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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 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.Β Goldin and R.Β Schiappa

Copyright Β© 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 𝑋ℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree β„“=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 𝑋ℓ 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 β„“ 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 𝑋ℓ polynomials, with β„“=1,2,…, were constructed in [3–5]. These systems were derived by deforming the radial oscillator potential and the Darboux-PΓΆschl-Teller (DPT) potential in terms of an eigen polynomial of degree β„“ (β„“=1,2,…). The lowest (β„“=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. [6–8]. By construction, these new orthogonal polynomials satisfy a second order differential equation (the SchrΓΆdinger equation) without contradicting Bochner’s theorem [11], since they start at degree β„“>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 𝑋ℓ 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 𝑋ℓ polynomials within the framework of the Fuchsian differential equations in the entire complex π‘₯-plane. They allow us to study in depth some important properties of the 𝑋ℓ polynomials, such as the actions of the forward and backward shift operators on the 𝑋ℓ polynomials, Gram-Schmidt orthonormalization for the algebraic construction of the 𝑋ℓ 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 β„“ 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 β„“. In Section 6, we consider behaviors of the zeros at large 𝑔 and/or β„Ž. 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 𝝀 and their shifts 𝜹 and a nonnegative integer β„“, which is the degree of the deforming polynomials. The two sets of exceptional Laguerre polynomials (β„“=1,2,…, 𝑛=0,1,2,…) are [9] 𝑃ℓ,𝑛(πœ‚;𝝀)def=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πœ‰β„“(πœ‚;𝝀+𝜹)𝑃𝑛(πœ‚;𝑔+β„“βˆ’1)βˆ’πœ‰β„“(πœ‚;𝝀)πœ•πœ‚π‘ƒπ‘›ξ‚€1(πœ‚;𝑔+β„“βˆ’1):L1𝑛+𝑔+2ξ‚βˆ’11𝑔+2ξ‚πœ‰β„“(πœ‚;𝝀+𝜹)𝑃𝑛(πœ‚;𝑔+β„“+1)+πœ‚πœ‰β„“(πœ‚;𝝀)πœ•πœ‚π‘ƒπ‘›(ξ‚‡πœ‚;𝑔+β„“+1):L2,(2.1) in which 𝝀def=𝑔>0, 𝜹def=1 and 𝑃𝑛(πœ‚;𝑔)def=𝐿𝑛(π‘”βˆ’(1/2))(πœ‚),πœ‰β„“(πœ‚;𝑔)def=⎧βŽͺ⎨βŽͺβŽ©πΏβ„“(𝑔+β„“βˆ’(3/2))𝐿(βˆ’πœ‚):L1β„“(βˆ’π‘”βˆ’β„“βˆ’(1/2))(πœ‚).:L2.(2.2) Here 𝐿𝑛(𝛼)(πœ‚) are the classical Laguerre polynomials (A.1). It is interesting to note the following alternative expressions for the L1 and L2 exceptional polynomials: 𝑃ℓ,𝑛(πœ‚;𝝀)def=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πΏβ„“(𝑔+β„“βˆ’(1/2))(βˆ’πœ‚)𝐿𝑛(𝑔+β„“βˆ’(1/2))(πœ‚)βˆ’πΏ(𝑔+β„“βˆ’(1/2))β„“βˆ’1(βˆ’πœ‚)𝐿(𝑔+β„“βˆ’(1/2))π‘›βˆ’1((πœ‚):L1𝑛+𝑔+(1/2))βˆ’1ξ‚€1𝑔+2𝐿ℓ(βˆ’π‘”βˆ’β„“βˆ’(3/2))(πœ‚)𝐿𝑛(𝑔+β„“+(3/2))(πœ‚)+(β„“+1)𝐿(βˆ’π‘”βˆ’β„“βˆ’(3/2))β„“+1(πœ‚)𝐿(𝑔+β„“+(3/2))π‘›βˆ’1(πœ‚):L2.(2.3) The two sets of exceptional Jacobi polynomials (β„“=1,2,…, 𝑛=0,1,2,…) are [9] 𝑃ℓ,𝑛(πœ‚;𝝀)def=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©ξ‚€ξ‚€1𝑛+β„Ž+2ξ‚ξ‚βˆ’1ξ€½(β„Ž+(1/2))πœ‰β„“(πœ‚;𝝀+𝜹)𝑃𝑛(πœ‚;𝑔+β„“βˆ’1,β„Ž+β„“+1)+(1+πœ‚)πœ‰β„“(πœ‚;𝝀)πœ•πœ‚π‘ƒπ‘›ξ€Ύξ‚€ξ‚€1(πœ‚;𝑔+β„“βˆ’1,β„Ž+β„“+1):J1𝑛+𝑔+2ξ‚ξ‚βˆ’1ξ€½(𝑔+(1/2))πœ‰β„“(πœ‚;𝝀+𝜹)𝑃𝑛(πœ‚;𝑔+β„“+1,β„Ž+β„“βˆ’1)βˆ’(1βˆ’πœ‚)πœ‰β„“(πœ‚;𝝀)πœ•πœ‚π‘ƒπ‘›ξ€Ύ(πœ‚;𝑔+β„“+1,β„Ž+β„“βˆ’1):J2,(2.4) in which 𝝀def=(𝑔,β„Ž), 𝑔>0, β„Ž>0, 𝜹def=(1,1) and 𝑃𝑛(πœ‚;𝑔,β„Ž)def=𝑃𝑛(π‘”βˆ’(1/2),β„Žβˆ’(1/2))(πœ‚),πœ‰β„“(πœ‚;𝑔,β„Ž)def=⎧βŽͺ⎨βŽͺβŽ©π‘ƒβ„“(𝑔+β„“βˆ’(3/2),βˆ’β„Žβˆ’β„“βˆ’(1/2))𝑃(πœ‚),𝑔>β„Ž>0:J1β„“(βˆ’π‘”βˆ’β„“βˆ’(1/2),β„Ž+β„“βˆ’(3/2))(πœ‚),β„Ž>𝑔>0:J2,(2.5) where 𝑃𝑛(𝛼,𝛽)(πœ‚) 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 β„“, and the two polynomials πœ‰β„“(πœ‚;𝝀) and πœ‰β„“(πœ‚;𝝀+𝜹) played the role of the deforming polynomials.

