𝑛++21(+(1/2))𝜉(𝜂;𝝀+𝜹)𝑃𝑛(𝜂;𝑔+1,++1)+(1+𝜂)𝜉(𝜂;𝝀)𝜕𝜂𝑃𝑛1(𝜂;𝑔+1,++1):J1𝑛+𝑔+21(𝑔+(1/2))𝜉(𝜂;𝝀+𝜹)𝑃𝑛(𝜂;𝑔++1,+1)(1𝜂)𝜉(𝜂;𝝀)𝜕𝜂𝑃𝑛(𝜂;𝑔++1,+1):J2,(2.4) in which 𝝀def=(𝑔,), 𝑔>0, >0, 𝜹def=(1,1) and 𝑃𝑛(𝜂;𝑔,)def=𝑃𝑛(𝑔(1/2),(1/2))(𝜂),𝜉(𝜂;𝑔,)def=𝑃(𝑔+(3/2),(1/2))𝑃(𝜂),𝑔>>0:J1(𝑔(1/2),+(3/2))(𝜂),>𝑔>0:J2,(2.5) where 𝑃𝑛(𝛼,𝛽)(𝜂) 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 𝜉(𝜂;𝝀) and 𝜉(𝜂;𝝀+𝜹) played the role of the deforming polynomials.

The differential equations satisfied by these new polynomials are as follows: L1𝜂𝜕2𝜂𝑃,𝑛1(𝜂;𝝀)+𝑔++2𝜂2𝜂𝜕𝜂𝜉(𝜂;𝝀)𝜉𝜕(𝜂;𝝀)𝜂𝑃,𝑛+2(𝜂;𝝀)𝜂𝜕𝜂𝜉(𝜂;𝝀+𝜹)𝜉𝑃(𝜂;𝝀)+𝑛,𝑛(𝜂;𝝀)=0,(2.6)L2𝜂𝜕2𝜂𝑃,𝑛1(𝜂;𝝀)+𝑔++2𝜂2𝜂𝜕𝜂𝜉(𝜂;𝝀)𝜉𝜕(𝜂;𝝀)𝜂𝑃,𝑛+((𝜂;𝝀)2𝑔+(1/2))𝜕𝜂𝜉(𝜂;𝝀+𝜹)𝜉𝑃(𝜂;𝝀)+𝑛+,𝑛(𝜂;𝝀)=0,(2.7)J11𝜂2𝜕2𝜂𝑃,𝑛(𝜂;𝝀)+𝑔(𝑔++2+1)𝜂21𝜂2𝜕𝜂𝜉(𝜂;𝝀)𝜉𝜕(𝜂;𝝀)𝜂𝑃,𝑛(+𝜂;𝝀)2(+(1/2))(1𝜂)𝜕𝜂𝜉(𝜂;𝝀+𝜹)𝜉𝑃(𝜂;𝝀)+(+𝑔1)+𝑛(𝑛+𝑔++2),𝑛(𝜂;𝝀)=0,(2.8)J21𝜂2𝜕2𝜂𝑃,𝑛(𝜂;𝝀)+𝑔(𝑔++2+1)𝜂21𝜂2𝜕𝜂𝜉(𝜂;𝝀)𝜉𝜕(𝜂;𝝀)𝜂𝑃,𝑛+(𝜂;𝝀)2(𝑔+(1/2))(1+𝜂)𝜕𝜂𝜉(𝜂;𝝀+𝜹)𝜉𝑃(𝜂;𝝀)+(+𝑔1)+𝑛(𝑛+𝑔++2),𝑛(𝜂;𝝀)=0.(2.9)

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 𝑋 polynomial 𝑃,𝑛(𝜂;𝝀), it has 𝑛 zeros in the (ordinary) domain, where the weight function is defined, that is, (0,) for the L1 and L2 polynomials and (1,1) 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 [1720]. We will say more about these zeros in Section 4. Besides these 𝑛 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: 𝜉𝑘():zerosof𝜉𝜉(𝜂;𝝀+𝜹),𝑘=1,2,,;(2.10)𝑘():zerosof𝜉(𝜂;𝝀),𝑘=1,2,,;(2.11)𝜂𝑘(,𝑛):extrazerosof𝑃,𝑛𝜂,𝑘=1,2,,;(2.12)𝑗(,𝑛):ordinaryzerosof𝑃,𝑛,𝑗=1,2,,𝑛.(2.13) We emphasize that 𝜂𝑗(,𝑛)(0,) for the L1 and L2 Laguerre polynomials, and 𝜂𝑗(,𝑛)(1,1) for the J1 and J2 Jacobi polynomials.

