# Zeros of the Exceptional Laguerre and Jacobi Polynomials

Authors: Choon-Lin Ho; Ryu Sasaki
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

An interesting discovery in the last two years in the field of mathematical physics has been the exceptionalXℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=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.


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## Body

## 1. Introduction

The discovery of new types of orthogonal polynomials, called the exceptionalXℓ 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 ℓ 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 ℓ=1,2,…, 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 ℓ (ℓ=1,2,…). The lowest (ℓ=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 Schrödinger equation) without contradicting Bochner’s theorem [11], since they start at degree ℓ>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-typeXℓ 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 x-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 Section2, 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 ℓ 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 ℓ. In Section 6, we consider behaviors of the zeros at large g and/or h. 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 ℓ, which is the degree of the deforming polynomials. The two sets of exceptional Laguerre polynomials (ℓ=1,2,…, n=0,1,2,…) are [9]
(2.1)Pℓ,n(η;λ)=def{ξℓ(η;λ+δ)Pn(η;g+ℓ-1)-ξℓ(η;λ)∂ηPn(η;g+ℓ-1):L1(n+g+12)-1{(g+12)ξℓ(η;λ+δ)Pn(η;g+ℓ+1)+ηξℓ(η;λ)∂ηPn(η;g+ℓ+1)(g+12)}:L2,
in which λ=defg>0, δ=def1 and
(2.2)Pn(η;g)=defLn(g-(1/2))(η),ξℓ(η;g)=def{Lℓ(g+ℓ-(3/2))(-η):L1Lℓ(-g-ℓ-(1/2))(η).:L2.
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:
(2.3)Pℓ,n(η;λ)=def{Lℓ(g+ℓ-(1/2))(-η)Ln(g+ℓ-(1/2))(η)-Lℓ-1(g+ℓ-(1/2))(-η)Ln-1(g+ℓ-(1/2))(η):L1(n+g+(1/2))-1{(g+(12))Lℓ(-g-ℓ-(3/2))(η)Ln(g+ℓ+(3/2))(η)+(ℓ+1)Lℓ+1(-g-ℓ-(3/2))(η)Ln-1(g+ℓ+(3/2))(η)(g+(12))}:L2.
The two sets of exceptional Jacobi polynomials (ℓ=1,2,…, n=0,1,2,…) are [9]
(2.4)Pℓ,n(η;λ)=def{(n+h+(12))-1{(h+(1/2))ξℓ(η;λ+δ)Pn(η;g+ℓ-1,h+ℓ+1)+(1+η)ξℓ(η;λ)∂ηPn(η;g+ℓ-1,h+ℓ+1)}:J1(n+g+(12))-1{(g+(1/2))ξℓ(η;λ+δ)Pn(η;g+ℓ+1,h+ℓ-1)-(1-η)ξℓ(η;λ)∂ηPn(η;g+ℓ+1,h+ℓ-1)}:J2,
in which λ=def(g,h), g>0, h>0, δ=def(1,1) and
(2.5)Pn(η;g,h)=defPn(g-(1/2),h-(1/2))(η),ξℓ(η;g,h)=def{Pℓ(g+ℓ-(3/2),-h-ℓ-(1/2))(η),g>h>0:J1Pℓ(-g-ℓ-(1/2),h+ℓ-(3/2))(η),h>g>0:J2,
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 ℓ, and the two polynomials ξℓ(η;λ) and ξℓ(η;λ+δ) played the role of the deforming polynomials.The differential equations satisfied by these new polynomials are as follows:(2.6)L1:η∂η2Pℓ,n(η;λ)+(g+ℓ+12-η-2η∂ηξℓ(η;λ)ξℓ(η;λ))∂ηPℓ,n(η;λ)+(2η∂ηξℓ(η;λ+δ)ξℓ(η;λ)+n-ℓ)Pℓ,n(η;λ)=0,(2.7)L2:η∂η2Pℓ,n(η;λ)+(g+ℓ+12-η-2η∂ηξℓ(η;λ)ξℓ(η;λ))∂ηPℓ,n(η;λ)+(-2(g+(1/2))∂ηξℓ(η;λ+δ)ξℓ(η;λ)+n+ℓ)Pℓ,n(η;λ)=0,(2.8)J1:(1-η2)∂η2Pℓ,n(η;λ)+(h-g-(g+h+2ℓ+1)η-2(1-η2)∂ηξℓ(η;λ)ξℓ(η;λ))∂ηPℓ,n(η;λ)+(-2(h+(1/2))(1-η)∂ηξℓ(η;λ+δ)ξℓ(η;λ)+ℓ(ℓ+g-h-1)+n(n+g+h+2ℓ)-2(h+(1/2))(1-η)∂ηξℓ(η;λ+δ)ξℓ(η;λ))Pℓ,n(η;λ)=0,(2.9)J2:(1-η2)∂η2Pℓ,n(η;λ)+(h-g-(g+h+2ℓ+1)η-2(1-η2)∂ηξℓ(η;λ)ξℓ(η;λ))∂ηPℓ,n(η;λ)+(2(g+(1/2))(1+η)∂ηξℓ(η;λ+δ)ξℓ(η;λ)+ℓ(ℓ+h-g-1)+n(n+g+h+2ℓ)-2(h+(1/2))(1-η)∂ηξℓ(η;λ+δ)ξℓ(η;λ))Pℓ,n(η;λ)=0.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 ofXℓ polynomial Pℓ,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 [17–20]. We will say more about these zeros in Section 4. Besides these n zeros, there are extra ℓ zeros outside the ordinary domain. For convenience, we will adopt the following notation for the zeros of the various polynomials involved:
(2.10)ξ-k(ℓ):zerosofξℓ(η;λ+δ),k=1,2,…,ℓ;(2.11)ξk(ℓ):zerosofξℓ(η;λ),k=1,2,…,ℓ;(2.12)η-k(ℓ,n):extrazerosofPℓ,n,k=1,2,…,ℓ;(2.13)ηj(ℓ,n):ordinaryzerosofPℓ,n,j=1,2,…,n.
We emphasize that ηj(ℓ,n)∈(0,∞) for the L1 and L2 Laguerre polynomials, and ηj(ℓ,n)∈(-1,1) for the J1 and J2 Jacobi polynomials.Figures1–10 depict the distribution of the zeros for some representative parameters of the systems, namely, n,ℓ,g, and h. From these figures, one can deduce certain patterns of the distribution of the zeros as those parameters vary. We will discuss these behaviors below.L1: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the L1 Laguerre polynomials, with g=0.5 and ℓ=2. The three diagrams correspond to n=1(a), 2(b), and 3(c), respectively. The ordinary zeros ηj(ℓ,n) lie in (0,∞).
(a)
(b)
(c)L2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the L2 Laguerre polynomials, with g=0.5 and ℓ=3. The three diagrams correspond to n=1(a), 2(b), and 5(c), respectively. The ordinary zeros ηj(ℓ,n) lie in (0,∞).
(a)
(b)
(c)J2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the J2 Jacobi polynomials, with g=3,h=4, and ℓ=3. The three diagrams correspond to n=1(a), 2(b), and 5(c), respectively. The ordinary zeros ηj(ℓ,n) lie in (-1,1).
(a)
(b)
(c)L1: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the L1 Laguerre polynomials, with g=0.5 and n=2. The three diagrams correspond to ℓ=1(a), 2(b), and 3(c), respectively.
(a)
(b)
(c)L1: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the L1 Laguerre polynomials, with g=1.5 and n=2. The three diagrams correspond to ℓ=1(a), 2(b), and 3(c), respectively.
(a)
(b)
(c)L2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the L2 Laguerre polynomials, with g=2 and n=2. The four diagrams correspond to ℓ=1(a), 2(b), 3(c), and 20(d), respectively.
(a)
(b)
(c)
(d)L2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the L2 Laguerre polynomials, with g=5 and n=2. The four diagrams correspond to ℓ=1(a), 2(b), 3(c), and 20(d), respectively.
(a)
(b)
(c)
(d)J2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the J2 Jacobi polynomials, with g=3, h=4, and n=4. The four diagrams correspond to ℓ=1(a), 2(b), 3(c), and 20(d), respectively.
(a)
(b)
(c)
(d)J2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the J2 Jacobi polynomials, with g=7, h=8, and n=4. The four diagrams correspond to ℓ=1(a), 2(b), 3(c), and 20(d), respectively.
(a)
(b)
(c)
(d)J2: distributions of the zerosη-k(ℓ,n),ηj(ℓ,n) (♦), ξ-k(ℓ) (○), and ξk(ℓ) (■) for the J2 Jacobi polynomials, with g=2, ℓ=10, and n=4. The two diagrams correspond to h=50(a) and 100(b), respectively.
(a)
(b)


