Orthogonal polynomials and M\"obius transformations

TL;DR

This paper explores how M"obius transformations applied to classical orthogonal polynomials produce new polynomial sequences with similar properties, including orthogonality on complex curves, recurrence relations, and differential equations, revealing interrelations among various classical polynomials.

Contribution

It introduces a framework for understanding M"obius-transformed orthogonal polynomials, establishing their properties and connections among classical families, and presents new orthogonality relations.

Findings

M"obius-transformed polynomials satisfy orthogonality on complex curves.

They obey three-term recurrence relations and Christoffel-Darboux identities.

New orthogonality relations for Bessel and Romanovski polynomials are derived.

Abstract

Given an orthogonal polynomial sequence on the real line, another sequence of polynomials can be found by composing these polynomials with a general M\"obius transformation. In this work, we study the properties of such M\"obius-transformed polynomials. We show that they satisfy an orthogonality relation in given curve of the complex plane with respect to a varying weight function and that they also enjoy several properties common to the orthogonal polynomial sequences on the real line --- e.g. a three-term recurrence relation, Christoffel-Darboux type identities, their zeros are simple, lie on the support of orthogonality and have the interlacing property, etc. Moreover, we also show that the M\"obius-transformed polynomials obtained from classical orthogonal polynomials also satisfy a second-order differential equation, a Rodrigues' type formula and generating functions. As an application, we show that Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are all related to each other by a suitable M\"obius transformation. New orthogonality relations for Bessel and Romanovski polynomials are also presented.

Tables

Table 1. Table 1: The first Möbius-transformed orthogonal polynomials 𝒯 n ​ ( x ) = x n ​ T n ​ ( M ​ ( x ) ) subscript 𝒯 𝑛 𝑥 superscript 𝑥 𝑛 subscript 𝑇 𝑛 𝑀 𝑥 \mathcal{T}_{n}(x)=x^{n}T_{n}(M(x)) , where T n ​ ( x ) subscript 𝑇 𝑛 𝑥 T_{n}(x) are the Chebyshev polynomials and M ​ ( x ) = 1 / x 𝑀 𝑥 1 𝑥 M(x)=1/x is the inversion transformation. Notice that the polynomials T n ​ ( x ) subscript 𝑇 𝑛 𝑥 T_{n}(x) for odd n 𝑛 n have a zero at x = 0 𝑥 0 x=0 , which implies that the Möbius-transformed polynomials 𝒯 n ​ ( x ) subscript 𝒯 𝑛 𝑥 \mathcal{T}_{n}(x) for odd n 𝑛 n will have even degree n ′ = n − 1 superscript 𝑛 ′ 𝑛 1 n^{\prime}=n-1 . Notwithstanding, these polynomials are orthogonal on the interval 𝒥 = ( − ∞ , − 1 ) ∪ ( 1 , ∞ ) 𝒥 1 1 \mathcal{J}=\left(-\infty,-1\right)\cup\left(1,\infty\right) of the real line with respect to the varying weight function ω m , n ​ ( x ) = x − m − n − 2 / 1 − x − 2 subscript 𝜔 𝑚 𝑛 𝑥 superscript 𝑥 𝑚 𝑛 2 1 superscript 𝑥 2 \omega_{m,n}(x)=x^{-m-n-2}/\sqrt{1-x^{-2}} .

Chebyshev polynomials	Inversion-transformed Chebyshev polynomials
$T_0\left(x\right)=1$	${𝒯}_0\left(x\right)=1$
$T_1\left(x\right)=x$	${𝒯}_1\left(x\right)=1$
$T_2\left(x\right)=-1+2x^2$	${𝒯}_2\left(x\right)=2-x^2$
$T_3\left(x\right)=-3x+4x^3$	${𝒯}_3\left(x\right)=4-3x^2$
$T_4\left(x\right)=1-8x^2+8x^4$	${𝒯}_4\left(x\right)=8-8x^2+x^4$
$T_5\left(x\right)=5x-20x^3+16x^5$	${𝒯}_5\left(x\right)=16-20x^2+5x^4$
$T_6\left(x\right)=-1+18x^2-48x^4+32x^6$	${𝒯}_6\left(x\right)=32-48x^2+18x^4-x^6$
$T_7\left(x\right)=-7x+56x^3-112x^5+64x^7$	${𝒯}_7\left(x\right)=64-112x^2+56x^4-7x^6$
$T_8\left(x\right)=1-32x^2+160x^4-256x^6+128x^8$	${𝒯}_8\left(x\right)=128-256x^2+160x^4-32x^6+x^8$

Table 2. Table 2: The first generalized Bessel polynomials and Bessel polynomials (normalized with the constant terms equal to unity).

Generalized Bessel polynomials	Bessel polynomials
${ℬ}_0^{(\gamma ,\beta )}\left(x\right)=1$	${ℬ}_0\left(x\right)=1$
${ℬ}_1^{(\gamma ,\beta )}\left(x\right)=1+\frac{\gamma }{\beta }x$	${ℬ}_1\left(x\right)=1+x$
${ℬ}_2^{(\gamma ,\beta )}\left(x\right)=1+\frac{2\left(1+\gamma \right)}{\beta }x+\frac{\left(1+\gamma \right)\left(2+\gamma \right)}{{\beta }^2}{x}^2$	${ℬ}_2\left(x\right)=1+3x+3x^2$
${ℬ}_3^{(\gamma ,\beta )}\left(x\right)=1+\frac{3\left(2+\gamma \right)}{\beta }x+\frac{3\left(2+\gamma \right)\left(3+\gamma \right)}{{\beta }^2}{x}^2+\frac{\left(2+\gamma \right)\left(3+\gamma \right)\left(4+\gamma \right)}{{\beta }^3}{x}^3$	${ℬ}_3\left(x\right)=1+6x+15x^2+15x^3$

Table 3. Table 3: The first Romanovski polynomials ℛ n ( γ , δ ) ​ ( x ) = ( − i ) n ​ J n ( γ + i ​ δ , γ − i ​ δ ) ​ ( i ​ x ) superscript subscript ℛ 𝑛 𝛾 𝛿 𝑥 superscript 𝑖 𝑛 superscript subscript 𝐽 𝑛 𝛾 𝑖 𝛿 𝛾 𝑖 𝛿 𝑖 𝑥 \mathcal{R}_{n}^{\left(\gamma,\delta\right)}(x)=\left(-i\right)^{n}J_{n}^{\left(\gamma+i\delta,\gamma-i\delta\right)}(ix) .

Romanovski polynomials
${ℛ}_0^{(\gamma ,\delta )}\left(x\right)=1$
${ℛ}_1^{(\gamma ,\delta )}\left(x\right)=\delta +\left(1+\gamma \right)x$
${ℛ}_2^{(\gamma ,\delta )}\left(x\right)=\frac{1}{4}\left(2+\gamma +2{\delta }^2\right)+\frac{1}{2}\delta \left(3+2\gamma \right)x+\frac{1}{4}\left(2+\gamma \right)\left(3+2\gamma \right)x^2$

