# Some New Connection Relations Related to Classical Orthogonal Polynomials

Authors: Wathek Chammam; Wasim Ul-Haq
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we deal with a problem of positivity of linear functionals in the linear spaceℙ of polynomials in one variable with complex coefficients. Some new results of connection relations between the corresponding sequences of monic orthogonal polynomials of classical character are established.


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## Body

## 1. Introduction

The criterion of positivity is important in special functions theory, such as approximation theory, moment problem, and orthogonal polynomials and applications, among others, because the solutions of many problems depend on the determination of when a specific function is positive or nonnegative. For example, several authors have focused on the study of linear positive operators and their applications in problems and inequalities. Here, we can mention Aloui [1, 2], Bennett [3], and others. Also, we quote the work of Chen et al. [4] in the total positivity of Riordan arrays.The linear functionalU is positive definite if and only if ΔnU>0, for every integer n≥0, where ΔnU is the Hankel determinants of order n of U (see [5]). In order to provide a construction process of positive-definite linear functional from positive-definite linear data, Sfaxi has proved a new construction process and he also gave an example a second-order positive-definite linear functional from a Laguerre positive-definite linear functional [6].The outline of this paper is as follows. In Section2, we introduce the basic background and notations to be used throughout the paper. In Section 3, we present our main results. Indeed, using Oppenheim’s inequality we prove that, for any pair μ,c∈]0,+∞[×ℝ and any positive-definite linear functional U, the linear functional Uμ,c given by μ−1Uμ,c−x−cUμ,c′=μU is also positive definite. Finally, some new results of connection relations between the corresponding sequences of monic classical orthogonal polynomials are presented.


## 2. Orthogonality and Positive-Definite Linear Functional Character

Letℙ be the linear space of polynomials in one variable with complex coefficients and ℙ′ its algebraic dual space. We denote by U,p the action of U∈ℙ′ on p∈ℙ and by Un:=U,xn, n≥0, the sequence of moments of U with respect to the polynomial sequence xnn≥0. Let us define the following operations in ℙ′. For any linear functionals U and V, any polynomial q, and any a,b,c∈ℂ∗×ℂ2, let DU=U′,qU, x−c−1U, τ−bU, haU, and UV be the linear functionals defined by duality (see [7, 8]):(1)U′,p:=−U,p′,qU,p:=U,qp,x−c−1U,p:=U,θcp=U,px−pcx−c,τ−bU,p:=U,τbp=U,px−b,haU,p:=U,hap=U,pax,UV,p:=U,Vp,p∈ℙ,where the right-multiplication of V by p is a polynomial given by(2)Vpx:=Vy,xpx−ypyx−y,p∈ℙ.For anyc∈ℂ, x−c−1x−cU=U−U0δc, where δc is the Dirac functional defined by δc,p=pc,p∈ℙ.Notice that ifU∈ℙ′ is such that U′=0, then U=0.A linear functionalU is said to be quasi-definite (regular) if we can associate with it a monic polynomial sequence (MPS) Bnn≥0 with degBn=n,n≥0, such that(3)U,BnBm=rnδn,m,m,n≥0,and wherern≠0,n≥0,where δn,m is the Kronecker delta.In such a situation,Bnn≥0 is said to be the monic orthogonal polynomial sequence (MOPS) with respect to U. The MOPS Bnn≥0 satisfies a three-term recurrence relation (TTRR):(4)B−1x=0,B0x=1,Bn+1x=x−βnBnx−γnBn−1x,n≥0,whereβn∈ℂ and γn∈ℂ∗ for every integer n≥0 and with γ0=U0. Furthermore, we need the following formula:(5)βn=U,xBn2U,Bn2,n≥0,γn+1=U,Bn+12U,Bn2,n≥0,U,Bn2=∏ν=0nγν,n≥0,γ0=U0.WhenU is quasi-definite and A∈ℙ such that AU=0, then A=0.A linear functionalU is said to be positive-definite, if it satisfies (3) with rn>0, for every integer n≥0.The quasi-definiteness of a linear functionalU is equivalent to (see [9, 10])(6)ΔnU:=U0U1U2…UnU1U2U3…Un+1U2U3U4…Un+2⋮⋮⋮⋮UnUn+1Un+2…U2n≠0,∀n∈ℕ0,where ΔnU is the Hankel determinant of order n+1 of U.In this way, we have the following formula [5]:(7)U,Bn2=ΔnUΔn−1U,n≥0,Δ−1U=1.As a consequence,U is positive-definite if and only if ΔnU>0, for every integer n≥0.Lemma 1.
(see [8]). Let Bnn≥0 be a MOPS with respect to the linear functional U and let A be a monic polynomial with degA=t. The following statements are equivalent:(i)
The linear functionalAU is quasi-definite.(ii)
There exists a MPSQnn≥0 and parameters λn,ν satisfying(8)AQnx=∑ν=nn+tλn,νBνx,n≥0,λn,n≠0,n≥0.In this case,Qnn≥0 is the MOPS with respect to AU.The Hadamard product of twon×n matrices A=aij and bij is defined by A∘B≡aijbij. The Hadamard product arises in a wide variety of ways. It was perhaps the first significant result published about the Hadamard product that the class of positive semidefinite matrices of a given size is closed under the Hadamard product. For more information, the authors may refer to [11]. Oppenheim in [12] proved the following:Lemma 2.
(see [12]). Let A=aij and B=aij be two n×n positive semidefinite matrices. Then,(9)detA∘B≥detA∏i=1nbii.