## 3. Extraℓ Zeros of Xℓ Polynomials

Here we discuss the locations of the extraℓ 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.Theℓ extra zeros of L1 polynomials are on the negative real line (-∞,0). The L2 Xℓ:odd polynomials have one real negative zero which lies to the left of the remaining (1/2)(ℓ-1) pairs of complex conjugate roots. The L2 Xℓ:even polynomials have (1/2)ℓ pairs of complex conjugate roots.The situations for theXℓ Jacobi polynomials are a bit more complicated. The J1 Xℓ:odd polynomials have one real negative root which lies to the left of the remaining (1/2)(ℓ-1) pairs of complex conjugate roots with negative real parts. The J1 Xℓ:even polynomials have (1/2)ℓ pairs of complex conjugate roots with negative real parts. The J2 Xℓ:odd polynomials have one real positive root which lies to the right of the remaining (1/2)(ℓ-1) pairs of complex conjugate roots with positive real parts. The J2 Xℓ:even polynomials have (1/2)ℓ 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 senseη↔-η and g↔h, as exemplified by the relation ξℓJ2(η;g,h)=(-1)ℓξℓJ1(-η;h,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 asn Increases at Fixed ℓ

In all cases, we have(3.1)Pℓ,0(η;λ)∝ξℓ(η;λ+δ).
This implies that the zeros of Pℓ,0 coincide with those of ξℓ(η;λ+δ), namely, ξ-k(ℓ),k=1,2,…,ℓ.At fixedℓ, all the η-k(ℓ,n) move from ξ-k(ℓ) at n=0 to ξk(ℓ) as n→∞. 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 zerosξ-k(ℓ), ξk(ℓ), and η-k(ℓ,n) for the L1 Laguerre polynomials with g=2, ℓ=5, and n=0,10,20,…,60 (k=1,2,…,ℓ). It can be seen that when n=0, η-k(ℓ,n=0)=ξ-k(ℓ). As n increases, η-k(ℓ,n) approaches to ξk(ℓ).
ξ
-
k
(
ℓ
):
−22.4802
−15.2391
−10.1403
−6.2977
−3.3427
n
=
0
−22.4802
−15.2391
−10.1403
−6.2977
−3.3427
10
−22.0686
−14.8767
−9.8314
−6.0505
−3.1698
20
−21.8830
−14.7189
−9.7004
−5.9469
−3.0962
η
-
k
(
ℓ
,
n
):
30
−21.7717
−14.6253
−9.6233
−5.8862
−3.0529
40
−21.6954
−14.5617
−9.5711
−5.8452
−3.0237
50
−21.6390
−14.5148
−9.5327
−5.8152
−3.0022
60
−21.5951
−14.4784
−9.5030
−5.7919
−2.9856
ξ
k
(
ℓ
):
−21.0456
−14.0274
−9.1375
−5.5071
−2.7824Table 2
Same as Table1 for L1 Laguerre polynomials with g=8 and ℓ=5.
ξ
-
k
(
ℓ
):
−30.7592
−22.3415
−16.1499
−11.2032
−7.0462
n
=
0
−30.7592
−22.3415
−16.1499
−11.2032
−7.0462
10
−30.4724
−22.0859
−15.9269
−11.0165
−6.9029
20
−30.3144
−21.9474
−15.8074
−10.9169
−6.8255
η
-
k
(
ℓ
,
n
):
30
−30.2107
−21.8574
−15.7301
−10.8525
−6.7752
40
−30.1361
−21.7928
−15.6748
−10.8065
−6.7393
50
−30.0791
−21.7436
−15.6328
−10.7715
−6.7119
60
−30.0336
−21.7046
−15.5994
−10.7438
−6.6902
ξ
k
(
ℓ
):
−29.4106
−21.1735
−15.1488
−10.3703
−6.3968Table 3
Same as Table1 but for L2 Laguerre polynomials with g=3 and ℓ=4.
ξ
-
k
(
ℓ
):
-
5.29007
±
1.65310
i
-
3.70993
±
5.05130
i
n
=
0
-
5.29007
±
1.65310
i
-
3.70993
±
5.05130
i
10
-
4.84198
±
1.57129
i
-
3.25524
±
4.78004
i
20
-
4.71299
±
1.54888
i
-
3.12839
±
4.70776
i
η
-
k
(
ℓ
,
n
):
30
-
4.64523
±
1.53732
i
-
3.06246
±
4.67065
i
40
-
4.60183
±
1.52998
i
-
3.02046
±
4.64713
i
50
-
4.57100
±
1.52479
i
-
2.99074
±
4.63053
i
60
-
4.54766