Equations

$$M(x)=-i\frac{x+1}{x-1},W(x)=\frac{x-i}{x+i},$$

$$M(x)=\frac{ax+b}{cx+d},$$

$$W(x)=-\frac{dx-b}{cx-a}.$$

$$M(-d/c)$$

$$r(x)=p_0+p_1\left(\frac{ax+b}{cx+d}\right)+\cdots +p_{n-1}{\left(\frac{ax+b}{cx+d}\right)}^{n-1}+p_n{\left(\frac{ax+b}{cx+d}\right)}^n,$$

$$\begin{gathered} q(x)={\left(cx+d\right)}^{n}r(x)={\left(cx+d\right)}^{n}p(M(x)) \\ =p_0{\left(cx+d\right)}^n+p_1\left(ax+b\right){\left(cx+d\right)}^{n-1}+\cdots +p_{n-1}{\left(ax+b\right)}^{n-1}\left(cx+d\right)+p_n{\left(ax+b\right)}^n. \end{gathered}$$

$$p(x)=r(W(x))=\frac{q(W(x))}{{\left(cW(x)+d\right)}^n}=\frac{{\left(-1\right)}^n}{{\Delta }^n}{\left(cx-a\right)}^{n}q(W(x)).$$

$$q_n=\sum _{k=0}^n{p}_{n-k}{a}^{n-k}{c}^k=c^n\sum _{k=0}^n{p}_{n-k}{\left(\frac{a}{c}\right)}^{n-k}=c^n\left[p_n{\left(\frac{a}{c}\right)}^n+\cdots +p_0\right]=c^{n}p\left(\frac{a}{c}\right),$$

$${\zeta }_1=W\left({\xi }_1\right),\dots ,{\zeta }_n=W({\xi }_n).$$

$$p\left({\xi }_k\right)=\frac{{\left(-1\right)}^n}{{\Delta }^n}{\left(c{\xi }_k-a\right)}^{n}q\left(W\left({\xi }_k\right)\right)=0,1⩽k⩽n.$$

$${\int }_l^r{P}_m(x)P_n(x)w(x)\mathrm{d}x=K_m{\delta }_{m,n},$$

$$R_n(x)=P_n\left(M(x)\right),n⩾0.$$

$$Q_n(x)={\left(cx+d\right)}^n{R}_n(x)={\left(cx+d\right)}^n{P}_n(M(x)),n⩾0.$$

$$P_n(x)=R_n(W(x))=\frac{Q_n(W(x))}{{\left(cW(x)+d\right)}^n}=\frac{{\left(-1\right)}^n}{{\Delta }^n}{\left(cx-a\right)}^n{Q}_n(W(x)),n⩾0.$$

$${\int }_{\Gamma }{R}_m(x)R_n(x)\omega (x)\mathrm{d}x=K_m{\delta }_{m,n},$$

$$\omega (x)=w(M(x))M^{\prime }(x)=\Delta \frac{w(M(x))}{{\left(cx+d\right)}^2}.$$

$${\int }_{\Gamma }{Q}_m(x)Q_n(x){\omega }_{m,n}(x)\mathrm{d}x=K_m{\delta }_{m,n}\text{where}{\omega }_{m,n}(x)=\frac{\omega (x)}{{\left(cx+d\right)}^{m+n}}.$$

$$\Gamma =\{z\in \mathbb{C}:z=W(x),\lambda <z<\rho \},\text{where}\lambda =W(l)\text{and}\rho =W(r),$$

$${\int }_l^r{P}_m^{{\alpha }_m}(x)P_n^{{\alpha }_n}(x)w_{{\alpha }_m,{\alpha }_n}(x)\mathrm{d}x=K_m{\delta }_{m,n}\text{where},w_{{\alpha }_m,{\alpha }_n}(x)=\{\begin{array}{ll}{w}_{{\alpha }_n}(x),& m<n,\\ w_{{\alpha }_m}(x),& m>n,\\ w_{\alpha }(x),& m=n,\end{array}$$

$$P_n(x)=\left(A_{n}x-B_n\right)P_{n-1}(x)-C_n{P}_{n-2}(x),n⩾1,$$

$$R_n(x)=\left[A_{n}M(x)-B_n\right]R_{n-1}(x)-C_n{R}_{n-2}(x)=\left[A_n\left(\frac{ax+b}{cx+d}\right)-B_n\right]R_{n-1}(x)-C_n{R}_{n-2}(x),n⩾1,$$

$$Q_n(x)=\left({\mathcal{A}}_{n}x-{\mathcal{B}}_n\right)Q_{n-1}(x)-{\mathcal{C}}_n{\left(cx+d\right)}^2{Q}_{n-2}(x)n⩾1,$$

$${\mathcal{A}}_n=\left(a{A}_n-c{B}_n\right),{\mathcal{B}}_n=\left(d{B}_n-b{A}_n\right),{\mathcal{C}}_n=C_n.$$

$$R_n(W(x))=\left(A_{n}x-B_n\right)R_{n-1}(W(x))-C_n{R}_{n-2}(W(x)).$$

$$R_n(y)$$

$$\frac{Q_n(y)}{{\left(cy+d\right)}^n}=\left[A_n\left(\frac{ay+b}{cy+d}\right)-B_n\right]\frac{Q_{n-1}(y)}{{\left(cy+d\right)}^{n-1}}-C_n\frac{Q_{n-2}(y)}{{\left(cy+d\right)}^{n-2}},$$

$$Q_n(y)$$

$$=\left[A_n\left(ay+b\right)-B_n\left(cy+d\right)\right]Q_{n-1}(y)-C_n{\left(cy+d\right)}^2{Q}_{n-2}(y),$$

$$=\left[\left(a{A}_n-c{B}_n\right)y-\left(d{B}_n-b{A}_n\right)\right]Q_{n-1}(y)-C_n{\left(cy+d\right)}^2{Q}_{n-2}(y),$$

$$\sum _{k=0}^n\frac{A_{k+1}}{C_0\cdots C_{k+1}}{P}_k(x)P_k(y)=\frac{1}{C_0\cdots C_{n+1}}\frac{P_{n+1}(x)P_n(y)-P_n(x)P_{n+1}(y)}{x-y},$$

$$\sum _{k=0}^n\frac{A_{k+1}}{C_0\cdots C_{k+1}}{P}_k^2(x)=\frac{P_{n+1}^{\prime }(x)P_n(x)-P_n^{\prime }(x)P_{n+1}(x)}{C_0\cdots C_{n+1}}.$$

Full text

1. 1 Introduction

2. 2 Möbius-transformed polynomials

3. 3 Möbius-transformed orthogonal polynomials

4. 3.1 Orthogonality relations

5. 3.2 Three-term recurrence relations

6. 3.3 Christoffel-Darboux identities

7. 3.4 Properties of the zeros.

8. 4 Classical Möbius-transformed orthogonal polynomials

9. 4.1 Second-order differential equation

10. 4.2 Weight function differential equations

11. 4.3 Rodrigues’ type formulas

12. 4.4 Generating functions

13. 5 Applications

14. 5.1 From Laguerre to Bessel polynomials

15. 5.2 From Jacobi to Romanovski polynomials

16. 5.3 From Jacobi to Laguerre polynomials

17. 5.4 From Jacobi to Hermite polynomials

1 Introduction

Since its introduction in mathematical sciences, the concept of orthogonal polynomials has been extended and generalized in many ways. The first concept of polynomial orthogonality referred to sequences of polynomials orthogonal with respect to a real, non-negative and continuous weight function (the classical orthogonal polynomials of Hermite, Laguerre and Jacobi are the most known examples). This concept was soon after generalized in order to cover the cases of discrete measures (the Charlier polynomials being the prototype in this class). The range of known families of orthogonal polynomials increased exponentially after Askey showed that hypergeometric functions lead to several families of orthogonal polynomials; together with their $q$-analogues, a concept that was introduced by Hahn, this resulted in Askey scheme for the classification of hypergeometric orthogonal polynomials [1]. All those polynomials are orthogonal on the real line; of course, an obvious generalization is to consider polynomials that are orthogonal in other curves of the complex plane. This was already taken into account, the most known example being the orthogonal polynomial sequences on the unit circle [2, 3]. Since then, many other concepts of polynomial orthogonality were defined: we can cite, for instance, quasi-orthogonal polynomials [4], polynomials that are orthogonal with respect to a varying measure [5], exceptional orthogonal polynomials [6, 7], para-orthogonal polynomials [8], among others.

In this work, we present another generalized class of orthogonal polynomials which we call Möbius-transformed orthogonal polynomials. As the name says, these sequences of polynomials are generated by the action of a general Möbius transformation over an orthogonal polynomial sequence on the real line. In what follows, we shall show that such sequences of polynomials are orthogonal in a given curve of the complex plane, with respect to a simple varying weight function. Besides, these Möbius-transformed orthogonal polynomials enjoy many properties of the usual orthogonal polynomials on the real line as, for example, their zeros are all simple, lie on the support of orthogonality and have the interlacing property; they also satisfy a three-term recurrence relation and identities similar to those of Christoffel-Darboux can be derived. In particular, the Möbius-transformed polynomials obtained from a classical orthogonal polynomial sequence also satisfy second-order differential equations, their weight function also satisfy a first-order differential equation of Pearson’s type, there also exist Rodrigues’ type formula for them and they can also be found through simple generating functions.