The differential equations satisfied by these new polynomials are as follows: L1βˆΆπœ‚πœ•2πœ‚π‘ƒβ„“,𝑛1(πœ‚;𝝀)+𝑔+β„“+2βˆ’πœ‚βˆ’2πœ‚πœ•πœ‚πœ‰β„“(πœ‚;𝝀)πœ‰β„“ξ‚Άπœ•(πœ‚;𝝀)πœ‚π‘ƒβ„“,𝑛+ξ‚΅2(πœ‚;𝝀)πœ‚πœ•πœ‚πœ‰β„“(πœ‚;𝝀+𝜹)πœ‰β„“ξ‚Άπ‘ƒ(πœ‚;𝝀)+π‘›βˆ’β„“β„“,𝑛(πœ‚;𝝀)=0,(2.6)L2βˆΆπœ‚πœ•2πœ‚π‘ƒβ„“,𝑛1(πœ‚;𝝀)+𝑔+β„“+2βˆ’πœ‚βˆ’2πœ‚πœ•πœ‚πœ‰β„“(πœ‚;𝝀)πœ‰β„“ξ‚Άπœ•(πœ‚;𝝀)πœ‚π‘ƒβ„“,𝑛+ξ‚΅((πœ‚;𝝀)βˆ’2𝑔+(1/2))πœ•πœ‚πœ‰β„“(πœ‚;𝝀+𝜹)πœ‰β„“ξ‚Άπ‘ƒ(πœ‚;𝝀)+𝑛+β„“β„“,𝑛(πœ‚;𝝀)=0,(2.7)J1∢1βˆ’πœ‚2ξ€Έπœ•2πœ‚π‘ƒβ„“,𝑛(ξƒ©ξ€·πœ‚;𝝀)+β„Žβˆ’π‘”βˆ’(𝑔+β„Ž+2β„“+1)πœ‚βˆ’21βˆ’πœ‚2ξ€Έπœ•πœ‚πœ‰β„“(πœ‚;𝝀)πœ‰β„“ξƒͺπœ•(πœ‚;𝝀)πœ‚π‘ƒβ„“,𝑛(+ξ‚΅βˆ’πœ‚;𝝀)2(β„Ž+(1/2))(1βˆ’πœ‚)πœ•πœ‚πœ‰β„“(πœ‚;𝝀+𝜹)πœ‰β„“ξ‚Άπ‘ƒ(πœ‚;𝝀)+β„“(β„“+π‘”βˆ’β„Žβˆ’1)+𝑛(𝑛+𝑔+β„Ž+2β„“)β„“,𝑛(πœ‚;𝝀)=0,(2.8)J2∢1βˆ’πœ‚2ξ€Έπœ•2πœ‚π‘ƒβ„“,𝑛(πœ‚;𝝀)+β„Žβˆ’π‘”βˆ’(𝑔+β„Ž+2β„“+1)πœ‚βˆ’21βˆ’πœ‚2ξ€Έπœ•πœ‚πœ‰β„“(πœ‚;𝝀)πœ‰β„“ξƒͺπœ•(πœ‚;𝝀)πœ‚π‘ƒβ„“,𝑛+ξ‚΅(πœ‚;𝝀)2(𝑔+(1/2))(1+πœ‚)πœ•πœ‚πœ‰β„“(πœ‚;𝝀+𝜹)πœ‰β„“ξ‚Άπ‘ƒ(πœ‚;𝝀)+β„“(β„“+β„Žβˆ’π‘”βˆ’1)+𝑛(𝑛+𝑔+β„Ž+2β„“)β„“,𝑛(πœ‚;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 𝑋ℓ polynomial 𝑃ℓ,𝑛(πœ‚;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 (βˆ’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 [17–20]. We will say more about these zeros in Section 4. Besides these 𝑛 zeros, there are extra β„“ zeros outside the ordinary domain. For convenience, we will adopt the following notation for the zeros of the various polynomials involved: πœ‰π‘˜(β„“):zerosofπœ‰β„“πœ‰(πœ‚;𝝀+𝜹),π‘˜=1,2,…,β„“;(2.10)π‘˜(β„“):zerosofπœ‰β„“(πœ‚;𝝀),π‘˜=1,2,…,β„“;(2.11)πœ‚π‘˜(β„“,𝑛):extrazerosof𝑃ℓ,π‘›πœ‚,π‘˜=1,2,…,β„“;(2.12)𝑗(β„“,𝑛):ordinaryzerosof𝑃ℓ,𝑛,𝑗=1,2,…,𝑛.(2.13) We emphasize that πœ‚π‘—(β„“,𝑛)∈(0,∞) for the L1 and L2 Laguerre polynomials, and πœ‚π‘—(β„“,𝑛)∈(βˆ’1,1) for the J1 and J2 Jacobi polynomials.

Figures 1–10 depict the distribution of the zeros for some representative parameters of the systems, namely, 𝑛,β„“,𝑔, and β„Ž. 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 πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the L1 Laguerre polynomials, with 𝑔=0.5 and β„“=2. The three diagrams correspond to 𝑛=1(a), 2(b), and 3(c), respectively. The ordinary zeros πœ‚π‘—(β„“,𝑛) lie in (0,∞).
fig2
Figure 2: L2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the L2 Laguerre polynomials, with 𝑔=0.5 and β„“=3. The three diagrams correspond to 𝑛=1(a), 2(b), and 5(c), respectively. The ordinary zeros πœ‚π‘—(β„“,𝑛) lie in (0,∞).
fig3
Figure 3: J2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the J2 Jacobi polynomials, with 𝑔=3,β„Ž=4, and β„“=3. The three diagrams correspond to 𝑛=1(a), 2(b), and 5(c), respectively. The ordinary zeros πœ‚π‘—(β„“,𝑛) lie in (βˆ’1,1).
fig4
Figure 4: L1: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the L1 Laguerre polynomials, with 𝑔=0.5 and 𝑛=2. The three diagrams correspond to β„“=1(a), 2(b), and 3(c), respectively.
fig5
Figure 5: L1: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the L1 Laguerre polynomials, with 𝑔=1.5 and 𝑛=2. The three diagrams correspond to β„“=1(a), 2(b), and 3(c), respectively.
fig6
Figure 6: L2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the L2 Laguerre polynomials, with 𝑔=2 and 𝑛=2. The four diagrams correspond to β„“=1(a), 2(b), 3(c), and 20(d), respectively.
fig7
Figure 7: L2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the L2 Laguerre polynomials, with 𝑔=5 and 𝑛=2. The four diagrams correspond to β„“=1(a), 2(b), 3(c), and 20(d), respectively.
fig8
Figure 8: J2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the J2 Jacobi polynomials, with 𝑔=3, β„Ž=4, and 𝑛=4. The four diagrams correspond to β„“=1(a), 2(b), 3(c), and 20(d), respectively.
fig9
Figure 9: J2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the J2 Jacobi polynomials, with 𝑔=7, β„Ž=8, and 𝑛=4. The four diagrams correspond to β„“=1(a), 2(b), 3(c), and 20(d), respectively.
fig10
Figure 10: J2: distributions of the zeros πœ‚π‘˜(β„“,𝑛),πœ‚π‘—(β„“,𝑛) (♦), πœ‰π‘˜(β„“) (β—‹), and πœ‰π‘˜(β„“) (β– ) for the J2 Jacobi polynomials, with 𝑔=2, β„“=10, and 𝑛=4. The two diagrams correspond to β„Ž=50(a) and 100(b), respectively.