## 3. Main Results

For anyμ,c∈ℂ2 with μ≠−n,n≥0 and U∈ℙ′, let us consider the linear functional Uμ,c where its moments with respect to the MPS x−cnn≥0 that we denote by Uμ,cn,c,n≥0, are given by(10)Uμ,cn,c=μn+μUn,c,n≥0.Equivalently,Uμ,c satisfies the distributional equation:(11)μ−1Uμ,c−x−cUμ,c′=μU.


### 3.1. Preservation of the Positive-Definite Property

Assuming thatU is positive definite, our purpose is to find the necessary and sufficient condition on the real parameter μ, in order that the linear functional Uμ,c to be also positive definite.Theorem 1.
Letμ be a real number with μ≠−n,n≥0, let c∈ℝ, and let U be a positive-definite linear functional. The following statements are equivalent:(i)
μ∈]0,+∞[.(ii)
the linear functionalUμ,c satisfying(12)μ−1Uμ,c−x−cUμ,c′=μUis positive definite.Proof.
Taking (10) into account, we can always decompose the determinant of Hankel associate to linear functional Uμ,c thanks to Hadamard’s product as follows:(13)ΔnUμ,c:=Uμ,c0,cUμ,c1,cUμ,c2,c…Uμ,cn,cUμ,c1,cUμ,c2,cUμ,c3,c…Uμ,cn+1,cUμ,c2,cUμ,c3,cUμ,c4,c…Uμ,cn+2,c⋮⋮⋮⋮Uμ,cn,cUμ,cn+1,cUμ,cn+2,c…Uμ,c2n,c=U0,cU1,cU2,c…Un,cU1,cU2,cU3,c…Un+1,cU2,cU3,cU4,c…Un+2,c⋮⋮⋮⋮Un,cUn+1,cUn+2,c…U2n,c°1μ1+μμ2+μ…μn+μμ1+μμ2+μμ3+μ…μn+μ+1μ2+μμ3+μμ4+μ…μn+μ+2⋮⋮⋮⋮μn+μμn+μ+1μn+μ+2…μ2n+μ.
Using Lemma2 (Oppenheim’s inequality) and ΔnU>0, for every integer n≥0 (U be a positive-definite linear functional), we find ∏k=0nμ/2k+μ>0, if and only if μ>0. Hence, the required result is as follows.