±
1.52087
i
-
2.96828
±
4.61801
i
ξ
k
(
ℓ
):
-
4.28361
±
1.47684
i
-
2.71639
±
4.47739
iTable 4
Same as Table3 for L2 Laguerre polynomials with g=10 and ℓ=5.
ξ
-
k
(
ℓ
):
−12.8111
-
12.2115
±
4.7185
i
-
10.1329
±
9.7965
i
n
=
0
−12.8111
-
12.2115
±
4.7185
i
-
10.1329
±
9.7965
i
10
−12.5476
-
11.9465
±
4.6639
i
-
9.8622
±
9.6780
i
20
−12.4210
-
11.8198
±
4.6384
i
-
9.7348
±
9.6233
i
η
-
k
(
ℓ
,
n
):
30
−12.3430
-
11.7418
±
4.6229
i
-
9.6570
±
9.5901
i
40
−12.2888
-
11.6877
±
4.6122
i
-
9.6032
±
9.5673
i
50
−12.2483
-
11.6473
±
4.6043
i
-
9.5632
±
9.5504
i
60
−12.2165
-
11.6157
±
4.5981
i
-
9.5319
±
9.5372
i
ξ
k
(
ℓ
):
−11.8092
-
11.2107
±
4.5195
i
-
9.1347
±
9.3702
iTable 5
Same as Table1 but for J2 Jacobi polynomials with g=3,h=4, and ℓ=4.
ξ
-
k
(
ℓ
):
1.56846
±
2.10278
i
3.00297
±
0.91199
i
n
=
0
1.56846
±
2.10278
i
3.00297
±
0.91199
i
10
1.45201
±
1.89890
i
2.76834
±
0.82626
i
20
1.42407
±
1.85433
i
2.71360
±
0.80733
i
η
-
k
(
ℓ
,
n
):
30
1.41139±1.83435i
2.68882±0.79884i
40
1.40414
±
1.82297
i
2.67466
±
0.79401
i
50
1.39944
±
1.81561
i
2.66550
±
0.79088
i
60
1.39615
±
1.81046
i
2.65907
±
0.78869
i
ξ
k
(
ℓ
):
1.37745±1.78118i
2.62255±0.77624iTable 6
Same as Table5 for J2 Jacobi polynomials with g=3,h=4, and ℓ=5.
ξ
-
k
(
ℓ
):
1.19188
±
1.85256
i
2.38851±1.21416i
2.83923
n
=
0
1.19188±1.85256i
2.38851±1.21416i
2.83923
10
1.11856±1.68660i
2.22979±1.11021i
2.64753
20
1.09936±1.64851i
2.18998±1.08600i
2.59983
η
-
k
(
ℓ
,
n
):
30
1.09041±1.63110i
2.17151±1.07491i
2.57771
40
1.08522±1.62106i
2.16081±1.06852i
2.56490
50
1.08184±1.61453i
2.15382±1.06436i
2.55654
60
1.07945±1.60993i
2.14890±1.06143i
2.55066
ξ
k
(
ℓ
):
1.06566±1.58339i
2.12047±1.04452i
2.51663Table 7
Same as Table5 for J2 Jacobi polynomials with g=8,h=9, and ℓ=3.
ξ
-
k
(
ℓ
):
3.90615
±
4.35051
i
6.58770
n
=
0
3.90615±4.35051i
6.58770
10
3.74981±4.16635i
6.32188
20
3.69527±4.10323i
6.22948
η
-
k
(
ℓ
,
n
):
30
3.66745±4.07116i
6.18240
40
3.65057±4.05174i
6.15383
50
3.63924±4.03870i
6.13465
60
3.63110±4.02935i
6.12088
ξ
k
(
ℓ
):
3.58151±3.97238i
6.03699


### 3.2. Behaviors asℓ Increases at Fixed n

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


#### 3.2.1. L1 Laguerre

Asℓ changes to ℓ+1, the zeros of ξℓ(η;g+1) and ξℓ(η;g) decrease (move to the left), and a new set of zeros appear from the right. 										(3.2)ξ-k(ℓ+1)<ξ-k(ℓ),ξk(ℓ+1)<ξk(ℓ),ξ-k(ℓ)<ξk(ℓ)<ξ-k+1(ℓ)<ξk+1(ℓ),
for k=1,2,…,ℓ-1,ℓ.We show these patterns for some representative parameters in Figures4 and 5.


#### 3.2.2. L2 Laguerre

Forℓ=1, there is one real root each for ξℓ(η;g+1) and ξℓ(η;g), with ξ-1(ℓ)<ξ1(ℓ)<0.Forℓ=2, the above two roots bifurcate into two complex roots, with ℜξ-(ℓ)<ℜξ(ℓ), |ℑξ-(ℓ)|>|ℑξ(ℓ)|.Generally, for evenℓ, there are ℓ complex zeros with
(3.3)ℜξ-k(ℓ)<ℜξk(ℓ),|ℑξ-k(ℓ)|>|ℑξk(ℓ)|,k=1,2,…,ℓ2.
All η-k(ℓ,n) are sandwiched between ξ-k(ℓ) and ξk(ℓ). As an even ℓ changes to ℓ+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 ℓ 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 ℓ increases.Figures6 and 7 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.