3. Extra β„“ Zeros of 𝑋ℓ Polynomials

Here we discuss the locations of the extra β„“ 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 β„“ extra zeros of L1 polynomials are on the negative real line (βˆ’βˆž,0). The L2 π‘‹β„“βˆΆodd polynomials have one real negative zero which lies to the left of the remaining (1/2)(β„“βˆ’1) pairs of complex conjugate roots. The L2 π‘‹β„“βˆΆeven polynomials have (1/2)β„“ pairs of complex conjugate roots.

The situations for the 𝑋ℓ Jacobi polynomials are a bit more complicated. The J1 π‘‹β„“βˆΆodd polynomials have one real negative root which lies to the left of the remaining (1/2)(β„“βˆ’1) pairs of complex conjugate roots with negative real parts. The J1 π‘‹β„“βˆΆeven polynomials have (1/2)β„“ pairs of complex conjugate roots with negative real parts. The J2 π‘‹β„“βˆΆodd polynomials have one real positive root which lies to the right of the remaining (1/2)(β„“βˆ’1) pairs of complex conjugate roots with positive real parts. The J2 π‘‹β„“βˆΆeven polynomials have (1/2)β„“ 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 πœ‚β†”βˆ’πœ‚ and π‘”β†”β„Ž, as exemplified by the relation πœ‰J2β„“(πœ‚;𝑔,β„Ž)=(βˆ’1)β„“πœ‰J1β„“(βˆ’πœ‚;β„Ž,𝑔) [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 𝑛 Increases at Fixed β„“

In all cases, we have 𝑃ℓ,0(πœ‚;𝝀)βˆπœ‰β„“(πœ‚;𝝀+𝜹).(3.1) This implies that the zeros of 𝑃ℓ,0 coincide with those of πœ‰β„“(πœ‚;𝝀+𝜹), namely, πœ‰π‘˜(β„“),π‘˜=1,2,…,β„“.

At fixed β„“, all the πœ‚π‘˜(β„“,𝑛) move from πœ‰π‘˜(β„“) at 𝑛=0 to πœ‰π‘˜(β„“) as π‘›β†’βˆž. This can be seen from Figures 1–3 and in Tables 1–7. We will provide heuristic arguments for this result in Section 5.

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

The discussions in the last subsection show that πœ‚π‘˜(β„“,𝑛) by proper numbering are sandwiched between πœ‰π‘˜(β„“) and πœ‰π‘˜(β„“). Thus to know how πœ‚π‘˜(β„“,𝑛) behave as β„“ increases at fixed 𝑛, we only need to study how the zeros πœ‰π‘˜(β„“) and πœ‰π‘˜(β„“) flow as β„“ increases.

3.2.1. L1 Laguerre

As β„“ changes to β„“+1, the zeros of πœ‰β„“(πœ‚;𝑔+1) and πœ‰β„“(πœ‚;𝑔) decrease (move to the left), and a new set of zeros appear from the right. πœ‰π‘˜(β„“+1)<πœ‰π‘˜(β„“),πœ‰π‘˜(β„“+1)<πœ‰π‘˜(β„“),πœ‰π‘˜(β„“)<πœ‰π‘˜(β„“)<πœ‰(β„“)π‘˜+1<πœ‰(β„“)π‘˜+1,(3.2) for π‘˜=1,2,…,β„“βˆ’1,β„“.

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

3.2.2. L2 Laguerre

For β„“=1, there is one real root each for πœ‰β„“(πœ‚;𝑔+1) and πœ‰β„“(πœ‚;𝑔), with πœ‰1(β„“)<πœ‰1(β„“)<0.

For β„“=2, the above two roots bifurcate into two complex roots, with β„œπœ‰(β„“)<β„œπœ‰(β„“), |β„‘πœ‰(β„“)|>|β„‘πœ‰(β„“)|.

Generally, for even β„“, there are β„“ complex zeros with β„œπœ‰π‘˜(β„“)<β„œπœ‰π‘˜(β„“),|||β„‘πœ‰π‘˜(β„“)|||>||β„‘πœ‰π‘˜(β„“)||β„“,π‘˜=1,2,…,2.(3.3) All πœ‚π‘˜(β„“,𝑛) are sandwiched between πœ‰π‘˜(β„“) and πœ‰π‘˜(β„“). As an even β„“ changes to β„“+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 β„“ 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 β„“ increases.

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

3.2.3. J2 Jacobi

For β„“=1, there is one real root each for πœ‰β„“(πœ‚;𝑔+1) and πœ‰β„“(πœ‚;𝑔), with πœ‰1(β„“)>πœ‰1(β„“)>1.

For β„“=2, the above two roots bifurcate into two complex roots, with β„œπœ‰(β„“)>β„œπœ‰(β„“), |β„‘πœ‰(β„“)|>|β„‘πœ‰(β„“)|.

Generally, for even β„“, there are β„“ complex zeros with β„œπœ‰π‘˜(β„“)>β„œπœ‰π‘˜(β„“),|||β„‘πœ‰π‘˜(β„“)|||>||β„‘πœ‰π‘˜(β„“)||β„“,π‘˜=1,2,…,2.(3.4) As β„“ changes to β„“+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 π‘₯-axis. As β„“ 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 β„“ increases. This pattern continues as β„“ increases.

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

4. Ordinary Zeros of 𝑋ℓ Polynomials

In the case of 𝑋ℓ polynomials 𝑃ℓ,𝑛(πœ‚;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 (βˆ’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 β„“

This is guaranteed by the oscillation theorem of the one-dimensional quantum mechanics, since 𝑃ℓ,𝑛(πœ‚;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 𝑃ℓ,𝑛 decrease, and a new zero appears from the right. Thus the 𝑛 zeros of 𝑃ℓ,𝑛(πœ‚;𝝀) and the 𝑛+1 zeros of 𝑃ℓ,𝑛+1(πœ‚;𝝀) interlace with each other: each zero of 𝑃ℓ,𝑛(πœ‚;𝝀) is surrounded by two zeros of 𝑃ℓ,𝑛+1(πœ‚;𝝀).

Figures 1–3 show these behaviors for selected parameters.

4.2. Behaviors as β„“ Increases at Fixed 𝑛

From Figures 4–7, 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 β„“ increases.

