Threefold symmetric Hahn-classical multiple orthogonal polynomials

Threefold symmetric Hahn-classical multiple orthogonal polynomials

Ana Loureiro School of Mathematics, Statistics and Actuarial Science (SMSAS), Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF (U.K.) Walter Van Assche Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Threefold symmetric Hahn-classical multiple orthogonal polynomials

Abstract. We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as -orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a -star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them -orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni. Keywords. orthogonal polynomials, multiple orthogonal polynomials, classical polynomials, recurrence relation, differential equation

AMS classification. 33C45; 42C05Email addresses: a.loureiro@kent.ac.uk (corresponding author) and   walter.vanassche@kuleuven.beVersion date: July 20, 2019

1 Introduction and motivation

In this paper we investigate and characterize all the multiple orthogonal polynomials of type II that are threefold symmetric and are such that the polynomial sequence of its derivatives is also a multiple orthogonal sequence of type II. Within the standard orthogonality context there are only four families satisfying this property, commonly referred to as the Hahn property, and they are the Hermite, Laguerre, Jacobi and Bessel, collectively known as the classical orthogonal polynomials. These four families of polynomials also share a number of analytic and algebraic properties. Several studies are dedicated to extensions of those properties to the context of multiple orthogonality. However, those extensions give rise to completely different sequences of multiple orthogonal polynomials. For the usual orthogonal polynomials the Hahn property is equivalent with the Bochner characterization (polynomials satisfy a second order differential equation of Sturm-Liouville type) and the existence of a Rodrigues formula. This is no longer true for multiple orthogonal polynomials since there are families of multiple orthogonal polynomials with a Rodrigues type formula [1] that do not satisfy the Hahn property, and various examples of multiple orthogonal polynomials satisfy a higher-order linear recurrence relation, which is not of Sturm-Liouville type [11]. Hence characterizations for multiple orthogonal polynomials based on the Hahn property, the Bochner property or the Rodrigues formula give different families of polynomials.

A sequence of monic polynomials with is orthogonal with respect to a Borel measure whenever

 ∫Pn(x)xkdμ(x)=0 if  k=0,1,…,n−1, n⩾0

and

 ∫Pn(x)xndμ(x)≠0 for  n⩾0.

Without loss of generality, often we normalize so that it is a probability measure. Obviously, an orthogonal polynomial sequence forms a basis of the vector space of polynomials . The measures described above can be represented via a linear functional , defined on , the dual space of , and it is understood that the action of over a polynomial corresponds to . Throughout, we denote this action as

 ⟨L,f(x)⟩\coloneqq∫f(x)dμ(x).

The derivative of a function is denoted by or . Properties on , such as differentiation or multiplication by a polynomial, can be defined by duality. A detailed explanation can be found in [29] [31]. In particular, given and a functional , we define

 ⟨g(x)L,f(x)⟩\coloneqq⟨L,g(x)f(x)⟩and⟨L′,f(x)⟩\coloneqq−⟨L,f′(x)⟩ (1.1)

for any polynomial .

From the definition it is straightforward that an orthogonal polynomial sequence satisfies a second order recurrence relation

 Pn+1(x)=(x−βn)Pn(x)−γnPn−1(x), n⩾1, (1.2)

with , and for all integers . This relation is often called the three-term recurrence relation, but we will avoid this terminology as we will be dealing with three-term recurrence relations which are of higher order. There is an important converse of this connection between orthogonal polynomials and second order recurrence relations, known as the Shohat-Favard theorem or spectral theorem for orthogonal polynomials. It states that any sequence of monic polynomials with , satisfying the recurrence relation (1.2) with for all and initial conditions and , is always an orthogonal polynomial sequence with respect to some measure and, if, in addition, and for all , then is a positive measure on the real line. So, basically, the orthogonality conditions and the second order recurrence relations are two equivalent ways to characterize an orthogonal polynomial sequence.

