Zeros of the exceptional Laguerre and Jacobi polynomials

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree $\ell=1,2,...$, 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.


Introduction
The discovery of new types of orthogonal polynomials, called the exceptional X ℓ polynomials, has been the most interesting development in the area of exactly solvable models in quantum mechanics in the last two years [1,2,3,4,5]. 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 X ℓ polynomials, with ℓ = 1, 2, . . ., were constructed in [3]. These systems were derived by deforming the radial oscillator potential and the Darboux-Pöschl-Teller (DPT) potential in terms of an eigenpolynomial of degree ℓ (ℓ = 1, 2, . . .).
The lowest (ℓ = 1) examples, the X 1 -Laguerre and X 1 -Jacobi polynomials, are equivalent to those introduced in the pioneering work of Gomez-Ullate et al. [1] within the Sturm-Liouville theory. The results in [1] were reformulated in the framework of quantum mechanics and shape-invariant potentials by Quesne et al. [2]. By construction these new orthogonal polynomials satisfy a second order differential equation (the Schrödinger equation) without contradicting Bochner's theorem [6], 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 [7].
Later, equivalent but much simpler looking forms of the Laguerre-and Jacobi-type X ℓ polynomials than those originally presented in [3] were given in [4]. These nice forms were derived based on an analysis of the second order differential equations for the X ℓ 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 ℓ polynomials, such as the actions of the forward and backward shift operators on the X ℓ polynomials, Gram-Schmidt orthonormalization for the algebraic construction of the X ℓ 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 [8,9]. Physical models which may involve these new polynomials were considered in [10].
One important aspect related to these new polynomials, which was only briefly mentioned in [4] 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.
The plan of this paper is as follows. In Sect. 2 we briefly review the forms of the exceptional polynomials. Sect. 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. Sect. 5 presents analytical proofs that explain the movements of the extra zeros of the exceptional polynomials as n changes at fixed ℓ. In Sect. 6 we consider behaviors of the zeros at large g and/or h. Sect. 7 summarizes the paper.

Exceptional orthogonal polynomials
Four sets of infinitely many exceptional orthogonal polynomials were derived in [3], 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 [4], 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 non-negative 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: in which λ def = g > 0 and δ def = 1 and The two sets of exceptional Jacobi polynomials (ℓ = 1, 2, . . ., n = 0, 1, 2, . . .) are : The new exceptional orthogonal polynomials can be viewed as deformations of the classical orthogonal polynomials by the parameter ℓ, and the two polynomials ξ ℓ (η; λ) and ξ ℓ (η; λ+δ) played the role of the deforming polunomials.
The zeros of orthogonal polynomials have always attracted the interest of researchers. In this paper we shall study the properties of the zeros of these new exceptional polynomials as some of their basic parameters change.
In the case of X ℓ polynomial P ℓ,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 shall call the ordinary zeros, are the same as those of other ordinary orthogonal polynomials. We shall say more about these zeros in sect. 4. Besides these n zeros, there are extra ℓ zeros outside the ordinary domain. For convenience, we shall adopt the following notation for the zeros of the various polynomials involved: k : zeros of ξ ℓ (η; λ), k = 1, 2, . . . , ℓ; : extra zeros of P ℓ,n , k = 1, 2, . . . , ℓ; η (ℓ,n) j : ordinary zeros of P ℓ,n , j = 1, 2, . . . , n.
Figs. 1-10 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.
3 Extra ℓ zeros of X ℓ polynomials Here we discuss the location of the extra ℓ zeros of the exceptional orthogonal polynomials, which lie in various different positions for the different types of polynomials. From our numerical analysis, we can summarize the trend as follows.
The ℓ extra zeros of L1 polynomials are on the negative real line (−∞, 0). The L2 X ℓ: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 X ℓ:even polynomials have 1 2 ℓ pairs of complex conjugate roots.
The situations for the X ℓ Jacobi polynomials are a bit more complicated. The J1 X ℓ: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 X ℓ:even polynomials have 1 2 ℓ pairs of complex conjugate roots with negative real parts. The J2 X ℓ: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 X ℓ: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 sense η ↔ −η and g ↔ h, as exemplified by the relation ξ J2 [3,4]. 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 shall only discuss the behaviors of the zeros of the J2 Jacobi polynomials in this paper.

L1 Laguerre
As ℓ changes to ℓ + 1, the zeros of ξ ℓ (η; g + 1) and ξ ℓ (η; g) decrease (move to the left), and a new set of zeros appear from the right.
We show these patterns for some representative parameters in Figs. 4 and 5.
Generally, for even ℓ, there are ℓ complex zeros with Allη (ℓ,n) k are sandwiched betweenξ (ℓ) k and ξ (ℓ) k . 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.
Figs. 6 and 7 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.
Generally, for even ℓ, there are ℓ complex zeros with 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.
Figs. 8 and 9 show these behaviors for some selected parameters. For large ℓ, these zeros distribute in a horse-shoe pattern.

Ordinary zeros of X ℓ polynomials
In the case of X ℓ polynomials P ℓ,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 are the same as those of other ordinary orthogonal polynomials.

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

Behaviors as ℓ increases at fixed n
From Figs. 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 ℓ increases.
For J2 Jacobi polynomials, the positive (negative) zeros shift left (right) as ℓ increases, i.e., they move toward the origin η = 0. This is illustrated in Figs. 8 and 9.

Additional observation for the L1 case
Using the well-known derivative relation and we get Hence when n = ℓ, the L1 Laguerre is an even function of η, and its zeros are symmetric w.r.t η = 0.

Proof thatη
As mentioned before, for n = 0, we haveη k , as P n=0 = 1. We shall show that as n → ∞,η k . This amounts to showing that in this limit, ∂ η P n dominates over P n
6 Behaviors at large g and /or h 6.1 L1 Laguerre As g increases, we have all increase. That is, all the zeros move away from the y-axis. This can be seen from Figs. 4 and 5.

Additional observation: h ≫ g
For h ≫ g, all zeros, i.e.,ξ , gather around η = 1. This can be understood as follows. From the series expansion of the Jacobi polynomials, Eq. (E.11) of [4], one sees that, for h ≫ g, the absolute value of P ℓ,n (η; g, h) is large near η = −1 and small at η = 1. Hence, in this limit, the zeros of P ℓ,n (η; g, h) distribute very near η = 1. We show this in Fig. 10 for certain parameters.

Summary
The discovery of new types of orthogonal polynomials, called the exceptional X ℓ 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 ℓ,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 [4].
In this paper, we have considered the distributions of the zeros of these new polynomials as some parameters of the Hamiltonians change. The X ℓ polynomials P ℓ,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 as n 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 P ℓ,n+1 (η; λ) interlace with the n zeros of P ℓ,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.