For J2 Jacobi polynomials, the positive (negative) zeros shift left (right) as β„“ increases, that is, they move toward the origin πœ‚=0. This is illustrated in Figures 8 and 9.

4.3. Additional Observation for the L1 Case

Using the well-known derivative relation πœ•πœ‚πΏπ‘›(𝛼)(πœ‚)=βˆ’πΏ(𝛼+1)π‘›βˆ’1(πœ‚),(4.1) and 𝐿𝑛(𝛼)(πœ‚)βˆ’πΏπ‘›(π›Όβˆ’1)(πœ‚)=𝐿(𝛼)π‘›βˆ’1(πœ‚),(4.2) we get 𝑃ℓ,β„“(πœ‚;𝑔)=𝐿ℓ(𝑔+β„“βˆ’(1/2))(βˆ’πœ‚)𝐿ℓ(𝑔+β„“βˆ’(3/2))(πœ‚)+𝐿ℓ(𝑔+β„“βˆ’(1/2))(πœ‚)𝐿ℓ(𝑔+β„“βˆ’(3/2))(βˆ’πœ‚)βˆ’πΏβ„“(𝑔+β„“βˆ’(3/2))(πœ‚)𝐿ℓ(𝑔+β„“βˆ’(3/2))(βˆ’πœ‚).(4.3) Hence, when 𝑛=β„“, the L1 Laguerre is an even function of πœ‚, and its zeros are symmetric with respect to πœ‚=0.

5. Asymptotic Behavior of πœ‚π‘˜(β„“,𝑛)β†’πœ‰π‘˜(β„“) as π‘›β†’βˆž

Here we provide intuitive arguments for the above asymptotic behavior. As mentioned before, for 𝑛=0, we have πœ‚π‘˜(β„“,0)=πœ‰π‘˜(β„“), as 𝑃𝑛=0=1. We will show that as π‘›β†’βˆž, πœ‚π‘˜(β„“,𝑛)β†’πœ‰π‘˜(β„“). This amounts to showing that in this limit, πœ•πœ‚π‘ƒπ‘› 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(πœ‚)β‰…πœ‚/22βˆšπœ‹(βˆ’πœ‚)βˆ’((𝛼/2)βˆ’(1/4))𝑛((𝛼/2)βˆ’(1/4))𝑒2βˆšβˆ’π‘›πœ‚,π›Όβˆˆβ„,πœ‚βˆˆβ„‚β§΅(0,∞),(5.1) which gives the asymptotic form of 𝐿𝑛(𝛼)(πœ‚) for large 𝑛. For the L1 and L2 cases, we have 𝛼=𝑔+β„“βˆ’3/2 and 𝑔+β„“+1/2, respectively.

One finds |||||𝐿𝑛(𝛼)(πœ‚)πœ•πœ‚πΏπ‘›(𝛼)|||||∼|||||βˆ’1(πœ‚)βˆšπ‘›(βˆ’πœ‚)1/2|||||.(5.2)

For large 𝑛 with fixed πœ‚, πœ•πœ‚πΏπ‘›(𝛼)(πœ‚) dominates over 𝐿𝑛(𝛼)(πœ‚), and thus the zeros of 𝑃ℓ,𝑛 are determined by those of πœ‰β„“(πœ‚;𝑔) as π‘›β†’βˆž.

5.2. J2 Jacobi

For the asymptotic form of 𝑃𝑛(𝛼,𝛽)(πœ‚) for large 𝑛, we will make use of Theorem 8.21.7 of [17]: 𝑃𝑛(𝛼,𝛽)(πœ‚)β‰…(πœ‚βˆ’1)βˆ’π›Ό/2(πœ‚+1)βˆ’π›½/2ξ‚†βˆšβˆšπœ‚+1+ξ‚‡πœ‚βˆ’1𝛼+π›½Γ—ξ€·πœ‚2ξ€Έβˆ’1βˆ’1/4βˆšξ‚†βˆš2πœ‹π‘›πœ‚+πœ‚2ξ‚‡βˆ’1𝑛+(1/2)[],,𝛼,π›½βˆˆβ„,πœ‚βˆˆβ„‚β§΅βˆ’1,1(5.3) and πœ•πœ‚π‘ƒπ‘›(𝛼,𝛽)1(πœ‚)=2(𝑛+𝛼+𝛽+1)𝑃(𝛼+1,𝛽+1)π‘›βˆ’1(πœ‚).(5.4)

One finds 𝑃𝑛(𝛼,𝛽)(πœ‚)πœ•πœ‚π‘ƒπ‘›(𝛼,𝛽)∼2(πœ‚)ξ‚™(𝑛+𝛼+𝛽+1)π‘›βˆ’1π‘›ξ€·πœ‚2ξ€Έβˆ’11/2πœ‚+βˆšπœ‚2βˆ’1ξ‚€βˆšβˆšπœ‚+1+ξ‚πœ‚βˆ’12.(5.5)

Again, for large 𝑛 with fixed πœ‚, πœ•πœ‚π‘ƒπ‘›(𝛼,𝛽)(πœ‚) dominates over 𝑃𝑛(𝛼,𝛽)(πœ‚), and thus the zeros of 𝑃ℓ,𝑛 are determined by those of πœ‰β„“(πœ‚;𝑔) as π‘›β†’βˆž.

6. Behaviors at Large 𝑔 and/or β„Ž

6.1. L1 Laguerre

As 𝑔 increases, we have that |||πœ‰π‘˜(β„“)|||,||πœ‰π‘˜(β„“)||,|||πœ‚π‘˜(β„“,𝑛)|||,||πœ‚π‘˜(β„“,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 πœ‰β„“(πœ‚;𝑔+1)β‰ˆπœ‰β„“(πœ‚;𝑔). Hence 𝑃ℓ,𝑛(πœ‚;𝑔)β‰ˆπœ‰β„“ξ‚ƒπΏ(πœ‚;𝑔)𝑛(𝑔+β„“βˆ’(3/2))(πœ‚)βˆ’πœ•πœ‚πΏπ‘›(𝑔+β„“βˆ’(3/2))ξ‚„(πœ‚)β‰ˆπœ‰β„“(πœ‚;𝑔)𝐿𝑛(𝑔+β„“βˆ’(1/2))(πœ‚).(6.2) For 𝑔≫1, 𝑃ℓ,𝑛(πœ‚;𝑔) approaches 𝑃ℓ,𝑛(πœ‚;𝑔)β‰ˆπΏβ„“(𝑔+β„“)(βˆ’πœ‚)𝐿𝑛(𝑔+β„“)(πœ‚).(6.3) Thus the extra (πœ‚π‘˜(β„“,𝑛)) and the ordinary (πœ‚π‘˜(β„“,𝑛)) zeros of 𝑃ℓ,𝑛(πœ‚;𝑔) are given by the zeros of 𝐿ℓ(𝑔+β„“)(βˆ’πœ‚) and 𝐿𝑛(𝑔+β„“)(πœ‚), respectively.