#### 3.2.3. J2 Jacobi

Forℓ=1, there is one real root each for ξℓ(η;g+1) and ξℓ(η;g), with ξ-1(ℓ)>ξ1(ℓ)>1.Forℓ=2, the above two roots bifurcate into two complex roots, with ℜξ-(ℓ)>ℜξ(ℓ), |ℑξ-(ℓ)|>|ℑξ(ℓ)|.Generally, for evenℓ, there are ℓ complex zeros with
(3.4)ℜξ-k(ℓ)>ℜξk(ℓ),|ℑξ-k(ℓ)|>|ℑξk(ℓ)|,k=1,2,…,ℓ2.
As ℓ changes to ℓ+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 ℓ 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 ℓ increases. This pattern continues as ℓ increases.Figures8 and 9 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.


## 3.1. Behaviors asn Increases at Fixed ℓ

In all cases, we have(3.1)Pℓ,0(η;λ)∝ξℓ(η;λ+δ).
This implies that the zeros of Pℓ,0 coincide with those of ξℓ(η;λ+δ), namely, ξ-k(ℓ),k=1,2,…,ℓ.At fixedℓ, all the η-k(ℓ,n) move from ξ-k(ℓ) at n=0 to ξk(ℓ) as n→∞. 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 zerosξ-k(ℓ), ξk(ℓ), and η-k(ℓ,n) for the L1 Laguerre polynomials with g=2, ℓ=5, and n=0,10,20,…,60 (k=1,2,…,ℓ). It can be seen that when n=0, η-k(ℓ,n=0)=ξ-k(ℓ). As n increases, η-k(ℓ,n) approaches to ξk(ℓ).
ξ
-
k
(
ℓ
):
−22.4802
−15.2391
−10.1403
−6.2977
−3.3427
n
=
0
−22.4802
−15.2391
−10.1403
−6.2977
−3.3427
10
−22.0686
−14.8767
−9.8314
−6.0505
−3.1698
20
−21.8830
−14.7189
−9.7004
−5.9469
−3.0962
η
-
k
(
ℓ
,
n
):
30
−21.7717
−14.6253
−9.6233
−5.8862
−3.0529
40
−21.6954
−14.5617
−9.5711
−5.8452
−3.0237
50
−21.6390
−14.5148
−9.5327
−5.8152
−3.0022
60
−21.5951
−14.4784
−9.5030
−5.7919
−2.9856
ξ
k
(
ℓ
):
−21.0456
−14.0274
−9.1375
−5.5071
−2.7824Table 2
Same as Table1 for L1 Laguerre polynomials with g=8 and ℓ=5.
ξ
-
k
(
ℓ
):
−30.7592
−22.3415
−16.1499
−11.2032
−7.0462
n
=
0
−30.7592
−22.3415
−16.1499
−11.2032
−7.0462
10
−30.4724
−22.0859
−15.9269
−11.0165
−6.9029
20
−30.3144
−21.9474
−15.8074
−10.9169
−6.8255
η
-
k
(
ℓ
,
n
):
30
−30.2107
−21.8574
−15.7301
−10.8525
−6.7752
40
−30.1361
−21.7928
−15.6748
−10.8065
−6.7393
50
−30.0791
−21.7436
−15.6328
−10.7715
−6.7119
60
−30.0336
−21.7046
−15.5994
−10.7438
−6.6902
ξ
k
(
ℓ
):
−29.4106
−21.1735
−15.1488
−10.3703
−6.3968Table 3
Same as Table1 but for L2 Laguerre polynomials with g=3 and ℓ=4.
ξ
-
k
(
ℓ
):
-
5.29007
±
1.65310
i
-
3.70993
±
5.05130
i
n
=
0
-
5.29007
±
1.65310
i
-
3.70993
±
5.05130
i
10
-
4.84198
±
1.57129
i
-
3.25524
±
4.78004
i
20
-
4.71299
±
1.54888
i
-
3.12839
±
4.70776
i
η
-
k
(
ℓ
,
n
):
30
-
4.64523
±
1.53732
i
-
3.06246
±
4.67065
i
40
-
4.60183
±
1.52998
i
-
3.02046
±
4.64713
i
50
-
4.57100
±
1.52479
i
-
2.99074
±
4.63053
i
60
-
4.54766
±
1.52087
i
-
2.96828
±
4.61801
i
ξ
k
(
ℓ
):
-
4.28361
±
1.47684
i
-
2.71639
±
4.47739
iTable 4
Same as Table3 for L2 Laguerre polynomials with g=10 and ℓ=5.
ξ
-
k
(
ℓ
):
−12.8111
-
12.2115
±
4.7185
i
-
10.1329
±
9.7965
i
n
=
0
−12.8111
-
12.2115
±
4.7185
i
-
10.1329
±
9.7965
i
10
−12.5476
-
11.9465
±
4.6639
i
-
9.8622
±
9.6780
i
20
−12.4210
-
11.8198
±
4.6384
i
-
9.7348
±
9.6233
i
η
-
k
(
ℓ
,
n
):
30
−12.3430
-
11.7418
±
4.6229
i
-
9.6570
±
9.5901
i
40
−12.2888
-
11.6877
±
4.6122
i
-
9.6032
±
9.5673
i
50
−12.2483
-
11.6473
±
4.6043
i
-
9.5632
±
9.5504
i
60
−12.2165
-
11.6157
±
4.5981
i
-
9.5319
±
9.5372
i
ξ
k
(
ℓ
):
−11.8092
-
11.2107
±
4.5195
i
-
9.1347
±
9.3702
iTable 5
Same as Table1 but for J2 Jacobi polynomials with g=3,h=4, and ℓ=4.
ξ
-
k
(
ℓ
):
1.56846
±
2.10278
i
3.00297
±
0.91199
i
n
=
0
1.56846
±
2.10278
i
3.00297
±
0.91199
i
10
1.45201
±
1.89890
i
2.76834
±
0.82626
i
20
1.42407
±
1.85433
i
2.71360
±
0.80733
i
η
-
k
(
ℓ
,
n
):
30
1.41139±1.83435i
2.68882±0.79884i
40
1.40414
±
1.82297
i
2.67466
±
0.79401
i
50
1.39944
±
1.81561
i
2.66550
±
0.79088
i
60
1.39615
±
1.81046
i
2.65907
±
0.78869
i
ξ
k
(
ℓ
):
1.37745±1.78118i
2.62255±0.77624iTable 6
Same as Table5 for J2 Jacobi polynomials with g=3,h=4, and ℓ=5.
ξ
-
k
(
ℓ
):
1.19188
±
1.85256
i
2.38851±1.21416i
2.83923
n
=
0
1.19188±1.85256i
2.38851±1.21416i
2.83923
10
1.11856±1.68660i
2.22979±1.11021i
2.64753
20
1.09936±1.64851i
2.18998±1.08600i
2.59983
η
-
k
(
ℓ
,
n
):
30
1.09041±1.63110i
2.17151±1.07491i
2.57771
40
1.08522±1.62106i
2.16081±1.06852i
2.56490
50
1.08184±1.61453i
2.15382±1.06436i
2.55654
60
1.07945±1.60993i
2.14890±1.06143i
2.55066
ξ
k
(
ℓ
):
1.06566±1.58339i
2.12047±1.04452i
2.51663Table 7
Same as Table5 for J2 Jacobi polynomials with g=8,h=9, and ℓ=3.
ξ
-
k
(
ℓ
):
3.90615
±
4.35051
i
6.58770
n
=
0
3.90615±4.35051i
6.58770
10
3.74981±4.16635i
6.32188
20
3.69527±4.10323i
6.22948
η
-
k
(
ℓ
,
n
):
30
3.66745±4.07116i
6.18240
40
3.65057±4.05174i
6.15383
50
3.63924±4.03870i
6.13465
60
3.63110±4.02935i
6.12088
ξ
k
(
ℓ
):
3.58151±3.97238i
6.03699