We highlight that these Möbius-transformed orthogonal polynomials can be of importance in Applied Mathematics and Physics. In fact, the composition of polynomials with a Möbius transformation is already considered in many applications: we can cite, for example, its role in some of the fastest polynomial isolating and root-finding methods available to date (e.g., Akritas method [9]). We can also find the number of zeros that a complex polynomial has on the unit circle through Sturm theorem if we previously consider the composition of the testing polynomial with the following pair of Möbius transformations (often called Cayley transformations),

[TABLE]

that map, respectively, the real line onto the unit circle and vice versa [10]. Moreover, if we act with the first of these special Möbius transformations over an orthogonal polynomial sequence on the real line, then we can verify (from the results that will be reported in this work) that a new sequence of polynomials whose zeros lie all on the unit circle is obtained. Consequently, we shall get an infinite sequence of self-inversive or self-reciprocal polynomials (which are complex or real polynomials whose zeros are all symmetric with respect to the unit circle). It is worth to mention that self-inversive and self-reciprocal polynomials are important in many fields of Physics and Mathematics — see [11] and references therein. Finally, such Möbius-transformed polynomials are also orthogonal on the unit circle with respect to a varying measure; because of this, we can conclude that they are intimately related with the usual orthogonal polynomials on the unit circle and also with para-orthogonal polynomials. The study of this special case will be reported a separated paper due to particular importance (in this work we shall consider a general Möbius transformation).

Another field where Möbius-transformed polynomials can be of relevance is related to boundary value problems and second-order differential equations, which are usually found in physical problems. It is well known the role of classical orthogonal polynomials in Sturm-Liouville problems on the real line; because Möbius transformations preserve the main characteristics of the polynomials, we can conclude from this that the corresponding Möbius-transformed polynomials will be associated with boundary values problems defined in some curves (arc circles or line segments) of the complex plane and that they will also obey second-order differential equations with complex coefficients. We can cite the so-called relativistic classical orthogonal polynomials, which are orthogonal polynomials on the real line with respect to a varying measure that appear in some problems of relativistic quantum mechanics [12, 13, 14]. These relativistic classical orthogonal polynomials are actually special cases of Möbius-transformed orthogonal polynomials — in fact, it was showed by Ismail that they can be obtained from Jacobi polynomials through the action of a specific Möbius transformation with real parameters [15].

This work is organized as follows: the concept of Möbius-transformed polynomials is introduced in Section 2. In Section 3, we begin our study on Möbius-transformed orthogonal polynomials, where their orthogonality, the three-term recurrence relations and the corresponding Christoffel-Darboux type identities are discussed. The classical Möbius-transformed polynomials are studied in Section 4, where we show that they also satisfy second-order differential equations, that their weight functions satisfy a first-order Pearson’s type differential equation, we derive Rodrigues’ like formulas for them and the respective generating functions. Finally, in Section 5 we show, as an application, that all the sequences of Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are related to each other essentially by a Möbius transformation; this also provided new orthogonality relations for Bessel and Romanovski polynomials.

2 Möbius-transformed polynomials

A Möbius transformation is a conformal mapping defined of the extended complex plane $\mathbb{C}_{\infty}=\mathbb{C}\cup\{\infty\}$ (i.e., the Riemann sphere) by the formula [16, 17]:

[TABLE]

where $x\in\mathbb{C}_{\infty}$ and $a$, $b$, $c$ and $d$ are complex numbers such that $\varDelta=ad-bc\neq 0$. Its inverse, which is also a Möbius transformation, is given by the expression,

[TABLE]

Together with the relations

[TABLE]

transformations (2) and (3) become continuous on $\mathbb{C}_{\infty}$. The Möbius transformations include as special cases many important geometric transformations as, for instance, translations $M(x)=x+b$, rotations $M(x)=ax$ $(|a|=1)$, inversions $M(x)=1/x$ and linear transformations $M(x)=ax+b$. They also have many interesting properties which can be found in [16, 17].

Given a complex polynomial $p(x)=p_{0}+p_{1}x+\cdots+p_{n-1}x^{n-1}+p_{n}x^{n}$ of degree $n$, we can compose $p(x)$ with $M(x)$ so that rational function

[TABLE]

is is obtained as the result. Thus, multiplying $r(x)$ by $\left(cx+d\right)^{n}$, a new polynomial is obtained:

[TABLE]

We shall refer to $r(x)$ as a Möbius-transformed rational function and to $q(z)$ as a Möbius-transformed polynomial. Notice that the original polynomial $p(x)$ can be retrieved from $r(x)$ and $q(x)$ by the formula

[TABLE]

Not always the Möbius-transformed polynomial $q(x)$, as defined in (6), has the same degree as the original polynomial $p(x)$. The precise condition for this to be true is the following:

Theorem 1**.**

Let $p(x)=p_{n}x^{n}+\cdots+p_{0}$ be a complex polynomial of degree $n$. Then, the Möbius-transformed polynomial $q(x)$ will be polynomial of degree $n$ if, and only if, $c=0$ or, either, if $c\neq 0$ and $p(a/c)\neq 0$.

Proof.

If $c=0$ the proof is trivial; thereby, assume $c\neq 0$. Expanding the binomials in (2) we find that the leading coefficient $q_{n}$ of $q(x)$ is given by the expression,

[TABLE]

from which we conclude that $q(x)$ will be degree $n$ whenever $a/c$ is not a zero of $p(x)$. ∎

We can also find how the zeros of the Möbius-transformed polynomial $q(x)$ are related with the zeros of the original polynomial $p(x)$ through the following:

Theorem 2**.**

Let $\xi_{1},\ldots,\xi_{n}$ denote the zeros of a polynomial $p(x)$ of degree $n$ and $\zeta_{1},\ldots,\zeta_{n}$ the respective zeros of the Möbius-transformed polynomial $q(x)$. Assuming that $p(a/c)\neq 0$, we have that,

[TABLE]

Proof.

For any zero $\xi_{k}$ $\left(1\leqslant k\leqslant n\right)$ of $P_{n}(x)$ we have from (7) that:

[TABLE]

This plainly shows us that $\zeta_{k}=W(\xi_{k})$ will be a zero of $Q_{n}(x)$ whenever $\xi_{k}\neq a/c$. ∎

From this we can also conclude that the Möbius-transformed polynomial $q(x)$ will be a polynomial of degree $n-m$ whenever the original polynomial $p(x)$ of degree $n$ has a zero of multiplicity $m$ at the point $x=a/c$. This, of course, is due to the cancellation of the $m$ poles generated by the mapping $W\left(a/c\right)=\infty$ with the corresponding $m$ zeros of the factors $\left(c\xi-a\right)$ generated by the action of the Möbius transformation over $p(z)$.

3 Möbius-transformed orthogonal polynomials

Now, let us consider an orthogonal polynomial sequence defined in an given interval $(l,r)\subseteq\mathbb{R}$ with respect to a given weight function $w(x)$. That is, let us consider a sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ in which each polynomial $P_{n}(x)\in P$ has degree $n$ and such that the following orthogonality relation is satisfied [18, 19]:

[TABLE]

where $\delta_{m,n}$ is the Kronecker symbol, $K_{m}$ are positive constants and the interval $(l,r)\in\mathbb{R}$ can be either finite, semi-infinite or infinite111We remark that definition above can also be generalized to the case where the weight function $w(x)$ is discrete by interpreting the integral in the sense of Stieltjes or Lebesgue, although in this work we shall only consider the case where $w(x)$ is continuous. .

Composing each polynomial $P_{n}(x)\in P$ with the Möbius transformation (2), we shall obtain a sequence $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ of rational functions, where,

[TABLE]

Multiplying each rational function $R_{n}(x)$ by $\left(cx+d\right)^{n}$, a sequence of Möbius-transformed polynomials $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ is obtained:

[TABLE]

As before, the relationship between $P_{n}(x)$, $Q_{n}(x)$ and $R_{n}(x)$ is the following:

[TABLE]

The properties of such sequences of Möbius-transformed rational functions and polynomials will be analyzed in what follows.

3.1 Orthogonality relations

Let us begin our analysis by presenting the orthogonality relation satisfied by the Möbius-transformed rational functions $R_{n}(x)$ and polynomials $Q_{n}(x)$.