Figures 110 depict the distribution of the zeros for some representative parameters of the systems, namely, 𝑛,,𝑔, 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.

fig1
Figure 1: L1: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the L1 Laguerre polynomials, with 𝑔=0.5 and =2. The three diagrams correspond to 𝑛=1(a), 2(b), and 3(c), respectively. The ordinary zeros 𝜂𝑗(,𝑛) lie in (0,).
fig2
Figure 2: L2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the L2 Laguerre polynomials, with 𝑔=0.5 and =3. The three diagrams correspond to 𝑛=1(a), 2(b), and 5(c), respectively. The ordinary zeros 𝜂𝑗(,𝑛) lie in (0,).
fig3
Figure 3: J2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the J2 Jacobi polynomials, with 𝑔=3,=4, and =3. The three diagrams correspond to 𝑛=1(a), 2(b), and 5(c), respectively. The ordinary zeros 𝜂𝑗(,𝑛) lie in (1,1).
fig4
Figure 4: L1: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the L1 Laguerre polynomials, with 𝑔=0.5 and 𝑛=2. The three diagrams correspond to =1(a), 2(b), and 3(c), respectively.
fig5
Figure 5: L1: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the L1 Laguerre polynomials, with 𝑔=1.5 and 𝑛=2. The three diagrams correspond to =1(a), 2(b), and 3(c), respectively.
fig6
Figure 6: L2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the L2 Laguerre polynomials, with 𝑔=2 and 𝑛=2. The four diagrams correspond to =1(a), 2(b), 3(c), and 20(d), respectively.
fig7
Figure 7: L2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the L2 Laguerre polynomials, with 𝑔=5 and 𝑛=2. The four diagrams correspond to =1(a), 2(b), 3(c), and 20(d), respectively.
fig8
Figure 8: J2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the J2 Jacobi polynomials, with 𝑔=3, =4, and 𝑛=4. The four diagrams correspond to =1(a), 2(b), 3(c), and 20(d), respectively.
fig9
Figure 9: J2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the J2 Jacobi polynomials, with 𝑔=7, =8, and 𝑛=4. The four diagrams correspond to =1(a), 2(b), 3(c), and 20(d), respectively.
fig10
Figure 10: J2: distributions of the zeros 𝜂𝑘(,𝑛),𝜂𝑗(,𝑛) (), 𝜉𝑘() (○), and 𝜉𝑘() (■) for the J2 Jacobi polynomials, with 𝑔=2, =10, and 𝑛=4. The two diagrams correspond to =50(a) and 100(b), respectively.

3. Extra Zeros of 𝑋 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 (,0). The L2 𝑋odd polynomials have one real negative zero which lies to the left of the remaining (1/2)(1) pairs of complex conjugate roots. The L2 𝑋even polynomials have (1/2) pairs of complex conjugate roots.

The situations for the 𝑋 Jacobi polynomials are a bit more complicated. The J1 𝑋odd polynomials have one real negative root which lies to the left of the remaining (1/2)(1) pairs of complex conjugate roots with negative real parts. The J1 𝑋even polynomials have (1/2) pairs of complex conjugate roots with negative real parts. The J2 𝑋odd polynomials have one real positive root which lies to the right of the remaining (1/2)(1) pairs of complex conjugate roots with positive real parts. The J2 𝑋even polynomials have (1/2) 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 𝜂𝜂 and 𝑔, as exemplified by the relation 𝜉J2(𝜂;𝑔,)=(1)𝜉J1(𝜂;,𝑔) [35, 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 𝑛 Increases at Fixed

In all cases, we have 𝑃,0(𝜂;𝝀)𝜉(𝜂;𝝀+𝜹).(3.1) This implies that the zeros of 𝑃,0 coincide with those of 𝜉(𝜂;𝝀+𝜹), namely, 𝜉𝑘(),𝑘=1,2,,.

At fixed , all the 𝜂𝑘(,𝑛) move from 𝜉𝑘() at 𝑛=0 to 𝜉𝑘() as 𝑛. This can be seen from Figures 13 and in Tables 17. We will provide heuristic arguments for this result in Section 5.

tab1
Table 1: List of the zeros 𝜉𝑘(), 𝜉𝑘(), and 𝜂𝑘(,𝑛) for the L1 Laguerre polynomials with 𝑔=2, =5, and 𝑛=0,10,20,,60 (𝑘=1,2,,). It can be seen that when 𝑛=0, 𝜂𝑘(,𝑛=0)=𝜉𝑘(). As 𝑛 increases, 𝜂𝑘(,𝑛) approaches to 𝜉𝑘().
tab2
Table 2: Same as Table 1 for L1 Laguerre polynomials with 𝑔=8 and =5.
tab3
Table 3: Same as Table 1 but for L2 Laguerre polynomials with 𝑔=3 and =4.
tab4
Table 4: Same as Table 3 for L2 Laguerre polynomials with 𝑔=10 and =5.
tab5
Table 5: Same as Table 1 but for J2 Jacobi polynomials with 𝑔=3,=4, and =4.
tab6
Table 6: Same as Table 5 for J2 Jacobi polynomials with 𝑔=3,=4, and =5.
tab7
Table 7: Same as Table 5 for J2 Jacobi polynomials with 𝑔=8,=9, and =3.
3.2. Behaviors as Increases at Fixed 𝑛

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

3.2.1. L1 Laguerre

As changes to +1, the zeros of 𝜉(𝜂;𝑔+1) and 𝜉(𝜂;𝑔) decrease (move to the left), and a new set of zeros appear from the right. 𝜉𝑘(+1)<𝜉𝑘(),𝜉𝑘(+1)<𝜉𝑘(),𝜉𝑘()<𝜉𝑘()<𝜉()𝑘+1<𝜉()𝑘+1,(3.2) for 𝑘=1,2,,1,.