### 3.2. Classical Case

A quasi-definite linear functionalW is said to be classical if it satisfies a functional equation (Pearson equation):(14)ϕW′+ψW=0,where ϕ and ψ are polynomials such that ϕ is monic, degϕ=t≤2, and degψ=1.The correspondingBnn≥0 is said to be classical (for more details, see [8, 13] and the literature therein). The classical character of a linear functional is invariant by shifting. Indeed, if W is a classical linear functional satisfying (10), then for any pair a,b∈ℂ2 with a≠0, the shifted linear functional W˜=ha−1∘τ−bW is also classical and satisfies ϕ˜W˜′+ψ˜W˜=0 with ϕ˜x=a−tϕax+b and ψ˜x=a1−tψax+b, where t=degϕ. We find in [8] a description of the classical linear functional (Hermite, Laguerre, Bessel, and Jacobi).Proposition 1.
IfW is a classical linear functional with degϕ=1, then(15)μ−1ψ+μWμ,c=W.Proof.
degϕ=1. So, we can write ϕx=x−c. Let μ∈ℂ with μ≠−n,n∈ℕ. From (14), it follows that(16)−x−cW′+μ−1W=ψ+μW.
Clearly, we haveψ+μW0,c=μ≠0. Accordingly, one has(17)μ−1ψ+μWμ,c=W.Now, let us study the particular situation:Namely,c=0, μ=α+1, ψx=x−α−1, and W=ℒα. The Laguerre linear functional with parameter α≠−n,n≥1, which is the unique linear functional satisfying (see [8])(18)xℒα′+x−α−1ℒα=0.Sinceα+1−1xℒα=ℒα+1 and αℒα−xℒα′=α+1ℒα+1, accordingly we obtain(19)ℒα+1α+1,0=ℒα.The MOPSL^nαxn≥0 of Laguerre satisfies the first and second structure relations [6, 8]:(20)xL^nα′x=nL^nα+1x+nn+αL^n−1αx,n≥0,(21)L^nαx=L^nα+1x+nL^n−1α+1x,n≥0.Corollary 1.
The polynomialsL^nαx and L^nα+1x satisfy the following connection relation:(22)xL^nα+1′x+α+1L^nα+1x=n+α+1L^nαx,n≥0.Proof.
Using (20) and (21) and by taking (19) into account,(23)xL^nα+1′x+α+1L^nα+1x=n+α+1L^nα+1x+nL^n−1α+1x,=n+α+1L^nαx,n≥0.Proposition 2.
IfW is a classical linear functional with degϕ=2, then(24)μ−1ρWμ,c=θcϕW.Proof.
degϕ=2 and s=degϕ−2=degψ−1. Let c be a zero of ϕ. From the Euclidean division of ψ by θcϕ, we always have(25)ψx=−μθcϕx+ρx,where ρ∈ℙ with degρ≤degψ−1. Clearly, μ is given by(26)μ=−ψs+10s+1!≠0.
Sincedegψ=degϕ−1, then μ is the leading coefficient of −ψ.
From (14) and (25), we can write(27)μ−1θcϕW−x−cθcϕW′=ρW.
Notice thatdegρ≥0, otherwise the linear functional θcϕW will satisfy μ−1θcϕW−x−cθcϕW′=0 and then its moments with respect to the MPS x−cnn≥0 are such that n+μθcϕWn,c=0,n≥0. So, there exists an integer n0≥0 such that θcϕWn,c=0,n≥n0, i.e., x−cn0θcϕW=0 which contradicts the quasi-definiteness of W.
Directly, we have(28)μ−1ρWμ,c=θcϕW.Now, let us study some particular situations.I1 (Bessel case): namely, c=0, μ=2α, ψx=−2αx+1, and W=ℬα. The Bessel quasi-definite linear functional with parameter α≠−n/2,n≥0, which is the unique monic linear functional satisfying(29)x2ℬα′−2αx+1ℬα=0.We can easily show that(30)−αxℬα=ℬα+1/2.It follows that(31)2α−1ℬα+1/2−xℬα+1/2′=2αℬα,so that(32)ℬα2α,0=ℬα+1/2.Denote byB^nαxn≥0 the MOPS with respect to the Bessel linear functional ℬα. The MOPS B^nαxn≥0 satisfies the first structure relation [8]:(33)x2B^n+1α′x=n+1x−1n+αB^n+1αx−δnB^nα,n≥0,where(34)δn=n+1n+2α−12n+2α−1n+α2,n≥0.Corollary 2.
The polynomialsB^n+1αx and B^nα+1/2x satisfy the following relation:(35)xB^n+1α′x−n+1B^n+1αx=−n+1n+αB^nα+1/2x,n≥0.Proof.
Using (33), we obtain for any integer n≥0(36)xQnx=B^n+1αx−n+2α−12n+2α−1n+αB^nαx,n≥0,where Qnx=−n+1/n+αxB^n+1α′x−n+1B^n+1αx,n≥0.
By Lemma1, with Ax=x, Bn=B^nα, and λn,n=−n+2α−1/2n+2α−1n+α,n≥0, λ0,0=−1/α and by taking (22) and (24) into account, the MPS Qnxn≥0 is orthogonal with respect to the linear functional ℬα+1/2. Thus, Qnx=B^nα+1/2x, n≥0. This leads to (35).
I2 (Jacobi case): namely, c=1, μ=α+β+2, ψx=−α+β+2x+α−β, and W=Jα,β. The Jacobi quasi-definite linear functional with parameters α and β where α,β≠−n and α+β≠−n−1,n≥1, which is the unique monic linear functional satisfying(37)x2−1Jα,β′+−α+β+2x+α−βJα,β=0.
Since(38)α+β+22α+1x+1Jα,β=Jα+1,β,it follows that(39)α+β+1Jα+1,β−x−1Jα+1,β′=α+β+2Jα,β,
So, we can deduce that(40)Jα,βα+β+2,0=Jα+1,β.
Denote byJ^nα,βxn≥0 the MOPS with respect to the Jacobi linear functional Jα,β. The MOPS J^nα,βxn≥0 satisfies the first structure relation for n≥0 [8]:(41)x2−1J^n+1α,β′x=n+1x+α−β2n+α+β+2J^n+1α,βx−2n+α+β+3γn+1J^nα,β,where(42)γn+1=4n+1n+α+β+1n+α+1n+β+12n+α+β+12n+α+β22n+α+β+3,n≥0.Corollary 3.
The polynomialsJ^n+1α,βx and J^nα+1,βx satisfy the following relation:(43)x−1J^n+1α,β′x−n+1J^n+1α,βx=εnJ^nα+1,βx,where for any integer n≥0,(44)εn=−2n+1n+β+12n+α+β+2,n≥0.Proof.
Using (41), we obtain for any integer n≥0,(45)x+1Qnx=J^n+1α,βx+2n+α+1n+α+β+12n+α+β+12n+α+β+2J^nα,βx,n≥0,where(46)Qnx=−2n+α+β+22n+1n+β+1x−1J^n+1α,β′x−n+1J^n+1α,βx,n≥0.
By Lemma1, with Ax=x+1, Bn=J^nα,β, λn,n=2n+α+1n+α+β+1/2n+α+β+12n+α+β+2≠0,n≥0, and by taking (38) and (40) into account, xthe MPS Qnxn≥0 is orthogonal with respect the linear functional Jα+1,β. Thus, Qnx=J^nα+1,βx, n≥0. This leads to (43).