## 3.2. Behaviors asℓ Increases at Fixed n

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


### 3.2.1. L1 Laguerre

Asℓ changes to ℓ+1, the zeros of ξℓ(η;g+1) and ξℓ(η;g) decrease (move to the left), and a new set of zeros appear from the right. 										(3.2)ξ-k(ℓ+1)<ξ-k(ℓ),ξk(ℓ+1)<ξk(ℓ),ξ-k(ℓ)<ξk(ℓ)<ξ-k+1(ℓ)<ξk+1(ℓ),
for k=1,2,…,ℓ-1,ℓ.We show these patterns for some representative parameters in Figures4 and 5.


### 3.2.2. L2 Laguerre

Forℓ=1, there is one real root each for ξℓ(η;g+1) and ξℓ(η;g), with ξ-1(ℓ)<ξ1(ℓ)<0.Forℓ=2, the above two roots bifurcate into two complex roots, with ℜξ-(ℓ)<ℜξ(ℓ), |ℑξ-(ℓ)|>|ℑξ(ℓ)|.Generally, for evenℓ, there are ℓ complex zeros with
(3.3)ℜξ-k(ℓ)<ℜξk(ℓ),|ℑξ-k(ℓ)|>|ℑξk(ℓ)|,k=1,2,…,ℓ2.
All η-k(ℓ,n) are sandwiched between ξ-k(ℓ) and ξk(ℓ). As an even ℓ changes to ℓ+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 ℓ 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 ℓ increases.Figures6 and 7 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.


### 3.2.3. J2 Jacobi

Forℓ=1, there is one real root each for ξℓ(η;g+1) and ξℓ(η;g), with ξ-1(ℓ)>ξ1(ℓ)>1.Forℓ=2, the above two roots bifurcate into two complex roots, with ℜξ-(ℓ)>ℜξ(ℓ), |ℑξ-(ℓ)|>|ℑξ(ℓ)|.Generally, for evenℓ, there are ℓ complex zeros with
(3.4)ℜξ-k(ℓ)>ℜξk(ℓ),|ℑξ-k(ℓ)|>|ℑξk(ℓ)|,k=1,2,…,ℓ2.
As ℓ changes to ℓ+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 ℓ 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 ℓ increases. This pattern continues as ℓ increases.Figures8 and 9 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.


## 3.2.1. L1 Laguerre

Asℓ changes to ℓ+1, the zeros of ξℓ(η;g+1) and ξℓ(η;g) decrease (move to the left), and a new set of zeros appear from the right. 										(3.2)ξ-k(ℓ+1)<ξ-k(ℓ),ξk(ℓ+1)<ξk(ℓ),ξ-k(ℓ)<ξk(ℓ)<ξ-k+1(ℓ)<ξk+1(ℓ),
for k=1,2,…,ℓ-1,ℓ.We show these patterns for some representative parameters in Figures4 and 5.


## 3.2.2. L2 Laguerre

Forℓ=1, there is one real root each for ξℓ(η;g+1) and ξℓ(η;g), with ξ-1(ℓ)<ξ1(ℓ)<0.Forℓ=2, the above two roots bifurcate into two complex roots, with ℜξ-(ℓ)<ℜξ(ℓ), |ℑξ-(ℓ)|>|ℑξ(ℓ)|.Generally, for evenℓ, there are ℓ complex zeros with
(3.3)ℜξ-k(ℓ)<ℜξk(ℓ),|ℑξ-k(ℓ)|>|ℑξk(ℓ)|,k=1,2,…,ℓ2.
All η-k(ℓ,n) are sandwiched between ξ-k(ℓ) and ξk(ℓ). As an even ℓ changes to ℓ+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 ℓ 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 ℓ increases.Figures6 and 7 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.


## 3.2.3. J2 Jacobi

Forℓ=1, there is one real root each for ξℓ(η;g+1) and ξℓ(η;g), with ξ-1(ℓ)>ξ1(ℓ)>1.Forℓ=2, the above two roots bifurcate into two complex roots, with ℜξ-(ℓ)>ℜξ(ℓ), |ℑξ-(ℓ)|>|ℑξ(ℓ)|.Generally, for evenℓ, there are ℓ complex zeros with
(3.4)ℜξ-k(ℓ)>ℜξk(ℓ),|ℑξ-k(ℓ)|>|ℑξk(ℓ)|,k=1,2,…,ℓ2.
As ℓ changes to ℓ+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 ℓ 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 ℓ increases. This pattern continues as ℓ increases.Figures8 and 9 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.