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

3.2.2. L2 Laguerre

For =1, there is one real root each for 𝜉(𝜂;𝑔+1) and 𝜉(𝜂;𝑔), with 𝜉1()<𝜉1()<0.

For =2, the above two roots bifurcate into two complex roots, with 𝜉()<𝜉(), |𝜉()|>|𝜉()|.

Generally, for even , there are complex zeros with 𝜉𝑘()<𝜉𝑘(),|||𝜉𝑘()|||>||𝜉𝑘()||,𝑘=1,2,,2.(3.3) All 𝜂𝑘(,𝑛) are sandwiched between 𝜉𝑘() and 𝜉𝑘(). As an even changes to +1 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 =1, there is one real root each for 𝜉(𝜂;𝑔+1) and 𝜉(𝜂;𝑔), with 𝜉1()>𝜉1()>1.

For =2, the above two roots bifurcate into two complex roots, with 𝜉()>𝜉(), |𝜉()|>|𝜉()|.

Generally, for even , there are complex zeros with 𝜉𝑘()>𝜉𝑘(),|||𝜉𝑘()|||>||𝜉𝑘()||,𝑘=1,2,,2.(3.4) As changes to +1 which is odd, all zeros move toward the 𝑦-axis, with the real parts decreased, and a new real zero appears to the right of all the complex zeros on the real 𝑥-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 𝑋 Polynomials

In the case of 𝑋 polynomials 𝑃,𝑛(𝜂;𝝀), it has 𝑛 zeros in the (ordinary) domain, where the weight function is defined, that is, (0,) for the L1 and L2 polynomials and (1,1) 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 𝑛 Increases at Fixed

This is guaranteed by the oscillation theorem of the one-dimensional quantum mechanics, since 𝑃,𝑛(𝜂;𝝀) are obtained as the polynomial parts of the eigenfunctions of a shape invariant quantum mechanical problem. Explicitly, as 𝑛 changes to 𝑛+1, all zeros of 𝑃,𝑛 decrease, and a new zero appears from the right. Thus the 𝑛 zeros of 𝑃,𝑛(𝜂;𝝀) and the 𝑛+1 zeros of 𝑃,𝑛+1(𝜂;𝝀) interlace with each other: each zero of 𝑃,𝑛(𝜂;𝝀) is surrounded by two zeros of 𝑃,𝑛+1(𝜂;𝝀).

Figures 13 show these behaviors for selected parameters.

4.2. Behaviors as Increases at Fixed 𝑛

From Figures 47, one sees that for L1 and L2 Laguerre polynomials (whose zeros are positive in the ordinary domains), all the 𝑛 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 𝜂=0. This is illustrated in Figures 8 and 9.

4.3. Additional Observation for the L1 Case

Using the well-known derivative relation 𝜕𝜂𝐿𝑛(𝛼)(𝜂)=𝐿(𝛼+1)𝑛1(𝜂),(4.1) and 𝐿𝑛(𝛼)(𝜂)𝐿𝑛(𝛼1)(𝜂)=𝐿(𝛼)𝑛1(𝜂),(4.2) we get 𝑃,(𝜂;𝑔)=𝐿(𝑔+(1/2))(𝜂)𝐿(𝑔+(3/2))(𝜂)+𝐿(𝑔+(1/2))(𝜂)𝐿(𝑔+(3/2))(𝜂)𝐿(𝑔+(3/2))(𝜂)𝐿(𝑔+(3/2))(𝜂).(4.3) Hence, when 𝑛=, the L1 Laguerre is an even function of 𝜂, and its zeros are symmetric with respect to 𝜂=0.

5. Asymptotic Behavior of 𝜂𝑘(,𝑛)𝜉𝑘() as 𝑛

Here we provide intuitive arguments for the above asymptotic behavior. As mentioned before, for 𝑛=0, we have 𝜂𝑘(,0)=𝜉𝑘(), as 𝑃𝑛=0=1. We will show that as 𝑛, 𝜂𝑘(,𝑛)𝜉𝑘(). This amounts to showing that in this limit, 𝜕𝜂𝑃𝑛 dominates over 𝑃𝑛.

5.1. L1 and L2 Cases

We will make use of the above derivative relation (4.1) and (Perron) Theorem 8.22.3 of [17], namely, 𝐿𝑛(𝛼)𝑒(𝜂)𝜂/22𝜋(𝜂)((𝛼/2)(1/4))𝑛((𝛼/2)(1/4))𝑒2𝑛𝜂,𝛼,𝜂(0,),(5.1) which gives the asymptotic form of 𝐿𝑛(𝛼)(𝜂) for large 𝑛. For the L1 and L2 cases, we have 𝛼=𝑔+3/2 and 𝑔++1/2, respectively.