Theorem 3**.**

Let $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ be an orthogonal polynomial sequence on the real line whose polynomials $P_{n}(x)$ satisfy the orthogonality relation (11) for a given weight function $w(x)$ which does not vanish along the orthogonality interval $I=\left(l,r\right)\subseteq\mathbb{R}$. Then, the sequence $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ of the Möbius-transformed rational functions $R_{n}(x)$ defined in (12) will satisfy the following orthogonality relations:

[TABLE]

where

[TABLE]

Similarly, the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed polynomials $Q_{n}(x)$ defined in (13) will satisfy orthogonality relations of the form,

[TABLE]

In both cases, the path of integration is

[TABLE]

which corresponds to a curve (a straight line or a circle) in the complex plane whose initial and final points are $\lambda=W(l)$ and $\rho=W(r)$, respectively. Finally, the new weight functions $\omega(x)$ and $\omega_{m,n}(x)$ do not vanish along the curve $\varGamma$, except at infinity.

Proof.

The orthogonality relation (15) for the Möbius-transformed rational functions $R_{n}(x)$ follows from the change of variable $x=M(y)$ in the integral (11), taking into account that the Jacobian is $M^{\prime}(x)=\varDelta/\left(cx+d\right)^{2}$. The corresponding orthogonality condition (17) for the Möbius-transformed polynomials $Q_{n}(x)$ follows from their definitions given in (13) and from (15), after we absorb the factors $\left(cx+d\right)^{m}$ and $\left(cx+d\right)^{n}$ into the varying weight function $\omega_{m,n}(x)$. Besides, notice that the path of integration $I=\{x\in\mathbb{R}:l<x<r\}$ is mapped through the inverse Möbius-transformation, $W(x)$, onto the curve $\varGamma$, as defined in (18), as well as the limits $x=l$ and $x=r$ are respectively mapped to $\lambda=W(l)$ and $\rho=W(r)$. Finally, consider a point $y$ on the curve $\varGamma$ so that we can write $y=W(x)$ where $x\in I$; in this case, (16) simplifies to $\omega(W(x))=-w(x)\left(cx-a\right)$. Thus, provided that $w(x)$ does not vanish on $I$, we conclude that $\omega(W(x))$ can vanish only at $x=a/c$; but this means that $y=W(a/c)=\infty$ so that $\omega(y)$ can vanish only at $y=\infty$ along the curve $\varGamma$, and the same line of reasoning holds for $\omega_{m,n}(x)$. ∎

Notice that integrals (15) and (17) should be regarded as a complex contour integrals. Whenever the point $x=a/c$ is not on the orthogonality curve $I=\left(l,r\right)\subseteq\mathbb{R}$, there is nothing to worry about, as in this case the orthogonality curve $\varGamma$ will be finite, open and connected, so that these integrals will depend only on their extreme points $\lambda=W(l)$ and $\rho=W(r)$. However, some care should be taken otherwise. In fact, if $x=a/c$ is on $I$, then the curve $\varGamma$ will no longer be connected: instead, it will be constituted by two disjoint straight lines in the complex plane passing through the point $W(a/c)=\infty$ — the first one starts at $\lambda=W(l)$ and goes towards the complex infinity, while the other starts at infinity and ends at $\rho=W(r)$. Notice as well that if $I$ comprehends the whole real line (as in the case of Hermite polynomials, for instance), then $\varGamma$ become either an infinite straight line or a circle in the complex plane; in fact, we have that $W(-\infty)=W(\infty)=-d/c$. Notice that in the extended complex plane $\mathbb{C}_{\infty}$, $\varGamma$ will be always a connected curve.

The weight function $\omega_{m,n}(x)$ that appears in the orthogonality relation (17) is a special type of which is called a varying weight function. Varying weight functions are measures that depend on some parameters of the polynomials entering in the orthogonality relation. Such kind of weights arises naturally from sequences of orthogonal polynomials $P=\left\{P_{n}^{\alpha}(x)\right\}_{n=0}^{\infty}$ whose members depend on one (or more) parameter $\alpha$ (e.g., the associated Laguerre polynomials $L_{n}^{\alpha}(x)$). In fact, provided that, for fixed $\alpha$, these polynomials satisfy an orthogonality relation of the usual form (11) with respect with a given weight function $w_{\alpha}(x)$, we can construct from $P$ a modified sequence $P^{\left\{\alpha\right\}}=\left\{P_{n}^{\alpha_{n}}(x)\right\}_{n=0}^{\infty}$ in which each polynomial $P_{n}^{\alpha_{n}}(x)$ has a different parameter $\alpha_{n}$; it follows in this case that the polynomials $P_{n}^{\alpha_{n}}(x)$ will satisfy an orthogonality relation with respect to a varying weight function, namely:

[TABLE]

for some positive constants $K_{m}$, $m\geqslant 0$. This works because every polynomial of degree $n$ that belongs to a sequence of orthogonal polynomials on the real line is orthogonal to any polynomial of degree less than $n$ — thus, to establish the orthogonality of a sequence of polynomials with different parameters we just need to pick up the weight function belonging to the polynomial of higher degree appearing on the orthogonality relation; besides, when the two polynomials have the same degree, they will also have the same value of $\alpha$ so that the value of the integral is the same as that one for the fixed value $\alpha_{m}=\alpha_{n}$ of $\alpha$. Finally, we remark that sequences of polynomials orthogonal with respect to varying measures appear in several applications, both in mathematics [5] as in physics [20].

3.2 Three-term recurrence relations

It is well-known that any sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ of orthogonal polynomials on the real line satisfies a three-term recurrence relation of the form [18, 19]:

[TABLE]

with the initial conditions $P_{-1}(x)=0$ and $P_{0}(x)=1$, where $A_{n}$, $B_{n}$ and $C_{n}$ are real constants with $A_{n}>0$ and $C_{n}>0$ for each $n\geqslant 1$. In the following, we shall show that the sequences $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ and $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed rational functions and polynomials also satisfy three-term recurrence relation.

Theorem 4**.**

Let $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ be orthogonal polynomial sequence on the real line whose polynomials $P_{n}(x)$ satisfy the three-term recurrence condition (19). Then, the sequence $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ of rational functions $R_{n}(x)$ defined in (12) will satisfy a three term relation of the following form:

[TABLE]

with the initial conditions $R_{-1}(x)=0$ and $R_{1}(x)=1$. Similarly, the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed polynomials $Q_{n}(x)$ defined in (13), will satisfy the three-term recurrence relation:

[TABLE]

with the initial conditions $Q_{-1}(x)=0$ and $Q_{0}(x)=1$, where,

[TABLE]

Proof.

Inserting relation (14) into (19), we get that,

[TABLE]

Now, making the change of variable $x=M(y)$, we obtain, at once,

[TABLE]

It is clear that the relation above holds for any $n\geqslant 1$ provided we define $R_{-1}(x)=0$ and $R_{1}(x)=1$. Finally, (21) follows after we insert (13) into the above expression. In fact, we get in this way,

[TABLE]

so that,

[TABLE]

which holds for any $n\geqslant 1$ provided we define $Q_{-1}(x)=0$ and $Q_{0}(x)=1$. ∎

3.3 Christoffel-Darboux identities

Another important relation satisfied by any orthogonal polynomial sequence on the real line is the so-called Christoffel-Darboux identity [18, 19]:

[TABLE]

and its confluent form:

[TABLE]

Similar identities also exist for the sequences of Möbius-transformed rational functions and polynomials, as we shall describe in the following:

Theorem 5**.**

Let $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ be an orthogonal polynomial sequence on the real line that satisfies the Christoffel-Darboux identities (27) and (28). Then the sequence $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed rational functions $R_{n}(x)$ defined in (12) will satisfy the following identities:

[TABLE]

and

[TABLE]

Similarly, the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed polynomials $Q_{n}(x)$ defined in (13) will satisfy the identities:

[TABLE]

and

[TABLE]

Proof.