## 4. Ordinary Zeros ofXℓ Polynomials

In the case ofXℓ polynomials Pℓ,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 asn Increases at Fixed ℓ

This is guaranteed by the oscillation theorem of the one-dimensional quantum mechanics, sincePℓ,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 Pℓ,n decrease, and a new zero appears from the right. Thus the n zeros of Pℓ,n(η;λ) and the n+1 zeros of Pℓ,n+1(η;λ) interlace with each other: each zero of Pℓ,n(η;λ) is surrounded by two zeros of Pℓ,n+1(η;λ).Figures1–3 show these behaviors for selected parameters.


### 4.2. Behaviors asℓ Increases at Fixed n

From Figures4–7, 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 ℓ increases.For J2 Jacobi polynomials, the positive (negative) zeros shift left (right) asℓ 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(4.1)∂ηLn(α)(η)=-Ln-1(α+1)(η),
and
(4.2)Ln(α)(η)-Ln(α-1)(η)=Ln-1(α)(η),
we get
(4.3)Pℓ,ℓ(η;g)=Lℓ(g+ℓ-(1/2))(-η)Lℓ(g+ℓ-(3/2))(η)+Lℓ(g+ℓ-(1/2))(η)Lℓ(g+ℓ-(3/2))(-η)-Lℓ(g+ℓ-(3/2))(η)Lℓ(g+ℓ-(3/2))(-η).
Hence, when n=ℓ, the L1 Laguerre is an even function of η, and its zeros are symmetric with respect to η=0.


## 4.1. Behaviors asn Increases at Fixed ℓ

This is guaranteed by the oscillation theorem of the one-dimensional quantum mechanics, sincePℓ,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 Pℓ,n decrease, and a new zero appears from the right. Thus the n zeros of Pℓ,n(η;λ) and the n+1 zeros of Pℓ,n+1(η;λ) interlace with each other: each zero of Pℓ,n(η;λ) is surrounded by two zeros of Pℓ,n+1(η;λ).Figures1–3 show these behaviors for selected parameters.


## 4.2. Behaviors asℓ Increases at Fixed n

From Figures4–7, 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 ℓ increases.For J2 Jacobi polynomials, the positive (negative) zeros shift left (right) asℓ 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(4.1)∂ηLn(α)(η)=-Ln-1(α+1)(η),
and
(4.2)Ln(α)(η)-Ln(α-1)(η)=Ln-1(α)(η),
we get
(4.3)Pℓ,ℓ(η;g)=Lℓ(g+ℓ-(1/2))(-η)Lℓ(g+ℓ-(3/2))(η)+Lℓ(g+ℓ-(1/2))(η)Lℓ(g+ℓ-(3/2))(-η)-Lℓ(g+ℓ-(3/2))(η)Lℓ(g+ℓ-(3/2))(-η).
Hence, when n=ℓ, the L1 Laguerre is an even function of η, and its zeros are symmetric with respect to η=0.


## 5. Asymptotic Behavior ofη-k(ℓ,n)→ξk(ℓ) as n→∞

Here we provide intuitive arguments for the above asymptotic behavior. As mentioned before, forn=0, we have η-k(ℓ,0)=ξ-k(ℓ), as Pn=0=1. We will show that as n→∞, η-k(ℓ,n)→ξk(ℓ). 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,
(5.1)Ln(α)(η)≅eη/22π(-η)-((α/2)-(1/4))n((α/2)-(1/4))e2-nη,α∈ℝ,η∈ℂ∖(0,∞),
which gives the asymptotic form of Ln(α)(η) for large n. For the L1 and L2 cases, we have α=g+ℓ-3/2 and g+ℓ+1/2, respectively.One finds(5.2)|Ln(α)(η)∂ηLn(α)(η)|~|-1n(-η)1/2|.For largen with fixed η, ∂ηLn(α)(η) dominates over Ln(α)(η), and thus the zeros of Pℓ,n are determined by those of ξℓ(η;g) as n→∞.


### 5.2. J2 Jacobi

For the asymptotic form ofPn(α,β)(η) for large n, we will make use of Theorem 8.21.7 of [17]:
(5.3)Pn(α,β)(η)≅(η-1)-α/2(η+1)-β/2{η+1+η-1}α+β×(η2-1)-1/42πn{η+η2-1}n+(1/2),α,β∈ℝ,η∈ℂ∖[-1,1],
and
(5.4)∂ηPn(α,β)(η)=12(n+α+β+1)Pn-1(α+1,β+1)(η).One finds(5.5)Pn(α,β)(η)∂ηPn(α,β)(η)~2(n+α+β+1)n-1n(η2-1)1/2η+η2-1(η+1+η-1)2.Again, for largen with fixed η, ∂ηPn(α,β)(η) dominates over Pn(α,β)(η), and thus the zeros of Pℓ,n are determined by those of ξℓ(η;g) as n→∞.


## 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,
(5.1)Ln(α)(η)≅eη/22π(-η)-((α/2)-(1/4))n((α/2)-(1/4))e2-nη,α∈ℝ,η∈ℂ∖(0,∞),
which gives the asymptotic form of Ln(α)(η) for large n. For the L1 and L2 cases, we have α=g+ℓ-3/2 and g+ℓ+1/2, respectively.One finds(5.2)|Ln(α)(η)∂ηLn(α)(η)|~|-1n(-η)1/2|.For largen with fixed η, ∂ηLn(α)(η) dominates over Ln(α)(η), and thus the zeros of Pℓ,n are determined by those of ξℓ(η;g) as n→∞.