One finds |||||𝐿𝑛(𝛼)(𝜂)𝜕𝜂𝐿𝑛(𝛼)||||||||||1(𝜂)𝑛(𝜂)1/2|||||.(5.2)

For large 𝑛 with fixed 𝜂, 𝜕𝜂𝐿𝑛(𝛼)(𝜂) dominates over 𝐿𝑛(𝛼)(𝜂), and thus the zeros of 𝑃,𝑛 are determined by those of 𝜉(𝜂;𝑔) as 𝑛.

5.2. J2 Jacobi

For the asymptotic form of 𝑃𝑛(𝛼,𝛽)(𝜂) for large 𝑛, we will make use of Theorem 8.21.7 of [17]: 𝑃𝑛(𝛼,𝛽)(𝜂)(𝜂1)𝛼/2(𝜂+1)𝛽/2𝜂+1+𝜂1𝛼+𝛽×𝜂211/42𝜋𝑛𝜂+𝜂21𝑛+(1/2)[],,𝛼,𝛽,𝜂1,1(5.3) and 𝜕𝜂𝑃𝑛(𝛼,𝛽)1(𝜂)=2(𝑛+𝛼+𝛽+1)𝑃(𝛼+1,𝛽+1)𝑛1(𝜂).(5.4)

One finds 𝑃𝑛(𝛼,𝛽)(𝜂)𝜕𝜂𝑃𝑛(𝛼,𝛽)2(𝜂)(𝑛+𝛼+𝛽+1)𝑛1𝑛𝜂211/2𝜂+𝜂21𝜂+1+𝜂12.(5.5)

Again, for large 𝑛 with fixed 𝜂, 𝜕𝜂𝑃𝑛(𝛼,𝛽)(𝜂) dominates over 𝑃𝑛(𝛼,𝛽)(𝜂), and thus the zeros of 𝑃,𝑛 are determined by those of 𝜉(𝜂;𝑔) as 𝑛.

6. Behaviors at Large 𝑔 and/or

6.1. L1 Laguerre

As 𝑔 increases, we have that |||𝜉𝑘()|||,||𝜉𝑘()||,|||𝜂𝑘(,𝑛)|||,||𝜂𝑘(,𝑛)||(6.1) all increase. That is, all the zeros move away from the 𝑦-axis. This can be seen from Figures 4 and 5.

In fact, for large 𝑔, we have 𝜉(𝜂;𝑔+1)𝜉(𝜂;𝑔). Hence 𝑃,𝑛(𝜂;𝑔)𝜉𝐿(𝜂;𝑔)𝑛(𝑔+(3/2))(𝜂)𝜕𝜂𝐿𝑛(𝑔+(3/2))(𝜂)𝜉(𝜂;𝑔)𝐿𝑛(𝑔+(1/2))(𝜂).(6.2) For 𝑔1, 𝑃,𝑛(𝜂;𝑔) approaches 𝑃,𝑛(𝜂;𝑔)𝐿(𝑔+)(𝜂)𝐿𝑛(𝑔+)(𝜂).(6.3) Thus the extra (𝜂𝑘(,𝑛)) and the ordinary (𝜂𝑘(,𝑛)) zeros of 𝑃,𝑛(𝜂;𝑔) are given by the zeros of 𝐿(𝑔+)(𝜂) and 𝐿𝑛(𝑔+)(𝜂), respectively.

6.2. L2 Laguerre

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

In fact, for large 𝑔, we have 𝜉(𝜂;𝑔+1)𝜉(𝜂;𝑔). Hence 𝑃,𝑛(𝜂;𝑔)𝜉1(𝜂;𝑔)𝑔+2𝐿𝑛(𝑔++(1/2))(𝜂)+𝜂𝜕𝜂𝐿𝑛(𝑔++(1/2)).(𝜂)(6.4) Using (E.2), (E.10), and (E.9) of [9], we arrive at 𝑃,𝑛(𝜂;𝑔)𝜉1(𝜂;𝑔)𝑔++2𝐿+𝑛𝑛(𝑔+(1/2))(𝜂)𝐿𝑛(𝑔++(1/2)).(𝜂)(6.5) For 𝑔1, 𝑃,𝑛(𝜂;𝑔) approaches 𝑃,𝑛(𝜂;𝑔)𝐿(𝑔)(𝜂)𝐿𝑛(𝑔+)(𝜂).(6.6) Thus the extra (𝜂𝑘(,𝑛)) and the ordinary (𝜂𝑘(,𝑛)) zeros of 𝑃,𝑛(𝜂;𝑔) are given by the zeros of 𝐿(𝑔)(𝜂) and 𝐿𝑛(𝑔+)(𝜂), respectively.