From (14) we get that (27) becomes,

[TABLE]

Now, making the change of variables $x=M(z)$ and $t=M(y)$, we obtain,

[TABLE]

But $M(z)-M(t)=\varDelta(z-t)/\left[\left(ct+d\right)\left(cz+d\right)\right]$, so that we get,

[TABLE]

which is of the same form as (29). Now, to prove (30), we proceed in a similar fashion: using (14) into (28), we get,

[TABLE]

that is,

[TABLE]

Thus, making the change of variables $x=M(y)$ and using the identity $W^{\prime}(M(y))=\left(cy+d\right)^{2}/\varDelta$, we obtain,

[TABLE]

which is identical to (30). Finally, to prove (31) and (32), we just need to use (13). With that, (31) becomes:

[TABLE]

so that (30) follows after a simplification. Now, using (13) into (32), we get,

[TABLE]

and (32) follows after we expand the derivatives and simplify the resulting expression. ∎

3.4 Properties of the zeros.

It is well-known that the zeros of polynomials belonging to a sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ of orthogonal polynomials on the real line are all simple and lie in the orthogonality interval $\mathcal{I}=(l,r)\subseteq\mathbb{R}$; moreover, for $n>1$, the zeros of $P_{n}(x)$ interlace with the zeros of $P_{n+1}(x)$ [18, 19]. In this section, we shall show that the Möbius-transformed polynomials defined in (13) also present similar properties.

Theorem 6**.**

Let $P_{n}(x)$ be a polynomial of degree $n\geqslant 1$ that belongs to a sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ of orthogonal polynomials defined in a given interval $\mathcal{I}=(l,r)\subseteq\mathbb{R}$ of the real line and such that $P_{n}(a/c)\neq 0$ for all $n$. Then, the zeros of the corresponding Möbius-transformed polynomials $Q_{n}(x)$ defined in (13), for $n\geqslant 1$, will all lie on the curve of orthogonality $\varGamma$, as defined in (18). Moreover, these zeros are all simple.

Proof.

From Theorems 1 and 2, it follows that the zeros $\zeta_{1}^{n},\ldots,\zeta_{n}^{n}$ of the Möbius-transformed polynomials $Q_{n}(x)$ are related to the zeros $\xi_{1}^{n},\ldots,\xi_{n}^{n}$ of the original polynomials $P_{n}(x)$ by the relations,

[TABLE]

provided that $a/c$ is not a zero of $P_{n}(x)$. This shows us that zeros of $Q_{n}(x)$ will be all finite and will lie on the curve $\varGamma$ given in (18), which corresponds to the image of the interval $\mathcal{I}=(l,r)\subseteq\mathbb{R}$ under the action of the inverse Möbius-transformation $W(x)$. Now, these zeros must be also simple because the Möbius transformation is one-to-one and onto on the extended complex plane. ∎

Theorem 7**.**

The zeros of any two consecutive polynomials $Q_{n}(x)$ and $Q_{n+1}(x)$, for $n\geqslant 1$, that belong to the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed orthogonal polynomials $Q_{n}(x)$ defined in (13) interlace on the new curve of orthogonality $\varGamma$ given by (18).

Proof.

The Möbius transformations are one-to-one and onto on the extended complex plane $\mathbb{C}_{\infty}$. In particular, this implies that the image of the interval $\mathcal{I}=(l,r)\subseteq\mathbb{R}$ under the mapping $y=W(x)$, which is the curve $\varGamma$, is a not-self-intersecting curve. Thus, the ordination of any set of points lying in the interval $\mathcal{I}$ is either unaltered or only inverted by a Möbius transformation. Because the zeros of $P_{n+1}(x)$ interlaces with the zeros of $P_{n}(x)$ on $I$, we conclude from what was said that the zeros of the Möbius-transformed polynomial $Q_{n+1}(x)$ will also interlace with the zeros of the polynomial $Q_{n}(x)$ over $\varGamma$. ∎

We remark that the ordination above should be understood as performed in the extended complex plane $\mathbb{C}_{\infty}$. In the usual complex plane, the ordination should be interpreted with care because the curve $\varGamma$ can be disconnected, as already commented. In fact, this will occur whenever the point $x=a/c$ lies in the interval $\mathcal{I}=\left(l,r\right)\subseteq\mathbb{R}$ as, in this case, the curve $\varGamma$ will cross the infinite. Thus, in this case the curve $\varGamma$ will be constituted by to disconnected branches in the complex plane and the ordination described above should be understood as beginning in the point $\lambda=W(l)$, passing through the infinity and ending in point $\rho=W(r)$ along the curve $\varGamma$.

A very interesting case occurs when some of the polynomials belonging to the sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ of orthogonal polynomials on the real line have a zero at the point $x=a/c$. Indeed, according to Theorem 1, if $P_{n}(a/c)=0$ for some polynomials $P_{n}(x)\in P$ of degree $n$, then the corresponding Möbius-transformed polynomials $Q_{n}(x)\in Q$ will have degree $n^{\prime}=n-1$. Thus, $Q$ will be in this case a defective sequence of orthogonal Möbius-transformed polynomials, that is, a sequence containing polynomials of repeated degrees whereas other polynomials of specific degrees are absent. This particularity resembles what is called a sequence of exceptional orthogonal polynomials [6, 7], which is a sequence of orthogonal polynomials on the real line that does not contain some polynomials of specific degrees. A sequence $Q$ of defective orthogonal polynomials, however, differ from the sequences of exceptional orthogonal polynomials in two points: firstly they usually are orthogonal in curves of the complex plane (not necessarily on the real line); secondly, even though the sequence $Q$ contain polynomials of repeated degrees, the orthogonality condition (17) is always satisfied, even for the defective polynomials222Here we remark that the parameters $m$ and $n$ on the varying weight function $\omega_{m,n}(x)$ do not rely on the degrees of the polynomials $Q_{m}(x)$ and $Q_{n}(x)$, but only to the order in which they appear in the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed polynomials..

As an example of defective orthogonal polynomials arising from a Möbius transformation, we can consider any sequence $S$ of orthogonal polynomials defined in a symmetrical interval of the real line under the action of the inversion map $M(x)=1/x$, so that we have $a/c=0$ in this case. In fact, we know that every polynomial $P_{n}(x)\in S$ of odd degree (say, $n=2m+1$) will have a zero at $x=0$, thence the corresponding Möbius-transformed polynomials $Q_{2m+1}(x)\in Q$ will have actually an even degree (say, $n^{\prime}=2m$), i.e. they will be defective. The sequence $Q$ of Möbius-transformed orthogonal polynomials will be, therefore, constituted by polynomials of even degrees only, and in such a way that two any consecutive polynomials belonging to this sequence will have the same degree. In Table 1 we present the first orthogonal Möbius-transformed polynomials for the Chebyshev polynomials under the inversion map.

4 Classical Möbius-transformed orthogonal polynomials

Among all the sequences of orthogonal polynomials on the real line, the so-called classical orthogonal polynomial sequences — which include the sequences of Jacobi, Laguerre and Hermite polynomials — are of particular interest. In fact, the classical families of orthogonal polynomials were the first ones to be systematically studied, which is justified by their great importance in applied mathematics, physics and engineering. Besides, the classical orthogonal polynomials have many properties that are not satisfied by any other sequences of orthogonal polynomials: for example, they satisfy second-order differential equations, their weight function also satisfy a linear differential equation, there exist Rodrigues’ type relations and generating functions from which the classical orthogonal polynomials can be derived and so on [18, 19]. In this section, we shall show that the Möbius-transformed rational functions and polynomials that arise from a classical orthogonal polynomial sequence on the real line enjoy similar properties.

4.1 Second-order differential equation

It is a well-known result that any sequence $P=\{P_{n}(x)\}_{n=0}^{\infty}$ of classical orthogonal polynomial satisfy a second-order differential equation of the form [18, 19]:

[TABLE]

Routh [21], Bochner [22] and Brenke [23], have shown, independently, that polynomial solutions of (42) exist only when $f(x)$, $g(x)$ and $h_{n}(x)$ are polynomials of degree $2$, $1$ and [math], respectively, where $h_{n}(x)$ can be different for each $n$. Besides, they showed that the only polynomial solutions of (42), up to a linear transformation, are the classical orthogonal polynomial sequences of Jacobi, Laguerre and Hermite, besides other few sequences of polynomials which nevertheless are not orthogonal on the real line in the usual sense333The first of such non-orthogonal sequences is constituted by polynomials of the form $\pi_{n}(x)=x^{n}+\alpha x^{m}$, where $\alpha$ is a real parameter and $m$ any integer satisfying $0\leqslant m\leqslant n-1$. The second sequence is related to the so-called Bessel polynomials, which were consistently studied for the first time in [24]. Finally, the third sequence consists of the Romanovski polynomials [25]. .