## 3.1. Preservation of the Positive-Definite Property

Assuming thatU is positive definite, our purpose is to find the necessary and sufficient condition on the real parameter μ, in order that the linear functional Uμ,c to be also positive definite.Theorem 1.
Letμ be a real number with μ≠−n,n≥0, let c∈ℝ, and let U be a positive-definite linear functional. The following statements are equivalent:(i)
μ∈]0,+∞[.(ii)
the linear functionalUμ,c satisfying(12)μ−1Uμ,c−x−cUμ,c′=μUis positive definite.Proof.
Taking (10) into account, we can always decompose the determinant of Hankel associate to linear functional Uμ,c thanks to Hadamard’s product as follows:(13)ΔnUμ,c:=Uμ,c0,cUμ,c1,cUμ,c2,c…Uμ,cn,cUμ,c1,cUμ,c2,cUμ,c3,c…Uμ,cn+1,cUμ,c2,cUμ,c3,cUμ,c4,c…Uμ,cn+2,c⋮⋮⋮⋮Uμ,cn,cUμ,cn+1,cUμ,cn+2,c…Uμ,c2n,c=U0,cU1,cU2,c…Un,cU1,cU2,cU3,c…Un+1,cU2,cU3,cU4,c…Un+2,c⋮⋮⋮⋮Un,cUn+1,cUn+2,c…U2n,c°1μ1+μμ2+μ…μn+μμ1+μμ2+μμ3+μ…μn+μ+1μ2+μμ3+μμ4+μ…μn+μ+2⋮⋮⋮⋮μn+μμn+μ+1μn+μ+2…μ2n+μ.
Using Lemma2 (Oppenheim’s inequality) and ΔnU>0, for every integer n≥0 (U be a positive-definite linear functional), we find ∏k=0nμ/2k+μ>0, if and only if μ>0. Hence, the required result is as follows.