6.3. J2 Jacobi

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

In fact, for large 𝑔 and , we have (𝛼𝑔++(1/2), 𝛽+(3/2)) as 𝑃,𝑛(𝜂;𝑔,)𝜉1(𝜂;𝑔,)𝑔+2𝑃𝑛(𝛼,𝛽)(𝜂)(1𝜂)𝜕𝜂𝑃𝑛(𝛼,𝛽).(𝜂)(6.7) Using (E.13) and (E.23) of [9], we arrive at 𝑃,𝑛(𝜂;𝑔,)𝜉(𝜂;𝑔,)(𝛼+𝑛)𝑃𝑛(𝛼+1,𝛽+1)(𝜂)𝑃𝑛(𝛼,𝛽)(𝜂).(6.8) For 𝑔1 and 1, 𝑃,𝑛(𝜂;𝑔,) approaches 𝑃,𝑛(𝜂;𝑔)𝑃(𝑔,+)(𝜂)𝑃𝑛(𝑔+,+)(𝜂).(6.9) Thus the extra (𝜂𝑘(,𝑛)) and the ordinary (𝜂𝑘(,𝑛)) zeros of 𝑃,𝑛(𝜂;𝑔,) are given by the zeros of 𝑃(𝑔,+)(𝜂) and 𝑃𝑛(𝑔+,+)(𝜂), respectively.

6.3.1. Additional Observation: 𝑔

For 𝑔, all zeros, that is, 𝜉𝑗(), 𝜉𝑗(), 𝜂𝑘(,𝑛), 𝜂𝑘(,𝑛), gather around 𝜂=1. This can be understood as follows. From the series expansion of the Jacobi polynomials, (A.2) 𝑃𝑛(𝛼,𝛽)(𝜂)=(𝛼+1)𝑛𝑛!𝑛𝑘=01(𝑘!𝑛)𝑘(𝑛+𝛼+𝛽+1)𝑘(𝛼+1)𝑘1𝜂2𝑘,(6.10) one sees that, for 𝑔, the absolute value of 𝑃,𝑛(𝜂;𝑔,) is large near 𝜂=1 and small at 𝜂=1. Hence, in this limit, the zeros of 𝑃,𝑛(𝜂;𝑔,) distribute very near 𝜂=1. We show this in Figure 10 for certain parameters.

7. Summary

The discovery of new types of orthogonal polynomials, called the exceptional 𝑋 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 𝑃,𝑛(𝜂;𝝀) have the lowest degree =1,2,, 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 𝑋 polynomial 𝑃,𝑛(𝜂;𝝀) has 𝑛 zeros in the ordinary domain where the weight function is defined, that is, (0,) for the L1 and L2 polynomials and (1,1) 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 𝑛 zeros, there are extra zeros outside the ordinary domain.

For the ordinary zeros, their distribution as 𝑛 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 𝑛+1 zeros of 𝑃,𝑛+1(𝜂;𝝀) interlace with the 𝑛 zeros of 𝑃,𝑛(𝜂;𝝀). On the other hand, when increases at a fixed 𝑛, the type L1 and L2 Laguerre polynomials will have all their 𝑛 zeros shifted to the right. For the J1 and the J2 Jacobi polynomials, both the positive and negative zeros move toward the origin 𝜂=0 as increases.

For the extra zeros of 𝑃,𝑛(𝜂;𝝀), each and everyone of them is sandwiched between the corresponding zeros of the deforming polynomials 𝜉(𝜂;𝝀+𝜹) and 𝜉(𝜂;𝝀). As 𝑛 increases at a fixed , the extra zeros move from the zeros of 𝜉(𝜂;𝝀+𝜹) to those of 𝜉(𝜂;𝝀).

The behaviors of the extra zeros as increases at a fixed 𝑛 are more complex. For the L1 Laguerre polynomials, all the extra zeros lie on the negative 𝑥-axis. So as increases by one, the number of the extra zeros increases from to +1. For the L2 Laguerre and J1 and J2 Jacobi polynomials, they have /2 pairs of complex zeros for even , and (1)/2 pairs of complex zeros and a real zero outside the ordinary domains where the weight functions are defined. As increases, all the complex zeros move toward the right in the case of the L2 Laguerre and J1 Jacobi polynomials, and toward the left for the J2 Jacobi polynomials, while the extra real zeros bifurcate into new pairs of complex zeros. For large , these zeros appear to distribute symmetrically with respect to the 𝑥-axis in horse-shoe patterns. It is interesting to note that in the asymptotic regions of the parameters (𝑔1, 1), the exceptional polynomial 𝑃,𝑛(𝜂,𝝀) is expressed as the product of the original polynomial 𝑃𝑛(𝜂) and the deforming polynomial 𝜉(𝜂;𝝀), (6.3), (6.6), and (6.9). After completing this paper, we became aware of some new works on exceptional orthogonal polynomials [2123].

Appendix

Explicit Forms of Some Lower Degree Exceptional Orthogonal Polynomials

In the Appendix we provide, for self-containedness, the definitions of the classical Laguerre and Jacobi polynomials and the explicit forms of some lower degree members of the exceptional orthogonal polynomials. Here the independent variable is denoted by 𝑥.