In the following, we shall derive the differential equations that are satisfied by the sequences of the Möbius-transformed rational functions and polynomials when the original sequence of orthogonal polynomials is of the classical type.

Theorem 8**.**

Let $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ be orthogonal polynomial sequence on the real line in which each polynomial $P_{n}(x)$ for $n\geqslant 0$ satisfy a second order differential equation of the form (42). Then, each Möbius-transformed rational function $R_{n}(x)$, as defined by (12), belonging to the sequence $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ will satisfy a second-order differential equation of the form,

[TABLE]

where, $F(y)$ and $G(y)$ are polynomials independent of $n$ whose degree are at most $4$ and $3$, respectively, and $H_{n}$ is a constant which may depend on $n$. Similarly, each Möbius-transformed polynomial $Q_{n}(x)$, as defined by (13), belonging to the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ will satisfy a second-order differential equation of the form,

[TABLE]

where, now, $\mathcal{F}(x)$ has degree at most $4$ and does not depend on n, while $\mathcal{G}_{n}(x)$ and $\mathcal{H}_{n}(x)$ are polynomials with degree at most $3$ and $2$, respectively, that in general depend on $n$.

Proof.

Inserting relation (14) on the differential equation (42), we get,

[TABLE]

Computing the derivatives with the help of chain’s rule, we get,

[TABLE]

Now, the change of variables $x=M(y)$ provides,

[TABLE]

But,

[TABLE]

so that we get, after gathering the derivatives of $R_{n}(y)$,

[TABLE]

which is of the same form of (43), provided we define,

[TABLE]

Now, because $f(x)$ and $g(x)$ are polynomials with degree at most $2$ and $1$ respectively, it is convenient to introduce the polynomials

[TABLE]

which are, as a matter of a fact, the Möbius-transformed polynomials of $f(x)$ and $g(x)$, respectively. Notice that $\varPhi(x)$ and $\varGamma(x)$ have the same degree as $f(x)$ and $g(x)$, from which we conclude that $F(y)$ and $G(y)$ are polynomials with degree at most $4$ and $3$, respectively.

Let us now prove (44). For this we can proceed as follows: by inserting (13) into (43) we get,

[TABLE]

Computing the derivatives and simplifying, this becomes,

[TABLE]

that is, gathering the derivatives of $Q_{n}(x)$,

[TABLE]

which is of the same form than (44), provided we define,

[TABLE]

Notice that in terms of the Möbius-transformed polynomials $\varPhi(x)$ and $\varGamma(x)$ given by (51), we can rewritten (55), (56) and (57) as,

[TABLE]

which proves that $\mathcal{F}(x)$, $\mathcal{G}_{n}(x)$ and $\mathcal{H}_{n}(x)$ are polynomials with degree at most $4$, $3$ and $2$, respectively. This concludes the proofs. ∎

At first glance, Theorem 8 seems to contradicts Routh-Bochner-Brenke’s Theorem because we found polynomial solutions of a second-order differential equation whose coefficients are polynomials of degree greater than $2$. However, this contradiction is only apparent because of the following reason: while Routh-Bochner-Brenke’s theorem concerns with the solutions of the second-order differential equation (42), where only constant term $h_{n}$ is supposed to be dependent of $n$, both the coefficients $G_{n}(x)$ and $H_{n}(x)$ in the differential equation (44) can depend on $n$, so that (42) and (44) are actually of different forms. Furthermore, it is clear that no restriction whatsoever can be imposed on the degrees of the polynomial coefficients of a second-order differential equation when all these coefficients are allowed to depend on $n$.

4.2 Weight function differential equations

The weight function $w(x)$ of any classical orthogonal polynomial also satisfy a differential equation. In this case, however, we have the following first-order homogeneous differential equation [18, 19]:

[TABLE]

where $f(x)$ and $g(x)$ are the same polynomials appearing in (42). In fact, (61) is obtained as a condition to the differential equation (42) be written in the self-adjoint form.

[TABLE]

It is well-known that any second-order differential equation can be written in the self-adjoint form by multiplying it by some integrating factor $w(x)$. In fact, multiplying (42) by $w(x)$ and comparing with (62) we conclude that $p(x)=w(x)f(x)$, $p^{\prime}(x)=w(x)g(x)$ and $q_{n}(x)=w(x)h(x)$, so that the relation $w(x)g(x)=\left[w(x)f(x)\right]^{\prime}$ must hold, from which (61) immediately follows. Besides, whenever we have

[TABLE]

Sturm-Liouville theory will ensure that the sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ of polynomials (which we suppose to be the solutions of a well-behaved Sturm-Liouville problem), will indeed be an orthogonal polynomial sequence in the interval $I=\left(a,b\right)$ of the real line, with respect to the weight function $w(x)$.

In this section, we shall show that weight functions of the Möbius-transformed rational functions and polynomials also satisfy similar first-order differential equations.

Theorem 9**.**

Let $w(x)$ be the weight function of a classical orthogonal polynomial sequence $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$, so that it satisfies the homogeneous first-order differential equation (61). Then, the corresponding weight function $\omega(x)$, given by (16), of the sequence $R=\left\{R_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed rational functions, as defined in (12), will satisfy the following first-order differential equation:

[TABLE]

where $F(x)$ and $G(x)$ are the same coefficients that appear in the differential equation (43). Similarly, the degree-dependent weight function $\omega_{m,n}(x)$ given in (17), that is associated with the sequence $Q=\left\{Q_{n}(x)\right\}_{n=0}^{\infty}$ of Möbius-transformed polynomials, as defined in (13), will satisfy the following first-order differential equation:

[TABLE]

where $\mathcal{F}(x)$, $\mathcal{G}_{m}(x)$ and $\mathcal{G}_{n}(x)$ are the same polynomials defined in (55) and (56), respectively.

Proof.

Inverting relation (16), we can express $w(x)$ in terms of $\omega(W(x))$:

[TABLE]

Thereby, taking the derivative of $w(x)$, we obtain,

[TABLE]

Inserting (66) and (67) into (61) we shall get, after simplification,

[TABLE]

Now, making the change of variables $x=M(y)$, we shall find,

[TABLE]

so that,

[TABLE]

Now, using the relations

[TABLE]

we can verify, taking (50) into account, that

[TABLE]

which establishes (64). Finally, to prove (65) we can proceed as follows: from (17), we know that $\omega(x)=\omega_{m,n}(x)(cy+d)^{n+n}$. Thus, inserting this expression into (64) and computing the derivative, we obtain,

[TABLE]

Now, from (55) we get that $F(x)=\mathcal{F}(x)$ and, hence, $F^{\prime}(x)=\mathcal{F}^{\prime}(x)$ as well; besides, from (56) we can see that the term $c\left(m+n\right)/(cx+d)$ is exactly canceled by expressing $G(x)$ in terms of $\frac{1}{2}\left[\mathcal{G}_{m}(x)+\mathcal{G}_{n}(x)\right]$, from which (65) is obtained. This conclude the proof. ∎

4.3 Rodrigues’ type formulas

Another interesting property of the classical orthogonal polynomial sequences is that they can be defined through the so-called Rodrigues’ formulas [18, 19]. In fact, if $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ is such a sequence, then the polynomials $P_{n}(x)\in P$ can be found by taking derivatives:

[TABLE]

where $\epsilon_{n}$ are some standardizing constants, $w(x)$ the corresponding weight function and $f(x)$ the same polynomial appearing in the differential equation (42).

In this section, we shall show that the Möbius-transformed rational functions and polynomials also can be found by taking derivatives through a Rodrigues’-type formula.

Theorem 10**.**

Let $P=\left\{P_{n}(x)\right\}_{n=0}^{\infty}$ be a sequence of orthogonal polynomials on the real line satisfying a Rodrigues’ type formula as given in (73). Then, the rational functions $R_{n}(x)$, as defined in (12), and the Möbius-transformed polynomials $Q_{n}(x)$, as defined in (13), will satisfy, respectively, the following Rodrigues’ type formulas:

[TABLE]

and

[TABLE]

where $F(x)=\mathcal{F}(x)$ is the same function defined in (50), $\omega(x)$ and $\omega_{n,n}(x)$ are the new weight functions defined in (16) and (17), respectively, and we introduced the differential operator,

[TABLE]

Proof.