6.2. L2 Laguerre

As 𝑔 increases, we have β„œπœ‰π‘˜(β„“),β„œπœ‰π‘˜(β„“) decreased, |β„‘πœ‰π‘˜(β„“)|, |β„‘πœ‰π‘˜(β„“)| increased, and πœ‚π‘˜(β„“,𝑛) increased. This is easily seen from Figures 6 and 7. That is, the zeros πœ‰π‘˜(β„“),πœ‰π‘˜(β„“), and hence πœ‚π‘˜(β„“,𝑛) all are moving leftwards and away from the π‘₯-axis, while the ordinary zeros πœ‚π‘˜(β„“,𝑛) are moving towards the right.

In fact, for large 𝑔, we have πœ‰β„“(πœ‚;𝑔+1)β‰ˆπœ‰β„“(πœ‚;𝑔). Hence 𝑃ℓ,𝑛(πœ‚;𝑔)β‰ˆπœ‰β„“1(πœ‚;𝑔)𝑔+2𝐿𝑛(𝑔+β„“+(1/2))(πœ‚)+πœ‚πœ•πœ‚πΏπ‘›(𝑔+β„“+(1/2))ξ‚„.(πœ‚)(6.4) Using (E.2), (E.10), and (E.9) of [9], we arrive at 𝑃ℓ,𝑛(πœ‚;𝑔)β‰ˆπœ‰β„“1(πœ‚;𝑔)𝑔+β„“+2𝐿+𝑛𝑛(𝑔+β„“βˆ’(1/2))(πœ‚)βˆ’β„“πΏπ‘›(𝑔+β„“+(1/2))ξ‚„.(πœ‚)(6.5) For 𝑔≫1, 𝑃ℓ,𝑛(πœ‚;𝑔) approaches 𝑃ℓ,𝑛(πœ‚;𝑔)β‰ˆπΏβ„“(βˆ’π‘”βˆ’β„“)(πœ‚)𝐿𝑛(𝑔+β„“)(πœ‚).(6.6) Thus the extra (πœ‚π‘˜(β„“,𝑛)) and the ordinary (πœ‚π‘˜(β„“,𝑛)) zeros of 𝑃ℓ,𝑛(πœ‚;𝑔) are given by the zeros of 𝐿ℓ(βˆ’π‘”βˆ’β„“)(πœ‚) and 𝐿𝑛(𝑔+β„“)(πœ‚), respectively.

6.3. J2 Jacobi

As 𝑔,β„Ž increases, we have β„œπœ‰π‘˜(β„“), β„œπœ‰π‘˜(β„“), |β„‘πœ‰π‘˜(β„“)|, |β„‘πœ‰π‘˜(β„“)| increased, as is evident from Figures 8 and 9. The extra zeros πœ‚π‘˜(β„“,𝑛), being in between these zeros, follow the same pattern. That is, the zeros πœ‰π‘˜(β„“), πœ‰π‘˜(β„“), and hence πœ‚π‘˜(β„“,𝑛) all are moving away from the π‘₯ and 𝑦-axes. The ordinary zeros πœ‚π‘˜(β„“,𝑛) will have their norm |πœ‚π‘˜(β„“,𝑛)|decrease in general as 𝑔 increases. Thus these zeros move towards the 𝑦-axis.

In fact, for large 𝑔 and β„Ž, we have (𝛼≑𝑔+β„“+(1/2), π›½β‰‘β„Ž+β„“βˆ’(3/2)) as 𝑃ℓ,𝑛(πœ‚;𝑔,β„Ž)β‰ˆπœ‰β„“1(πœ‚;𝑔,β„Ž)𝑔+2𝑃𝑛(𝛼,𝛽)(πœ‚)βˆ’(1βˆ’πœ‚)πœ•πœ‚π‘ƒπ‘›(𝛼,𝛽)ξ‚„.(πœ‚)(6.7) Using (E.13) and (E.23) of [9], we arrive at 𝑃ℓ,𝑛(πœ‚;𝑔,β„Ž)β‰ˆπœ‰β„“ξ‚ƒ(πœ‚;𝑔,β„Ž)(𝛼+𝑛)𝑃𝑛(𝛼+1,𝛽+1)(πœ‚)βˆ’β„“π‘ƒπ‘›(𝛼,𝛽)ξ‚„(πœ‚).(6.8) For 𝑔≫1 and β„Žβ‰«1, 𝑃ℓ,𝑛(πœ‚;𝑔,β„Ž) approaches 𝑃ℓ,𝑛(πœ‚;𝑔)β‰ˆπ‘ƒβ„“(βˆ’π‘”βˆ’β„“,β„Ž+β„“)(πœ‚)𝑃𝑛(𝑔+β„“,β„Ž+β„“)(πœ‚).(6.9) Thus the extra (πœ‚π‘˜(β„“,𝑛)) and the ordinary (πœ‚π‘˜(β„“,𝑛)) zeros of 𝑃ℓ,𝑛(πœ‚;𝑔,β„Ž) are given by the zeros of 𝑃ℓ(βˆ’π‘”βˆ’β„“,β„Ž+β„“)(πœ‚) and 𝑃𝑛(𝑔+β„“,β„Ž+β„“)(πœ‚), respectively.

6.3.1. Additional Observation: β„Žβ‰«π‘”

For β„Žβ‰«π‘”, all zeros, that is, πœ‰π‘—(β„“), πœ‰π‘—(β„“), πœ‚π‘˜(β„“,𝑛), πœ‚π‘˜(β„“,𝑛), gather around πœ‚=1. This can be understood as follows. From the series expansion of the Jacobi polynomials, (A.2) 𝑃𝑛(𝛼,𝛽)(πœ‚)=(𝛼+1)𝑛𝑛!π‘›ξ“π‘˜=01(π‘˜!βˆ’π‘›)π‘˜(𝑛+𝛼+𝛽+1)π‘˜(𝛼+1)π‘˜ξ‚΅1βˆ’πœ‚2ξ‚Άπ‘˜,(6.10) one sees that, for β„Žβ‰«π‘”, the absolute value of 𝑃ℓ,𝑛(πœ‚;𝑔,β„Ž) is large near πœ‚=βˆ’1 and small at πœ‚=1. Hence, in this limit, the zeros of 𝑃ℓ,𝑛(πœ‚;𝑔,β„Ž) distribute very near πœ‚=1. We show this in Figure 10 for certain parameters.