The Classical Laguerre Polynomials
The degree 𝑛 classical Laguerre polynomial is defined by 𝐿𝑛(𝛼)1(𝑥)=𝑛!𝑛𝑘=0(𝑛)𝑘𝑘!(𝛼+𝑘+1)𝑛𝑘𝑥𝑘.(A.1)

The Classical Jacobi Polynomials
The degree 𝑛 classical Jacobi polynomial is defined by 𝑃𝑛(𝛼,𝛽)(𝑥)=(𝛼+1)𝑛𝑛!𝑛𝑘=01(𝑘!𝑛)𝑘(𝑛+𝛼+𝛽+1)𝑘(𝛼+1)𝑘1𝑥2𝑘.(A.2) In these formulas, (𝑎)𝑛def=𝑛1𝑘=0(𝑎+𝑘) is the shifted factorial (the Pochhammer symbol).

L1 Exceptional Laguerre Polynomials
𝑃𝐿11,01(𝑥;𝑔)=2𝑃(3+2𝑔)+𝑥,𝐿11,11(𝑥;𝑔)=4(1+2𝑔)(5+2𝑔)𝑥2,𝑃𝐿11,21(𝑥;𝑔)=116(1+2𝑔)(3+2𝑔)(7+2𝑔)81(1+2𝑔)(7+2𝑔)𝑥4(7+2𝑔)𝑥2+12𝑥3,𝑃𝐿12,01(𝑥;𝑔)=81(5+2𝑔)(7+2𝑔)+21(7+2𝑔)𝑥+2𝑥2,𝑃𝐿12,11(𝑥;𝑔)=116(3+2𝑔)(5+2𝑔)(9+2𝑔)+81(3+2𝑔)(9+2𝑔)𝑥4(9+2𝑔)𝑥212𝑥3,𝑃𝐿12,21(𝑥;𝑔)=64(3+2𝑔)(5+2𝑔)21(11+2𝑔)8(5+2𝑔)(11+2𝑔)𝑥2+14𝑥4,𝑃L13,0(1𝑥;𝑔)=(1487+2𝑔)(9+2𝑔)(11+2𝑔)+8(19+2𝑔)(11+2𝑔)𝑥+4(11+2𝑔)𝑥2+16𝑥3,𝑃𝐿13,11(𝑥;𝑔)=196(5+2𝑔)(7+2𝑔)(9+2𝑔)(13+2𝑔)+124(5+2𝑔)(9+2𝑔)(13+2𝑔)𝑥4(13+2𝑔)𝑥216(13+2𝑔)𝑥316𝑥4.(A.3)

L2 Exceptional Laguerre Polynomials
𝑃𝐿21,01(𝑥;𝑔)=2𝑃(3+2𝑔)𝑥,𝐿21,11(𝑥;𝑔)=4(1+2𝑔)(5+2𝑔)𝑥2,𝑃𝐿21,21(𝑥;𝑔)=116(1+2𝑔)(3+2𝑔)(7+2𝑔)+81(1+2𝑔)(7+2𝑔)𝑥+4(7+2𝑔)𝑥212𝑥3,𝑃𝐿22,01(𝑥;𝑔)=81(3+2𝑔)(5+2𝑔)+21(3+2𝑔)𝑥+2𝑥2,𝑃𝐿22,11(𝑥;𝑔)=116(1+2𝑔)(5+2𝑔)(7+2𝑔)+81(1+2𝑔)(7+2𝑔)𝑥4(1+2𝑔)𝑥212𝑥3,𝑃𝐿22,21(𝑥;𝑔)=164(1+2𝑔)(3+2𝑔)(7+2𝑔)(9+2𝑔)8(1+2𝑔)(9+2𝑔)𝑥2𝑥3+14𝑥4,𝑃𝐿23,0(1𝑥;𝑔)=(1483+2𝑔)(5+2𝑔)(7+2𝑔)8(13+2𝑔)(5+2𝑔)𝑥4(3+2𝑔)𝑥216𝑥3,𝑃𝐿23,11(𝑥;𝑔)=196(1+2𝑔)(5+2𝑔)(7+2𝑔)(9+2𝑔)124(1+2𝑔)(5+2𝑔)(9+2𝑔)𝑥4(1+2𝑔)𝑥2+16(1+2𝑔)𝑥3+16𝑥4.(A.4)