Formulas (74) and (75) follow in a straightforward way by making the change of variable $x=M(y)$ into (73) and using (12), (13), (16), (17) and (50). ∎

We remark that formulas (74) and (75) can be rewritten as well in the following equivalent form:

[TABLE]

They can also be rewritten in many other ways — for example, we can expand the derivatives by invoking Faà-di-Bruno formula for the $n$th derivative of a composed function, Leibniz formula for the derivative of a product, etc —, however, we shall not go through these lines because the resulting expressions are very cumbersome.

4.4 Generating functions

As a last property, let us discuss the existence of generating functions for the Möbius-transformed rational functions and polynomials. As it is well-known, the sequences of classical orthogonal polynomials can be generated by generating function of the following forms [18, 19]:

[TABLE]

Similar formulas hold for the Möbius-transformed rational functions and polynomials:

Theorem 11**.**

Let $P=\left\{P_{k}(x)\right\}_{k=0}^{\infty}$ be a classical orthogonal polynomial sequence whose generating function is one of those given in (79). Then, the corresponding generating functions of the sequence $R=\left\{R_{k}(x)\right\}_{k=0}^{\infty}$ of Möbius-transformed rational functions $R_{n}(x)$ defined in (12) are the following:

[TABLE]

respectively. Similarly, the corresponding generating functions of the sequence $Q=\left\{Q_{k}(x)\right\}_{k=0}^{\infty}$ of Möbius-transformed polynomials $Q_{n}(x)$ defined in (13) are:

[TABLE]

respectively.

Proof.

We have at once that,

[TABLE]

and

[TABLE]

In the same manner, we have that,

[TABLE]

and

[TABLE]

∎

5 Applications

We close this paper with some applications of the theory developed above. More specifically, we show that the sequences of Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are all related with each other by a Möbius transformation. We also show that generalized Bessel polynomials enjoy a finite orthogonality on the real line, as well as the Romanovski polynomials satisfy an orthogonality relation on the imaginary axis.

5.1 From Laguerre to Bessel polynomials

The associated Laguerre polynomials $L_{n}^{\alpha}(x)$ are polynomials orthogonal on the interval $I=\left(0,\infty\right)$ of the real line with respect to the weight function $w(x)=x^{\alpha}\mathrm{e}^{-x}$, for any $\alpha>-1$ [18, 19]. Performing the re-scaling transformation $M(x)=\beta x$, we can easily generate the so-called generalized Legendre polynomials $L_{n}^{\left(\alpha,\beta\right)}(x)=L_{n}^{\alpha}(\beta x)$, which are orthogonal on the same interval $I=\left(0,\infty\right)$ of the real line, for $\alpha>-1$ and $\beta>0$, but with respect the weight function $w(x)=x^{\alpha}\mathrm{e}^{-\beta x}$. Now let us consider the generalized Laguerre polynomials under the inversion map $M(x)=1/x$. The resulting transformed polynomials will have, therefore, the following form:

[TABLE]

From the results of Section 3, it follows that these polynomials are also orthogonal on the interval $I=\left(0,\infty\right)$ but with respect to a varying measure, provided that $\alpha>-1$ and $\beta>0$. Namely, we have the following:

[TABLE]

In the following, we shall show that the polynomials $\mathcal{L}_{n}^{\left(\alpha,\beta\right)}(x)$ are related with the so-called Bessel polynomials, which were studied extensively in [24]. To this end, notice that, according to the results of Section 4.1, the polynomials $\mathcal{L}_{n}^{\left(\alpha,\beta\right)}(x)=y(x)$ will satisfy the differential equation:

[TABLE]

Comparing with the differential equation given in for [24] the generalized Bessel polynomials $\mathcal{B}_{n}^{\left(\gamma,\beta\right)}$, namely,

[TABLE]

we see that a complete agreement between (87) and (88) is achieved when we set $\alpha=1-2n-\gamma$. Thus, the generalized Bessel polynomials are given by the formula

[TABLE]

up to a normalization. Thereby, each polynomial $\mathcal{B}_{n}^{\left(\gamma,\beta\right)}(x)$ is associated with a generalized Laguerre polynomial with a different value of $\alpha$. From the discussion given in Section 3.1, we can conclude that the polynomials $\mathcal{B}_{n}^{\left(\gamma,\beta\right)}(x)$ will be satisfy the orthogonality relation

[TABLE]

provided that $\gamma<2-2\max(m,n)$ and $\beta>0$. From this we can see that the polynomials $\mathcal{B}_{m}^{\left(\gamma,\beta\right)}(x)$ can at most enjoy a finite orthogonality on the interval $I=\left(0,\infty\right)$ of the real line444Notice as well that, for $\gamma$ a negative integer, the following identity holds: $\mathcal{B}_{n}^{\left(\gamma,\beta\right)}(x)=\mathcal{B}_{1-\gamma-n}^{\left(\gamma,\beta\right)}(x)$. Thus, in these cases the sequence $B=\{\mathcal{B}_{n}^{\left(\gamma,\beta\right)}(x)\}_{n=0}^{\infty}$ of generalized Bessel polynomials will be a defective sequence of orthogonal polynomials..

Following [24], the usual Bessel polynomials are obtained by setting $\gamma=2$ and $\beta=2$, from which we get that $B_{n}(x)=\mathcal{L}_{n}^{\left(-1-2n,2\right)}(x)$, up to a normalization, while the weight function becomes $\Omega_{m,n}(x)=x^{|m-n|-1}\mathrm{e}^{-2/x}$. In this case, however, the condition for the validity of the orthogonality relation (90) would hold only for $n<0$, which is not possible. Thus, we conclude that the Bessel polynomials are not orthogonal on the real line. The first generalized and usual Bessel polynomials are presented in Table 2.

5.2 From Jacobi to Romanovski polynomials

*Romanovski polynomials *consist of an infinite sequence of real polynomials studied by Romanovski in [25], although they have appeared before in Routh’s work [21]. Romanovski polynomials also appear in physical problems, which justified a more detailed study of them — see [26, 27] and references therein. In what follows we shall show that Romanovski polynomials are also related with Jacobi polynomials $J_{n}^{\left(\alpha,\beta\right)}(x)$ via a Möbius transformation. In this case, however, we should consider the mapping $M(x)=ix$ with the following values for the Möbius parameters: $a=1$, $b=0$, $c=0$ and $d=-i$.

Remember that Jacobi polynomials $J_{n}^{\left(\alpha,\beta\right)}(x)$ are orthogonal on the interval $I=(-1,1)$ of the real line with respect to the weight function $w(x)=\left(1-x\right)^{\alpha}\left(1+x\right)^{\beta}$ for $\alpha>-1$ and $\beta>-1$ [18, 19]. Thus, according with the results of Section 3, the Möbius-transformed polynomials

[TABLE]

will be orthogonal on the interval $J=\left(-i,i\right)$ along the imaginary line, provided that $\alpha>-1$ and $\beta>-1$. The polynomials $\mathcal{J}_{n}^{\left(\alpha,\beta\right)}(x)$, however, do not have real coefficients for real $\alpha$ and $\beta$; nonetheless, a real polynomial can be obtained if we allow $\alpha$ and $\beta$ to assume non-real values. The precise condition for this is that $\alpha$ be the complex conjugate of $\beta$, so that we can write $\alpha=\gamma+i\delta$, $\beta=\gamma-i\delta$ with $\gamma,\delta\in\mathbb{R}$ and $\delta\neq 0$. In fact, up to a normalization, this gives place to the Romanovski polynomials555We remark that some authors adopt different values for the parameters $\gamma$ and $\delta$ in the definition of Romanovski polynomials.:

[TABLE]

The first Romanovski polynomials are presented in Table 3.