## 5.2. J2 Jacobi

For the asymptotic form ofPn(α,β)(η) for large n, we will make use of Theorem 8.21.7 of [17]:
(5.3)Pn(α,β)(η)≅(η-1)-α/2(η+1)-β/2{η+1+η-1}α+β×(η2-1)-1/42πn{η+η2-1}n+(1/2),α,β∈ℝ,η∈ℂ∖[-1,1],
and
(5.4)∂ηPn(α,β)(η)=12(n+α+β+1)Pn-1(α+1,β+1)(η).One finds(5.5)Pn(α,β)(η)∂ηPn(α,β)(η)~2(n+α+β+1)n-1n(η2-1)1/2η+η2-1(η+1+η-1)2.Again, for largen with fixed η, ∂ηPn(α,β)(η) dominates over Pn(α,β)(η), and thus the zeros of Pℓ,n are determined by those of ξℓ(η;g) as n→∞.


## 6. Behaviors at Largeg and/or h

### 6.1. L1 Laguerre

Asg increases, we have that
(6.1)|ξ-k(ℓ)|,|ξk(ℓ)|,|η-k(ℓ,n)|,|ηk(ℓ,n)|
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 largeg, we have ξℓ(η;g+1)≈ξℓ(η;g). Hence
(6.2)Pℓ,n(η;g)≈ξℓ(η;g)[Ln(g+ℓ-(3/2))(η)-∂ηLn(g+ℓ-(3/2))(η)]≈ξℓ(η;g)Ln(g+ℓ-(1/2))(η).
For g≫1, Pℓ,n(η;g) approaches
(6.3)Pℓ,n(η;g)≈Lℓ(g+ℓ)(-η)Ln(g+ℓ)(η).
Thus the extra (η-k(ℓ,n)) and the ordinary (ηk(ℓ,n)) zeros of Pℓ,n(η;g) are given by the zeros of Lℓ(g+ℓ)(-η) and Ln(g+ℓ)(η), respectively.


### 6.2. L2 Laguerre

Asg increases, we have ℜξ-k(ℓ),ℜξk(ℓ) decreased, |ℑξ-k(ℓ)|, |ℑξk(ℓ)| increased, and ηk(ℓ,n) increased. This is easily seen from Figures 6 and 7. That is, the zeros ξ-k(ℓ),ξk(ℓ), and hence η-k(ℓ,n) all are moving leftwards and away from the x-axis, while the ordinary zeros ηk(ℓ,n) are moving towards the right.In fact, for largeg, we have ξℓ(η;g+1)≈ξℓ(η;g). Hence
(6.4)Pℓ,n(η;g)≈ξℓ(η;g)[(g+12)Ln(g+ℓ+(1/2))(η)+η∂ηLn(g+ℓ+(1/2))(η)].
Using (E.2), (E.10), and (E.9) of [9], we arrive at
(6.5)Pℓ,n(η;g)≈ξℓ(η;g)[(g+ℓ+12+n)Ln(g+ℓ-(1/2))(η)-ℓLn(g+ℓ+(1/2))(η)].
For g≫1, Pℓ,n(η;g) approaches
(6.6)Pℓ,n(η;g)≈Lℓ(-g-ℓ)(η)Ln(g+ℓ)(η).
Thus the extra (η-k(ℓ,n)) and the ordinary (ηk(ℓ,n)) zeros of Pℓ,n(η;g) are given by the zeros of Lℓ(-g-ℓ)(η) and Ln(g+ℓ)(η), respectively.


### 6.3. J2 Jacobi

Asg,h increases, we have ℜξ-k(ℓ), ℜξk(ℓ), |ℑξ-k(ℓ)|, |ℑξk(ℓ)| increased, as is evident from Figures 8 and 9. The extra zeros η-k(ℓ,n), being in between these zeros, follow the same pattern. That is, the zeros ξ-k(ℓ), ξk(ℓ), and hence η-k(ℓ,n) all are moving away from the x and y-axes. The ordinary zeros ηk(ℓ,n) will have their norm |ηk(ℓ,n)|decrease in general as g increases. Thus these zeros move towards the y-axis.In fact, for largeg and h, we have (α≡g+ℓ+(1/2), β≡h+ℓ-(3/2)) as
(6.7)Pℓ,n(η;g,h)≈ξℓ(η;g,h)[(g+12)Pn(α,β)(η)-(1-η)∂ηPn(α,β)(η)].
Using (E.13) and (E.23) of [9], we arrive at
(6.8)Pℓ,n(η;g,h)≈ξℓ(η;g,h)[(α+n)Pn(α+1,β+1)(η)-ℓPn(α,β)(η)].
For g≫1 and h≫1, Pℓ,n(η;g,h) approaches
(6.9)Pℓ,n(η;g)≈Pℓ(-g-ℓ,h+ℓ)(η)Pn(g+ℓ,h+ℓ)(η).
Thus the extra (η-k(ℓ,n)) and the ordinary (ηk(ℓ,n)) zeros of Pℓ,n(η;g,h) are given by the zeros of Pℓ(-g-ℓ,h+ℓ)(η) and Pn(g+ℓ,h+ℓ)(η), respectively.


#### 6.3.1. Additional Observation:h≫g

Forh≫g, all zeros, that is, ξ-j(ℓ), ξj(ℓ), η-k(ℓ,n), ηk(ℓ,n), gather around η=1. This can be understood as follows. From the series expansion of the Jacobi polynomials, (A.2)
(6.10)Pn(α,β)(η)=(α+1)nn!∑k=0n‍1k!(-n)k(n+α+β+1)k(α+1)k(1-η2)k,
one sees that, for h≫g, the absolute value of Pℓ,n(η;g,h) is large near η=-1 and small at η=1. Hence, in this limit, the zeros of Pℓ,n(η;g,h) distribute very near η=1. We show this in Figure 10 for certain parameters.


## 6.1. L1 Laguerre

Asg increases, we have that
(6.1)|ξ-k(ℓ)|,|ξk(ℓ)|,|η-k(ℓ,n)|,|ηk(ℓ,n)|
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 largeg, we have ξℓ(η;g+1)≈ξℓ(η;g). Hence
(6.2)Pℓ,n(η;g)≈ξℓ(η;g)[Ln(g+ℓ-(3/2))(η)-∂ηLn(g+ℓ-(3/2))(η)]≈ξℓ(η;g)Ln(g+ℓ-(1/2))(η).
For g≫1, Pℓ,n(η;g) approaches
(6.3)Pℓ,n(η;g)≈Lℓ(g+ℓ)(-η)Ln(g+ℓ)(η).
Thus the extra (η-k(ℓ,n)) and the ordinary (ηk(ℓ,n)) zeros of Pℓ,n(η;g) are given by the zeros of Lℓ(g+ℓ)(-η) and Ln(g+ℓ)(η), respectively.