7. Summary

The discovery of new types of orthogonal polynomials, called the exceptional 𝑋ℓ 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 𝑃ℓ,𝑛(πœ‚;𝝀) have the lowest degree β„“=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 𝑋ℓ polynomial 𝑃ℓ,𝑛(πœ‚;𝝀) has 𝑛 zeros in the ordinary domain where the weight function is defined, that is, (0,∞) for the L1 and L2 polynomials and (βˆ’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 β„“ zeros outside the ordinary domain.

For the ordinary zeros, their distribution as 𝑛 increases at a fixed β„“ 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 𝑃ℓ,𝑛+1(πœ‚;𝝀) interlace with the 𝑛 zeros of 𝑃ℓ,𝑛(πœ‚;𝝀). On the other hand, when β„“ 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 πœ‚=0 as β„“ increases.

For the β„“ extra zeros of 𝑃ℓ,𝑛(πœ‚;𝝀), each and everyone of them is sandwiched between the corresponding zeros of the deforming polynomials πœ‰β„“(πœ‚;𝝀+𝜹) and πœ‰β„“(πœ‚;𝝀). As 𝑛 increases at a fixed β„“, the extra zeros move from the zeros of πœ‰β„“(πœ‚;𝝀+𝜹) to those of πœ‰β„“(πœ‚;𝝀).

The behaviors of the extra zeros as β„“ increases at a fixed 𝑛 are more complex. For the L1 Laguerre polynomials, all the extra zeros lie on the negative π‘₯-axis. So as β„“ increases by one, the number of the extra zeros increases from β„“ to β„“+1. For the L2 Laguerre and J1 and J2 Jacobi polynomials, they have β„“/2 pairs of complex zeros for even β„“, and (β„“βˆ’1)/2 pairs of complex zeros and a real zero outside the ordinary domains where the weight functions are defined. As β„“ 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 β„“, these zeros appear to distribute symmetrically with respect to the π‘₯-axis in horse-shoe patterns. It is interesting to note that in the asymptotic regions of the parameters (𝑔≫1, β„Žβ‰«1), the exceptional polynomial 𝑃ℓ,𝑛(πœ‚,𝝀) is expressed as the product of the original polynomial 𝑃𝑛(πœ‚) and the deforming polynomial πœ‰β„“(πœ‚;𝝀), (6.3), (6.6), and (6.9). After completing this paper, we became aware of some new works on exceptional orthogonal polynomials [21–23].

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 π‘₯.

The Classical Laguerre Polynomials
The degree 𝑛 classical Laguerre polynomial is defined by 𝐿𝑛(𝛼)1(π‘₯)=𝑛!π‘›ξ“π‘˜=0(βˆ’π‘›)π‘˜π‘˜!(𝛼+π‘˜+1)π‘›βˆ’π‘˜π‘₯π‘˜.(A.1)

The Classical Jacobi Polynomials
The degree 𝑛 classical Jacobi polynomial is defined by 𝑃𝑛(𝛼,𝛽)(π‘₯)=(𝛼+1)𝑛𝑛!π‘›ξ“π‘˜=01(π‘˜!βˆ’π‘›)π‘˜(𝑛+𝛼+𝛽+1)π‘˜(𝛼+1)π‘˜ξ‚€1βˆ’π‘₯2ξ‚π‘˜.(A.2) In these formulas, (π‘Ž)𝑛def=βˆπ‘›βˆ’1π‘˜=0(π‘Ž+π‘˜) is the shifted factorial (the Pochhammer symbol).

L1 Exceptional Laguerre Polynomials
𝑃𝐿11,01(π‘₯;𝑔)=2𝑃(3+2𝑔)+π‘₯,𝐿11,11(π‘₯;𝑔)=4(1+2𝑔)(5+2𝑔)βˆ’π‘₯2,𝑃𝐿11,21(π‘₯;𝑔)=116(1+2𝑔)(3+2𝑔)(7+2𝑔)βˆ’81(1+2𝑔)(7+2𝑔)π‘₯βˆ’4(7+2𝑔)π‘₯2+12π‘₯3,𝑃𝐿12,01(π‘₯;𝑔)=81(5+2𝑔)(7+2𝑔)+21(7+2𝑔)π‘₯+2π‘₯2,𝑃𝐿12,11(π‘₯;𝑔)=116(3+2𝑔)(5+2𝑔)(9+2𝑔)+81(3+2𝑔)(9+2𝑔)π‘₯βˆ’4(9+2𝑔)π‘₯2βˆ’12π‘₯3,𝑃𝐿12,21(π‘₯;𝑔)=64(3+2𝑔)(5+2𝑔)21(11+2𝑔)βˆ’8(5+2𝑔)(11+2𝑔)π‘₯2+14π‘₯4,𝑃L13,0(1π‘₯;𝑔)=(1487+2𝑔)(9+2𝑔)(11+2𝑔)+8(19+2𝑔)(11+2𝑔)π‘₯+4(11+2𝑔)π‘₯2+16π‘₯3,𝑃𝐿13,11(π‘₯;𝑔)=196(5+2𝑔)(7+2𝑔)(9+2𝑔)(13+2𝑔)+βˆ’124(5+2𝑔)(9+2𝑔)(13+2𝑔)π‘₯4(13+2𝑔)π‘₯2βˆ’16(13+2𝑔)π‘₯3βˆ’16π‘₯4.(A.3)

L2 Exceptional Laguerre Polynomials
𝑃𝐿21,01(π‘₯;𝑔)=βˆ’2𝑃(3+2𝑔)βˆ’π‘₯,𝐿21,11(π‘₯;𝑔)=4(1+2𝑔)(5+2𝑔)βˆ’π‘₯2,𝑃𝐿21,21(π‘₯;𝑔)=βˆ’116(1+2𝑔)(3+2𝑔)(7+2𝑔)+81(1+2𝑔)(7+2𝑔)π‘₯+4(7+2𝑔)π‘₯2βˆ’12π‘₯3,𝑃𝐿22,01(π‘₯;𝑔)=81(3+2𝑔)(5+2𝑔)+21(3+2𝑔)π‘₯+2π‘₯2,𝑃𝐿22,11(π‘₯;𝑔)=116(1+2𝑔)(5+2𝑔)(7+2𝑔)+81(1+2𝑔)(7+2𝑔)π‘₯βˆ’4(1+2𝑔)π‘₯2βˆ’12π‘₯3,𝑃𝐿22,21(π‘₯;𝑔)=164(1+2𝑔)(3+2𝑔)(7+2𝑔)(9+2𝑔)βˆ’8(1+2𝑔)(9+2𝑔)π‘₯2βˆ’π‘₯3+14π‘₯4,𝑃𝐿23,0(1π‘₯;𝑔)=βˆ’(1483+2𝑔)(5+2𝑔)(7+2𝑔)βˆ’8(13+2𝑔)(5+2𝑔)π‘₯βˆ’4(3+2𝑔)π‘₯2βˆ’16π‘₯3,𝑃𝐿23,11(π‘₯;𝑔)=βˆ’196(1+2𝑔)(5+2𝑔)(7+2𝑔)(9+2𝑔)βˆ’βˆ’124(1+2𝑔)(5+2𝑔)(9+2𝑔)π‘₯4(1+2𝑔)π‘₯2+16(1+2𝑔)π‘₯3+16π‘₯4.(A.4)