J1 Exceptional Jacobi Polynomials
𝑃𝐽11,01(𝑥;𝑔,)=21(3+𝑔+)+2𝑃(𝑔)𝑥,𝐽11,11(𝑥;𝑔,)=41(𝑔)(3+𝑔+)+45+6𝑔+2𝑔2+6+22𝑥+14(𝑔)(3+𝑔+)𝑥2,𝑃𝐽11,21(𝑥;𝑔,)=1678𝑔+2𝑔2+𝑔388𝑔𝑔2+22𝑔2+3+116(𝑔)11+13𝑔+3𝑔2+13+2𝑔+32𝑥+116(4+𝑔+)7+8𝑔+3𝑔2+82𝑔+32𝑥2+116(𝑔)(4+𝑔+)(5+𝑔+)𝑥3,𝑃𝐽12,01(𝑥;𝑔,)=823+9𝑔+𝑔2+11+2𝑔+214+1(1𝑔+)(5+𝑔+)𝑥8(2𝑔+)(1𝑔+)𝑥2,𝑃𝐽12,11(𝑥;𝑔,)=168+23𝑔+9𝑔2+𝑔323+2𝑔+𝑔2112𝑔23+116(5+𝑔+)15+11𝑔+3𝑔2+92𝑔+32𝑥116(1𝑔+)42+22𝑔+3𝑔2+18+2𝑔+32𝑥2+116(2𝑔+)(1𝑔+)(5+𝑔+)𝑥3,𝑃𝐽12,21(𝑥;𝑔,)=648154𝑔+7𝑔2+8𝑔3+𝑔410080𝑔10𝑔2+5216𝑔22𝑔22+103+4132367𝑔61𝑔220𝑔32𝑔4+75+652+203+24𝑥+132252+291𝑔+153𝑔2+36𝑔3+3𝑔4+250+80𝑔+10𝑔2+1252+16𝑔2+2𝑔22+303+34𝑥21(321𝑔+)(6+𝑔+)27+14𝑔+2𝑔2+10+22𝑥3+164(2𝑔+)(1𝑔+)(6+𝑔+)(7+𝑔+)𝑥4,𝑃𝐽13,01(𝑥;𝑔,)=48(7+𝑔+)39+11𝑔+𝑔2+17+2𝑔+2116(2𝑔+)45+13𝑔+𝑔2+15+2𝑔+2𝑥+116(3𝑔+)(2𝑔+)(7+𝑔+)𝑥2148(4𝑔+)(3𝑔+)(2𝑔+)𝑥3,𝑃𝐽13,11(𝑥;𝑔,)=96(7+𝑔+)2439𝑔11𝑔2𝑔3+396𝑔𝑔2+172+𝑔2+3+1961395+1223𝑔+443𝑔2+70𝑔3+4𝑔4+905+286𝑔+54𝑔2+4𝑔3+3592+30𝑔2+703+4𝑔3+44𝑥132(2𝑔+)(7+𝑔+)37+16𝑔+2𝑔2+12+22𝑥2+196(3𝑔+)(2𝑔+)135+46𝑔+4𝑔2+38+4𝑔+42𝑥3196(4𝑔+)(3𝑔+)(2𝑔+)(7+𝑔+)𝑥4.(A.5)

J2 Exceptional Jacobi Polynomials
𝑃𝐽21,01(𝑥;𝑔,)=21(3+𝑔+)2𝑃(𝑔)𝑥,𝐽21,11(𝑥;𝑔,)=41(𝑔)(3+𝑔+)45+6𝑔+2𝑔2+6+22𝑥14(𝑔)(3+𝑔+)𝑥2,𝑃𝐽21,21(𝑥;𝑔,)=167+8𝑔2𝑔2𝑔3+8+8𝑔+𝑔222+𝑔23116(𝑔)11+13𝑔+3𝑔2+13+2𝑔+32𝑥116(4+𝑔+)7+8𝑔+3𝑔2+82𝑔+32𝑥2116(𝑔)(4+𝑔+)(5+𝑔+)𝑥3,𝑃𝐽22,01(𝑥;𝑔,)=823+11𝑔+𝑔2+9+2𝑔+2+14+1(1+𝑔)(5+𝑔+)𝑥8(2+𝑔)(1+𝑔)𝑥2,𝑃𝐽22,11(𝑥;𝑔,)=168+23𝑔+11𝑔2+𝑔3232𝑔+𝑔292𝑔23+116(5+𝑔+)15+9𝑔+3𝑔2+112𝑔+32𝑥+116(1+𝑔)42+18𝑔+3𝑔2+22+2𝑔+32𝑥2+116(2+𝑔)(1+𝑔)(5+𝑔+)𝑥3,𝑃𝐽22,21(𝑥;𝑔,)=6481100𝑔+5𝑔2+10𝑔3+𝑔45480𝑔16𝑔2+7210𝑔22𝑔22+83+4+1323+75𝑔+65𝑔2+20𝑔3+2𝑔46761220324𝑥+132252+250𝑔+125𝑔2+30𝑔3+3𝑔4+291+80𝑔+16𝑔2+1532+10𝑔2+2𝑔22+363+34𝑥2+1(321+𝑔)(6+𝑔+)27+10𝑔+2𝑔2+14+22𝑥3+164(2+𝑔)(1+𝑔)(6+𝑔+)(7+𝑔+)𝑥4,𝑃𝐽23,01(𝑥;𝑔,)=48(7+𝑔+)39+17𝑔+𝑔2+11+2𝑔+2116(2+𝑔)45+15𝑔+𝑔2+13+2𝑔+2𝑥116(3+𝑔)(2+𝑔)(7+𝑔+)𝑥2148(4+𝑔)(3+𝑔)(2+𝑔)𝑥3,𝑃𝐽23,11(𝑥;𝑔,)=96(7+𝑔+)24+39𝑔+17𝑔2+𝑔3396𝑔+𝑔2112𝑔231961395+905𝑔+359𝑔2+70𝑔3+4𝑔4+1223+286𝑔+30𝑔2+4𝑔3+4432+54𝑔2+703+4𝑔3+44𝑥1(322+𝑔)(7+𝑔+)37+12𝑔+2𝑔2+16+22𝑥2196(3+𝑔)(2+𝑔)135+38𝑔+4𝑔2+46+4𝑔+42𝑥3196(4+𝑔)(3+𝑔)(2+𝑔)(7+𝑔+)𝑥4.(A.6)

Acknowledgments

This work is supported in part by the National Science Council (NSC) of the Republic of China under Grant no. NSC NSC-99-2112-M-032-002-MY3 (C. Ho) and in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, no. 19540179 (R. Sasaki). R. Sasaki wishes to thank the R.O.C.’s National Center for Theoretical Sciences and National Taiwan University for the hospitality extended to him during his visit in which part of the work was done.