## 6.2. L2 Laguerre

Asg increases, we have ℜξ-k(ℓ),ℜξk(ℓ) decreased, |ℑξ-k(ℓ)|, |ℑξk(ℓ)| increased, and ηk(ℓ,n) increased. This is easily seen from Figures 6 and 7. That is, the zeros ξ-k(ℓ),ξk(ℓ), and hence η-k(ℓ,n) all are moving leftwards and away from the x-axis, while the ordinary zeros ηk(ℓ,n) are moving towards the right.In fact, for largeg, we have ξℓ(η;g+1)≈ξℓ(η;g). Hence
(6.4)Pℓ,n(η;g)≈ξℓ(η;g)[(g+12)Ln(g+ℓ+(1/2))(η)+η∂ηLn(g+ℓ+(1/2))(η)].
Using (E.2), (E.10), and (E.9) of [9], we arrive at
(6.5)Pℓ,n(η;g)≈ξℓ(η;g)[(g+ℓ+12+n)Ln(g+ℓ-(1/2))(η)-ℓLn(g+ℓ+(1/2))(η)].
For g≫1, Pℓ,n(η;g) approaches
(6.6)Pℓ,n(η;g)≈Lℓ(-g-ℓ)(η)Ln(g+ℓ)(η).
Thus the extra (η-k(ℓ,n)) and the ordinary (ηk(ℓ,n)) zeros of Pℓ,n(η;g) are given by the zeros of Lℓ(-g-ℓ)(η) and Ln(g+ℓ)(η), respectively.


## 6.3. J2 Jacobi

Asg,h increases, we have ℜξ-k(ℓ), ℜξk(ℓ), |ℑξ-k(ℓ)|, |ℑξk(ℓ)| increased, as is evident from Figures 8 and 9. The extra zeros η-k(ℓ,n), being in between these zeros, follow the same pattern. That is, the zeros ξ-k(ℓ), ξk(ℓ), and hence η-k(ℓ,n) all are moving away from the x and y-axes. The ordinary zeros ηk(ℓ,n) will have their norm |ηk(ℓ,n)|decrease in general as g increases. Thus these zeros move towards the y-axis.In fact, for largeg and h, we have (α≡g+ℓ+(1/2), β≡h+ℓ-(3/2)) as
(6.7)Pℓ,n(η;g,h)≈ξℓ(η;g,h)[(g+12)Pn(α,β)(η)-(1-η)∂ηPn(α,β)(η)].
Using (E.13) and (E.23) of [9], we arrive at
(6.8)Pℓ,n(η;g,h)≈ξℓ(η;g,h)[(α+n)Pn(α+1,β+1)(η)-ℓPn(α,β)(η)].
For g≫1 and h≫1, Pℓ,n(η;g,h) approaches
(6.9)Pℓ,n(η;g)≈Pℓ(-g-ℓ,h+ℓ)(η)Pn(g+ℓ,h+ℓ)(η).
Thus the extra (η-k(ℓ,n)) and the ordinary (ηk(ℓ,n)) zeros of Pℓ,n(η;g,h) are given by the zeros of Pℓ(-g-ℓ,h+ℓ)(η) and Pn(g+ℓ,h+ℓ)(η), respectively.


### 6.3.1. Additional Observation:h≫g

Forh≫g, all zeros, that is, ξ-j(ℓ), ξj(ℓ), η-k(ℓ,n), ηk(ℓ,n), gather around η=1. This can be understood as follows. From the series expansion of the Jacobi polynomials, (A.2)
(6.10)Pn(α,β)(η)=(α+1)nn!∑k=0n‍1k!(-n)k(n+α+β+1)k(α+1)k(1-η2)k,
one sees that, for h≫g, the absolute value of Pℓ,n(η;g,h) is large near η=-1 and small at η=1. Hence, in this limit, the zeros of Pℓ,n(η;g,h) distribute very near η=1. We show this in Figure 10 for certain parameters.


## 6.3.1. Additional Observation:h≫g

Forh≫g, all zeros, that is, ξ-j(ℓ), ξj(ℓ), η-k(ℓ,n), ηk(ℓ,n), gather around η=1. This can be understood as follows. From the series expansion of the Jacobi polynomials, (A.2)
(6.10)Pn(α,β)(η)=(α+1)nn!∑k=0n‍1k!(-n)k(n+α+β+1)k(α+1)k(1-η2)k,
one sees that, for h≫g, the absolute value of Pℓ,n(η;g,h) is large near η=-1 and small at η=1. Hence, in this limit, the zeros of Pℓ,n(η;g,h) 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 exceptionalXℓ 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(η;λ) have the lowest degree ℓ=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. TheXℓ polynomial Pℓ,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 ℓ zeros outside the ordinary domain.For the ordinary zeros, their distribution asn increases at a fixed ℓ 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(η;λ) interlace with the n zeros of Pℓ,n(η;λ). On the other hand, when ℓ 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 ℓ increases.For theℓ extra zeros of Pℓ,n(η;λ), each and everyone of them is sandwiched between the corresponding zeros of the deforming polynomials ξℓ(η;λ+δ) and ξℓ(η;λ). As n increases at a fixed ℓ, the extra zeros move from the zeros of ξℓ(η;λ+δ) to those of ξℓ(η;λ).The behaviors of the extra zeros asℓ 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 ℓ increases by one, the number of the extra zeros increases from ℓ to ℓ+1. For the L2 Laguerre and J1 and J2 Jacobi polynomials, they have ℓ/2 pairs of complex zeros for even ℓ, and (ℓ-1)/2 pairs of complex zeros and a real zero outside the ordinary domains where the weight functions are defined. As ℓ 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 ℓ, 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 (g≫1, h≫1), the exceptional polynomial Pℓ,n(η,λ) is expressed as the product of the original polynomial Pn(η) and the deforming polynomial ξℓ(η;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].



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*Source: 920475-2012-09-11.xml*