J1 Exceptional Jacobi Polynomials
𝑃𝐽11,01(π‘₯;𝑔,β„Ž)=21(3+𝑔+β„Ž)+2𝑃(π‘”βˆ’β„Ž)π‘₯,𝐽11,11(π‘₯;𝑔,β„Ž)=41(π‘”βˆ’β„Ž)(3+𝑔+β„Ž)+4ξ€·5+6𝑔+2𝑔2+6β„Ž+2β„Ž2ξ€Έπ‘₯+14(π‘”βˆ’β„Ž)(3+𝑔+β„Ž)π‘₯2,𝑃𝐽11,21(π‘₯;𝑔,β„Ž)=ξ€·16βˆ’7βˆ’8𝑔+2𝑔2+𝑔3βˆ’8β„Žβˆ’8π‘”β„Žβˆ’π‘”2β„Ž+2β„Ž2βˆ’π‘”β„Ž2+β„Ž3ξ€Έ+1ξ€·16(π‘”βˆ’β„Ž)11+13𝑔+3𝑔2+13β„Ž+2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯+1ξ€·16(4+𝑔+β„Ž)7+8𝑔+3𝑔2+8β„Žβˆ’2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯2+116(π‘”βˆ’β„Ž)(4+𝑔+β„Ž)(5+𝑔+β„Ž)π‘₯3,𝑃𝐽12,01(π‘₯;𝑔,β„Ž)=8ξ€·23+9𝑔+𝑔2+11β„Ž+2π‘”β„Ž+β„Ž2ξ€Έβˆ’14+1(βˆ’1βˆ’π‘”+β„Ž)(5+𝑔+β„Ž)π‘₯8(βˆ’2βˆ’π‘”+β„Ž)(βˆ’1βˆ’π‘”+β„Ž)π‘₯2,𝑃𝐽12,11(π‘₯;𝑔,β„Ž)=ξ€·168+23𝑔+9𝑔2+𝑔3βˆ’23β„Ž+2π‘”β„Ž+𝑔2β„Žβˆ’11β„Ž2βˆ’π‘”β„Ž2βˆ’β„Ž3ξ€Έ+1ξ€·16(5+𝑔+β„Ž)15+11𝑔+3𝑔2+9β„Žβˆ’2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯βˆ’1ξ€·16(βˆ’1βˆ’π‘”+β„Ž)42+22𝑔+3𝑔2+18β„Ž+2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯2+116(βˆ’2βˆ’π‘”+β„Ž)(βˆ’1βˆ’π‘”+β„Ž)(5+𝑔+β„Ž)π‘₯3,𝑃𝐽12,21(π‘₯;𝑔,β„Ž)=ξ€·64βˆ’81βˆ’54𝑔+7𝑔2+8𝑔3+𝑔4βˆ’100β„Žβˆ’80π‘”β„Žβˆ’10𝑔2β„Ž+5β„Ž2βˆ’16π‘”β„Ž2βˆ’2𝑔2β„Ž2+10β„Ž3+β„Ž4ξ€Έβˆ’1ξ€·323βˆ’67π‘”βˆ’61𝑔2βˆ’20𝑔3βˆ’2𝑔4+75β„Ž+65β„Ž2+20β„Ž3+2β„Ž4ξ€Έπ‘₯+1ξ€·32252+291𝑔+153𝑔2+36𝑔3+3𝑔4+250β„Ž+80π‘”β„Ž+10𝑔2β„Ž+125β„Ž2+16π‘”β„Ž2+2𝑔2β„Ž2+30β„Ž3+3β„Ž4ξ€Έπ‘₯2βˆ’1(ξ€·32βˆ’1βˆ’π‘”+β„Ž)(6+𝑔+β„Ž)27+14𝑔+2𝑔2+10β„Ž+2β„Ž2ξ€Έπ‘₯3+164(βˆ’2βˆ’π‘”+β„Ž)(βˆ’1βˆ’π‘”+β„Ž)(6+𝑔+β„Ž)(7+𝑔+β„Ž)π‘₯4,𝑃𝐽13,01(π‘₯;𝑔,β„Ž)=ξ€·48(7+𝑔+β„Ž)39+11𝑔+𝑔2+17β„Ž+2π‘”β„Ž+β„Ž2ξ€Έβˆ’1ξ€·16(βˆ’2βˆ’π‘”+β„Ž)45+13𝑔+𝑔2+15β„Ž+2π‘”β„Ž+β„Ž2ξ€Έπ‘₯+116(βˆ’3βˆ’π‘”+β„Ž)(βˆ’2βˆ’π‘”+β„Ž)(7+𝑔+β„Ž)π‘₯2βˆ’148(βˆ’4βˆ’π‘”+β„Ž)(βˆ’3βˆ’π‘”+β„Ž)(βˆ’2βˆ’π‘”+β„Ž)π‘₯3,𝑃𝐽13,11(π‘₯;𝑔,β„Ž)=βˆ’ξ€·96(7+𝑔+β„Ž)βˆ’24βˆ’39π‘”βˆ’11𝑔2βˆ’π‘”3+39β„Žβˆ’6π‘”β„Žβˆ’π‘”2β„Ž+17β„Ž2+π‘”β„Ž2+β„Ž3ξ€Έ+1ξ€·961395+1223𝑔+443𝑔2+70𝑔3+4𝑔4+905β„Ž+286π‘”β„Ž+54𝑔2β„Ž+4𝑔3β„Ž+359β„Ž2+30π‘”β„Ž2+70β„Ž3+4π‘”β„Ž3+4β„Ž4ξ€Έπ‘₯βˆ’1ξ€·32(βˆ’2βˆ’π‘”+β„Ž)(7+𝑔+β„Ž)37+16𝑔+2𝑔2+12β„Ž+2β„Ž2ξ€Έπ‘₯2+1ξ€·96(βˆ’3βˆ’π‘”+β„Ž)(βˆ’2βˆ’π‘”+β„Ž)135+46𝑔+4𝑔2+38β„Ž+4π‘”β„Ž+4β„Ž2ξ€Έπ‘₯3βˆ’196(βˆ’4βˆ’π‘”+β„Ž)(βˆ’3βˆ’π‘”+β„Ž)(βˆ’2βˆ’π‘”+β„Ž)(7+𝑔+β„Ž)π‘₯4.(A.5)