Now, in order to investigate the orthogonality of the polynomials $\mathcal{R}_{n}^{\left(\gamma,\delta\right)}(x)$ along the imaginary axis we need to verify under what conditions the Jacobi polynomials with complex conjugate parameters are orthogonal on the real line. It is not difficult to verify that Jacobi polynomials $J_{n}^{\left(\gamma+i\delta,\gamma-i\delta\right)}(x)$ are orthogonal on the interval $I=(-1,1)$ of the real line provided that $\gamma>-1$ and $\delta\in\mathbb{R}$. In fact, this follows because the limit relations (63) are satisfied for any non-negative integer $n$ whenever $\gamma>-1$ and $\delta$ any real number. Besides, we can verify that all the moments $\mu(n)=\int_{-1}^{-1}x^{n}\left(1-x\right)^{\gamma+i\delta}\left(1+x\right)^{\gamma-i\delta}\mathrm{d}x$ exist under these conditions as well, which is enough to guarantee the convergence of the integrals (11) for any $m,n\in\mathbb{N}$, for the same range of $\gamma$ and $\delta$.

Consequently, we conclude that Romanovski polynomials $\mathcal{R}_{n}^{\left(\gamma,\delta\right)}(x)$ will be orthogonal on the interval $J=(-i,i)$ of the imaginary axis whenever $\gamma>-1$ and $\delta$ is a real number666Notice that the zeros of Romanovski polynomials $\mathcal{R}_{n}^{\left(\gamma,\delta\right)}(x)$ usually do not lie on the imaginary axis, which can be explained by the fact that the Jacobi polynomials $J_{n}^{\left(\gamma+i\delta,\gamma-i\delta\right)}(x)$ are not real polynomials for $\gamma$ and $\delta$ real, so that their zeros usually do not lie on the real line either. . The corresponding weight function can be found either directly from (16) and (17) or by solving the differential equation (65). We have that, up to a multiplicative constant,

[TABLE]

Thus, the orthogonality relation reads:

[TABLE]

which holds for $\gamma>-1$ and real $\delta$.

Finally, it should be mentioned that the limit relations (63), if applied to this Möbius-transformed weight function $\omega(x)$, provide as well a finite orthogonality for the Romanovski polynomials on the real line. In fact, in this case we can verify that only those polynomials $\mathcal{R}_{m}^{\left(\gamma,\delta\right)}(x)$ and $\mathcal{R}_{n}^{\left(\gamma,\delta\right)}(x)$ satisfying the relation $m+n+2\gamma<-1$ will form an orthogonal polynomial sequence on the interval $J=(-\infty,\infty)$ of the real line.

5.3 From Jacobi to Laguerre polynomials

The Möbius transformation is rich enough to map any finite point to the infinite — in fact, on the extended complex plane $\mathbb{C}_{\infty}$ there is no special distinction between the infinite and any other point. Thereby, we can transform a given finite interval of the real line to a semi-infinite one. Following this idea, we shall show in the sequel that Laguerre polynomials can be obtained from Jacobi polynomials through a Möbius transformation followed by an adequate limit of their parameters.

The associated Laguerre polynomials are orthogonal polynomials on the interval $I=(0,\infty)$ of the real line with respect to the weight function $w(x)=x^{\alpha}\mathrm{e}^{-x}$ [18, 19]. Thus, let us define the the Möbius transformed polynomials $\mathcal{J}_{n}^{(\alpha,\beta)}(x)=\left(cx+d\right)^{n}J_{n}^{(\alpha,\beta)}\left(M(x)\right)$ and look for the transformation $W(x)$ that maps the points $l=-1$ and $r=1$ respectively to $\lambda=0$ and $\rho=\infty$. From (2), we plainly see that we should have $c=a$ and $d=-b$ to this end. With this, we have that the first Möbius-transformed polynomial becomes:

[TABLE]

Comparing this expression with the first Laguerre polynomial, $L_{1}^{\beta}(x)=\beta+1-x$, we see that we must have $a=-1/\left(\alpha+1\right)$ and $b=1$. Then we can verify that the other Möbius transformed polynomials $\mathcal{J}_{n}^{(\alpha,\beta)}(x)$ will match with the Laguerre polynomials $L_{n}^{\beta}(x)$ after we take the limit $\alpha\rightarrow\infty$, that is,

[TABLE]

Using the identity $J_{n}^{(\beta,\alpha)}(x)=\left(-1\right)^{n}J_{n}^{(\alpha,\beta)}(-x)$ we can also write:

[TABLE]

These expressions can be compared with Szegő’s identity [18]: $L_{n}^{\alpha}(x)=\lim_{\beta\rightarrow\infty}J_{n}^{(\alpha,\beta)}\left(1-2\beta^{-1}x\right)$ .

Finally, let us show that Laguerre’s weight function can also be obtained from Jacobi’s weight function through the same procedure. Notice that $\varOmega_{m,n}(x)\rightarrow\omega(x)$ as $\alpha\rightarrow\infty$ because $c\rightarrow 0$ in this limit. Thus, is enough to prove that $w(x)=\left(1-x\right)^{\alpha}\left(1+x\right)^{\beta}$ reduces to $\omega(x)=x^{\beta}\mathrm{e}^{-x}$ in the limit $\alpha\rightarrow\infty$ after we perform the Möbius transformation above. From (64) we can verify that the transformed weight function is,

[TABLE]

where $\kappa(\alpha,\beta)$ is the constant of integration. From this we can see that Laguerre’s weight function is obtained after we set $\kappa(\alpha,\beta)=\alpha^{\beta+1}(\alpha+1)^{\alpha+1}$ and take the limit $\alpha\rightarrow\infty$. All the other properties of Laguerre polynomials can be obtained from those of Jacobi by similar arguments.

5.4 From Jacobi to Hermite polynomials

Finally, we can show that Hermite polynomials can also be obtained from Jacobi polynomials through a Möbius transformation followed by a specific limit. At first sight, we would be tempted to map the finite interval $I=(-1,1)$ into the infinite interval $J=(-\infty,\infty)$. This, however, it is not possible to be done with a Möbius transformation because the points $x=\pm\infty$ are regarded as equals in the extended complex plane $\mathbb{C}_{\infty}$ and the Möbius transformation is one-to-one and onto in $\mathbb{C}_{\infty}$. Thus, to work this around, we shall proceed in a somewhat different way: the idea is to first find the Möbius transformation $W(x)$ that maps the interval $I=(-1,1)$ into the interval $J=(-\xi,\xi)$ for some $\xi>0$ and, then, to take the limit $\xi\rightarrow\infty$. Imposing further that the point [math] is mapped to [math], we find that the required Möbius transformation becomes $W(x)=a\xi$, whose inverse is $M(x)=a/\xi$ ($a$ is still arbitrary).

Now, according (16), (17) or (65), we get that Jacobi’s weight function $w(x)=\left(1-x\right)^{\alpha}\left(1+x\right)^{\beta}$ will be transformed into

[TABLE]

where $\kappa(\alpha,\beta)$ is the constant of integration. From this we can plainly see that Hermite’s weight function $\omega(x)=\mathrm{e}^{-x^{2}}$ is achieved by making $\beta=\alpha$, $\kappa(\alpha,\beta)=\xi=\sqrt{\alpha}$ and taking limit $\alpha\rightarrow\infty$ (so that we get $\xi\rightarrow\infty$ as well). With these identifications, the Möbius-transformed polynomials become $\mathcal{J}_{n}^{(\alpha)}(x)=\left(a^{n}\sqrt{\alpha}\right)^{n}J_{n}^{(\alpha,\alpha)}\left(x/\sqrt{\alpha}\right)$. Now, the limits of $\mathcal{J}_{n}^{(\alpha)}(x)$ for $\alpha\rightarrow\infty$ only exist if $a$ is inversely proportional to $\alpha$. In fact, if we set $a=2/\alpha$ and take the limit $\alpha\rightarrow\infty$ then we can verify that we get a complete correspondence between the Möbius-transformed polynomials $\mathcal{J}_{n}^{(\alpha)}(x)$ and Hermite polynomials $H_{n}(x)$:

[TABLE]

Acknowledgments

We kindly thank Professors C. F. Bracciali and A. Sri Ranga for the discussions we had about these ideas and also for their valuable comments. The work of RSV was supported by grants from Coordination for the Improvement of Higher Education Personnel (CAPES). The work of VB was supported by funds from São Paulo Research Foundation (FAPESP), grant #2016/02700-8.