## 3.2. Classical Case

A quasi-definite linear functionalW is said to be classical if it satisfies a functional equation (Pearson equation):(14)ϕW′+ψW=0,where ϕ and ψ are polynomials such that ϕ is monic, degϕ=t≤2, and degψ=1.The correspondingBnn≥0 is said to be classical (for more details, see [8, 13] and the literature therein). The classical character of a linear functional is invariant by shifting. Indeed, if W is a classical linear functional satisfying (10), then for any pair a,b∈ℂ2 with a≠0, the shifted linear functional W˜=ha−1∘τ−bW is also classical and satisfies ϕ˜W˜′+ψ˜W˜=0 with ϕ˜x=a−tϕax+b and ψ˜x=a1−tψax+b, where t=degϕ. We find in [8] a description of the classical linear functional (Hermite, Laguerre, Bessel, and Jacobi).Proposition 1.
IfW is a classical linear functional with degϕ=1, then(15)μ−1ψ+μWμ,c=W.Proof.
degϕ=1. So, we can write ϕx=x−c. Let μ∈ℂ with μ≠−n,n∈ℕ. From (14), it follows that(16)−x−cW′+μ−1W=ψ+μW.
Clearly, we haveψ+μW0,c=μ≠0. Accordingly, one has(17)μ−1ψ+μWμ,c=W.Now, let us study the particular situation:Namely,c=0, μ=α+1, ψx=x−α−1, and W=ℒα. The Laguerre linear functional with parameter α≠−n,n≥1, which is the unique linear functional satisfying (see [8])(18)xℒα′+x−α−1ℒα=0.Sinceα+1−1xℒα=ℒα+1 and αℒα−xℒα′=α+1ℒα+1, accordingly we obtain(19)ℒα+1α+1,0=ℒα.The MOPSL^nαxn≥0 of Laguerre satisfies the first and second structure relations [6, 8]:(20)xL^nα′x=nL^nα+1x+nn+αL^n−1αx,n≥0,(21)L^nαx=L^nα+1x+nL^n−1α+1x,n≥0.Corollary 1.
The polynomialsL^nαx and L^nα+1x satisfy the following connection relation:(22)xL^nα+1′x+α+1L^nα+1x=n+α+1L^nαx,n≥0.Proof.
Using (20) and (21) and by taking (19) into account,(23)xL^nα+1′x+α+1L^nα+1x=n+α+1L^nα+1x+nL^n−1α+1x,=n+α+1L^nαx,n≥0.Proposition 2.
IfW is a classical linear functional with degϕ=2, then(24)μ−1ρWμ,c=θcϕW.Proof.
degϕ=2 and s=degϕ−2=degψ−1. Let c be a zero of ϕ. From the Euclidean division of ψ by θcϕ, we always have(25)ψx=−μθcϕx+ρx,where ρ∈ℙ with degρ≤degψ−1. Clearly, μ is given by(26)μ=−ψs+10s+1!≠0.
Sincedegψ=degϕ−1, then μ is the leading coefficient of −ψ.
From (14) and (25), we can write(27)μ−1θcϕW−x−cθcϕW′=ρW.
Notice thatdegρ≥0, otherwise the linear functional θcϕW will satisfy μ−1θcϕW−x−cθcϕW′=0 and then its moments with respect to the MPS x−cnn≥0 are such that n+μθcϕWn,c=0,n≥0. So, there exists an integer n0≥0 such that θcϕWn,c=0,n≥n0, i.e., x−cn0θcϕW=0 which contradicts the quasi-definiteness of W.