References

  1. D. Gómez-Ullate, N. Kamran, and R. Milson, “An extended class of orthogonal polynomials defined by a Sturm-Liouville problem,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 352–367, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. D. Gómez-Ullate David, N. Kamran, and R. Milson, “An extension of Bochner's problem: exceptional invariant subspaces,” Journal of Approximation Theory, vol. 162, no. 5, pp. 987–1006, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. S. Odake and R. Sasaki, “Infinitely many shape invariant potentials and new orthogonal polynomials,” Physics Letters B, vol. 679, no. 4, pp. 414–417, 2009. View at Publisher · View at Google Scholar
  4. S. Odake and R. Sasaki, “Another set of infinitely many exceptional (X) Laguerre polynomials,” Physics Letters B, vol. 684, no. 2-3, pp. 173–176, 2009. View at Publisher · View at Google Scholar
  5. S. Odake and R. Sasaki, “Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials,” Journal of Mathematical Physics, vol. 51, no. 5, Article ID 053513, 9 pages, 2010. View at Publisher · View at Google Scholar
  6. C. Quesne, “Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry,” Journal of Physics A, vol. 41, no. 39, Article ID 392001, 6 pages, 2008. View at Publisher · View at Google Scholar
  7. C. Quesne, “Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics,” SIGMA, vol. 5, article 084, 24 pages, 2009. View at Publisher · View at Google Scholar
  8. B. Bagchi, C. Quesne, and R. Roychoudhury, “Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry,” Pramana—Journal of Physics, vol. 73, no. 2, pp. 337–347, 2009. View at Publisher · View at Google Scholar · View at Scopus
  9. C. L. Ho, S. Odake, and R. Sasaki, “Properties of the exceptional (X) Laguerre and Jacobi polynomials,” SIGMA, vol. 7, article 107, 24 pages, 2011.
  10. D. Dutta and P. Roy, “Conditionally exactly solvable potentials and exceptional orthogonal polynomials,” Journal of Mathematical Physics, vol. 51, no. 4, Article ID 042101, 9 pages, 2010. View at Publisher · View at Google Scholar
  11. S. Bochner, “Über Sturm-Liouvillesche Polynomsysteme,” Mathematische Zeitschrift, vol. 29, no. 1, pp. 730–736, 1929. View at Publisher · View at Google Scholar
  12. S. Odake and R. Sasaki, “Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials,” Physics Letters B, vol. 682, no. 1, pp. 130–136, 2009. View at Publisher · View at Google Scholar
  13. S. Odake and R. Sasaki, “The Exceptional (X) (q)-Racah Polynomials,” Progress of Theoretical Physics, vol. 125, no. 5, pp. 851–870, 2011. View at Publisher · View at Google Scholar
  14. D. Gómez-Ullate, N. Kamran, and R. Milson, “Exceptional orthogonal polynomials and the Darboux transformation,” Journal of Physics A, vol. 43, no. 43, Article ID 434016, 16 pages, 2010. View at Publisher · View at Google Scholar
  15. R. Sasaki, S. Tsujimoto, and A. Zhedanov, “Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations,” Journal of Physics A, vol. 43, no. 31, Article ID 315204, 20 pages, 2010. View at Publisher · View at Google Scholar
  16. C. L. Ho, “Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials,” Annals of Physics, vol. 326, no. 4, pp. 797–807, 2011. View at Publisher · View at Google Scholar
  17. G. Szego, Orthogonal Polynomials, vol. 23 of American Mathematical Society Colloquium Publications, American Mathematical Society, New York, NY, USA, 1939.
  18. T. S. Chihara, An Introduction to Orthogonal Polynomials, vol. 13 of Mathematics and its Applications, Gordon and Breach Science, New York, NY, USA, 1978.
  19. G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1999.
  20. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, vol. 98 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 2005.
  21. Y. Grandati and A. Bérard, “Solvable rational extension of translationally shape invariantpotentials,” http://128.84.158.119/abs/0912.3061v2.
  22. Y. Grandati, “Solvable rational extensions of the isotonic oscillator,” Annals of Physics, vol. 326, no. 8, pp. 2074–2090, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. Y. Grandati, “Solvable rational extensions of the Morse and Kepler-Coulomb potentials,” Journal of Mathematical Physics, vol. 52, no. 10, Article ID 103505, 12 pages, 2011. View at Publisher · View at Google Scholar