J2 Exceptional Jacobi Polynomials
𝑃𝐽21,01(π‘₯;𝑔,β„Ž)=βˆ’21(3+𝑔+β„Ž)βˆ’2𝑃(π‘”βˆ’β„Ž)π‘₯,𝐽21,11(π‘₯;𝑔,β„Ž)=βˆ’41(π‘”βˆ’β„Ž)(3+𝑔+β„Ž)βˆ’4ξ€·5+6𝑔+2𝑔2+6β„Ž+2β„Ž2ξ€Έπ‘₯βˆ’14(π‘”βˆ’β„Ž)(3+𝑔+β„Ž)π‘₯2,𝑃𝐽21,21(π‘₯;𝑔,β„Ž)=ξ€·167+8π‘”βˆ’2𝑔2βˆ’π‘”3+8β„Ž+8π‘”β„Ž+𝑔2β„Žβˆ’2β„Ž2+π‘”β„Ž2βˆ’β„Ž3ξ€Έβˆ’1ξ€·16(π‘”βˆ’β„Ž)11+13𝑔+3𝑔2+13β„Ž+2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯βˆ’1ξ€·16(4+𝑔+β„Ž)7+8𝑔+3𝑔2+8β„Žβˆ’2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯2βˆ’116(π‘”βˆ’β„Ž)(4+𝑔+β„Ž)(5+𝑔+β„Ž)π‘₯3,𝑃𝐽22,01(π‘₯;𝑔,β„Ž)=8ξ€·23+11𝑔+𝑔2+9β„Ž+2π‘”β„Ž+β„Ž2ξ€Έ+14+1(βˆ’1+π‘”βˆ’β„Ž)(5+𝑔+β„Ž)π‘₯8(βˆ’2+π‘”βˆ’β„Ž)(βˆ’1+π‘”βˆ’β„Ž)π‘₯2,𝑃𝐽22,11(π‘₯;𝑔,β„Ž)=ξ€·16βˆ’8+23𝑔+11𝑔2+𝑔3βˆ’23β„Žβˆ’2π‘”β„Ž+𝑔2β„Žβˆ’9β„Ž2βˆ’π‘”β„Ž2βˆ’β„Ž3ξ€Έ+1ξ€·16(5+𝑔+β„Ž)15+9𝑔+3𝑔2+11β„Žβˆ’2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯+1ξ€·16(βˆ’1+π‘”βˆ’β„Ž)42+18𝑔+3𝑔2+22β„Ž+2π‘”β„Ž+3β„Ž2ξ€Έπ‘₯2+116(βˆ’2+π‘”βˆ’β„Ž)(βˆ’1+π‘”βˆ’β„Ž)(5+𝑔+β„Ž)π‘₯3,𝑃𝐽22,21(π‘₯;𝑔,β„Ž)=ξ€·64βˆ’81βˆ’100𝑔+5𝑔2+10𝑔3+𝑔4βˆ’54β„Žβˆ’80π‘”β„Žβˆ’16𝑔2β„Ž+7β„Ž2βˆ’10π‘”β„Ž2βˆ’2𝑔2β„Ž2+8β„Ž3+β„Ž4ξ€Έ+1ξ€·323+75𝑔+65𝑔2+20𝑔3+2𝑔4βˆ’67β„Žβˆ’61β„Ž2βˆ’20β„Ž3βˆ’2β„Ž4ξ€Έπ‘₯+1ξ€·32252+250𝑔+125𝑔2+30𝑔3+3𝑔4+291β„Ž+80π‘”β„Ž+16𝑔2β„Ž+153β„Ž2+10π‘”β„Ž2+2𝑔2β„Ž2+36β„Ž3+3β„Ž4ξ€Έπ‘₯2+1(ξ€·32βˆ’1+π‘”βˆ’β„Ž)(6+𝑔+β„Ž)27+10𝑔+2𝑔2+14β„Ž+2β„Ž2ξ€Έπ‘₯3+164(βˆ’2+π‘”βˆ’β„Ž)(βˆ’1+π‘”βˆ’β„Ž)(6+𝑔+β„Ž)(7+𝑔+β„Ž)π‘₯4,𝑃𝐽23,01(π‘₯;𝑔,β„Ž)=βˆ’ξ€·48(7+𝑔+β„Ž)39+17𝑔+𝑔2+11β„Ž+2π‘”β„Ž+β„Ž2ξ€Έβˆ’1ξ€·16(βˆ’2+π‘”βˆ’β„Ž)45+15𝑔+𝑔2+13β„Ž+2π‘”β„Ž+β„Ž2ξ€Έπ‘₯βˆ’116(βˆ’3+π‘”βˆ’β„Ž)(βˆ’2+π‘”βˆ’β„Ž)(7+𝑔+β„Ž)π‘₯2βˆ’148(βˆ’4+π‘”βˆ’β„Ž)(βˆ’3+π‘”βˆ’β„Ž)(βˆ’2+π‘”βˆ’β„Ž)π‘₯3,𝑃𝐽23,11(π‘₯;𝑔,β„Ž)=βˆ’ξ€·96(7+𝑔+β„Ž)βˆ’24+39𝑔+17𝑔2+𝑔3βˆ’39β„Žβˆ’6π‘”β„Ž+𝑔2β„Žβˆ’11β„Ž2βˆ’π‘”β„Ž2βˆ’β„Ž3ξ€Έβˆ’1ξ€·961395+905𝑔+359𝑔2+70𝑔3+4𝑔4+1223β„Ž+286π‘”β„Ž+30𝑔2β„Ž+4𝑔3β„Ž+443β„Ž2+54π‘”β„Ž2+70β„Ž3+4π‘”β„Ž3+4β„Ž4ξ€Έπ‘₯βˆ’1(ξ€·32βˆ’2+π‘”βˆ’β„Ž)(7+𝑔+β„Ž)37+12𝑔+2𝑔2+16β„Ž+2β„Ž2ξ€Έπ‘₯2βˆ’1ξ€·96(βˆ’3+π‘”βˆ’β„Ž)(βˆ’2+π‘”βˆ’β„Ž)135+38𝑔+4𝑔2+46β„Ž+4π‘”β„Ž+4β„Ž2ξ€Έπ‘₯3βˆ’196(βˆ’4+π‘”βˆ’β„Ž)(βˆ’3+π‘”βˆ’β„Ž)(βˆ’2+π‘”βˆ’β„Ž)(7+𝑔+β„Ž)π‘₯4.(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.

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