Directly, we have(28)μ−1ρWμ,c=θcϕW.Now, let us study some particular situations.I1 (Bessel case): namely, c=0, μ=2α, ψx=−2αx+1, and W=ℬα. The Bessel quasi-definite linear functional with parameter α≠−n/2,n≥0, which is the unique monic linear functional satisfying(29)x2ℬα′−2αx+1ℬα=0.We can easily show that(30)−αxℬα=ℬα+1/2.It follows that(31)2α−1ℬα+1/2−xℬα+1/2′=2αℬα,so that(32)ℬα2α,0=ℬα+1/2.Denote byB^nαxn≥0 the MOPS with respect to the Bessel linear functional ℬα. The MOPS B^nαxn≥0 satisfies the first structure relation [8]:(33)x2B^n+1α′x=n+1x−1n+αB^n+1αx−δnB^nα,n≥0,where(34)δn=n+1n+2α−12n+2α−1n+α2,n≥0.Corollary 2.
The polynomialsB^n+1αx and B^nα+1/2x satisfy the following relation:(35)xB^n+1α′x−n+1B^n+1αx=−n+1n+αB^nα+1/2x,n≥0.Proof.
Using (33), we obtain for any integer n≥0(36)xQnx=B^n+1αx−n+2α−12n+2α−1n+αB^nαx,n≥0,where Qnx=−n+1/n+αxB^n+1α′x−n+1B^n+1αx,n≥0.
By Lemma1, with Ax=x, Bn=B^nα, and λn,n=−n+2α−1/2n+2α−1n+α,n≥0, λ0,0=−1/α and by taking (22) and (24) into account, the MPS Qnxn≥0 is orthogonal with respect to the linear functional ℬα+1/2. Thus, Qnx=B^nα+1/2x, n≥0. This leads to (35).
I2 (Jacobi case): namely, c=1, μ=α+β+2, ψx=−α+β+2x+α−β, and W=Jα,β. The Jacobi quasi-definite linear functional with parameters α and β where α,β≠−n and α+β≠−n−1,n≥1, which is the unique monic linear functional satisfying(37)x2−1Jα,β′+−α+β+2x+α−βJα,β=0.
Since(38)α+β+22α+1x+1Jα,β=Jα+1,β,it follows that(39)α+β+1Jα+1,β−x−1Jα+1,β′=α+β+2Jα,β,
So, we can deduce that(40)Jα,βα+β+2,0=Jα+1,β.
Denote byJ^nα,βxn≥0 the MOPS with respect to the Jacobi linear functional Jα,β. The MOPS J^nα,βxn≥0 satisfies the first structure relation for n≥0 [8]:(41)x2−1J^n+1α,β′x=n+1x+α−β2n+α+β+2J^n+1α,βx−2n+α+β+3γn+1J^nα,β,where(42)γn+1=4n+1n+α+β+1n+α+1n+β+12n+α+β+12n+α+β22n+α+β+3,n≥0.Corollary 3.
The polynomialsJ^n+1α,βx and J^nα+1,βx satisfy the following relation:(43)x−1J^n+1α,β′x−n+1J^n+1α,βx=εnJ^nα+1,βx,where for any integer n≥0,(44)εn=−2n+1n+β+12n+α+β+2,n≥0.Proof.
Using (41), we obtain for any integer n≥0,(45)x+1Qnx=J^n+1α,βx+2n+α+1n+α+β+12n+α+β+12n+α+β+2J^nα,βx,n≥0,where(46)Qnx=−2n+α+β+22n+1n+β+1x−1J^n+1α,β′x−n+1J^n+1α,βx,n≥0.
By Lemma1, with Ax=x+1, Bn=J^nα,β, λn,n=2n+α+1n+α+β+1/2n+α+β+12n+α+β+2≠0,n≥0, and by taking (38) and (40) into account, xthe MPS Qnxn≥0 is orthogonal with respect the linear functional Jα+1,β. Thus, Qnx=J^nα+1,βx, n≥0. This leads to (43).



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