Multiple orthogonal polynomials of type II correspond to a sequence of polynomials of a single variable that satisfy multiple orthogonality conditions with respect to measures. With the multi-index , type II multiple orthogonal polynomials correspond to a (multi-index) sequence of monic polynomials of degree for which there is a vector of measures such that

 ∫xkP→n(x)dμj(x)=0,0⩽k⩽nj−1 (1.3)

holds for every . By setting , we recover the usual orthogonal polynomials. Obviously, (1.3) amounts to the same as saying there exists a vector of linear functionals such that

 ⟨Lj,xkP→n(x)⟩=0,0⩽k⩽nj−1.

This polynomial set may not exist or is not unique unless we impose some extra conditions on the measures , but under appropriate conditions the polynomial set satisfies a system of recurrence relations, relating with its nearest neighbors and , where consists of the -dimensional unit vector with zero entries except for the th component which is (see [37] for further details). The focus of the present work is on the polynomials whose multi-indices lie on the step-line near the diagonal , and are defined by

 Prn(x)=P→n(x),Prn+1(x)=P→n+→e1(x),…,Prn+j(x)=P→n+→ej(x),j=0,1,…,r−1.

This step-line polynomial sequence satisfies a th order recurrence relation (involving up to consecutive terms) and it can be written as

 xPn(x)=Pn+1(x)+r∑j=0ζn,jPn−j(x),n⩾1,P0(x)=1,P−j(x)=0,j=1,…,r, (1.4)

where for all . Such a step-line polynomial sequence corresponds to so-called -orthogonal polynomials (with ), see [28, 15, 32] and Figure 1.1 for the case of . So, basically, if there exists a vector of linear functionals , any polynomial sequence satisfying

then the polynomials are related by (1.4). There is a converse result, which is a natural generalization of the Shohat-Favard theorem, in the sense that if a polynomial sequence satisfies (1.4) with for all , then there is a vector of linear functionals with respect to which is -orthogonal. Such a vector of linear functionals is not unique, but we can consider its components to be the first elements of the dual sequence associated to , which always exist and which are defined by

 ⟨un,Pm⟩=δn,m\coloneqq{0ifn≠m1ifn=m,m,n⩾0,

where represents the Kronecker symbol. The remaining elements of the dual sequence of a -orthogonal polynomial sequence can be generated from the first ones: for each pair of integers there exist polynomials such that [28]

 udn+j = d−1∑ν=0qn,j,ν(x)uν,for0⩽j⩽d−1,

where , for if and for if . Unlike the standard orthogonality (case ), this result only provides structural properties for the dual sequence and in practical terms it can be used in that sense.

In this paper we mainly deal with -orthogonal polynomial sequences, but our results and ideas can be extended to the -orthogonal case. Here, we provide a characterization of all the -orthogonal polynomial sequence that are threefold symmetric and possess the Hahn property in the context of -orthogonality, i.e., the sequence of monic derivatives is also -orthogonal. The notion and properties of threefold symmetric -orthogonal polynomial sequences are revised and discussed in Section 2, while we also bring a new result (Theorem 2.6) relating the asymptotic behavior of the recurrence coefficient to the asymptotic behavior of the absolute value of the largest zero. Section 3 is dedicated to a detailed characterization of all the threefold symmetric -orthogonal polynomials. We bring together old and new results in a self-contained and complete analysis to fully characterize this type of sequences in terms of their explicit recurrence relations, weight functions and a differential equation of third order. In that regard, we complete the study initiated by Douak and Maroni in [14, 15] by providing sufficient conditions and taking the study based only on the properties of the zeros in combination with the weights. We end up this coherent description with a detailed analysis of the four distinct families of threefold symmetric -orthogonal polynomials. The first case, treated in Section 3.1 corresponds polynomials with the Appell property whose -orthogonality weights are supported on the three-starlike (set as in Fig.2.2) represented via Airy function and its derivative. These polynomials have been studied in [13], but the explicit representation of the weights supported on a set containing all the zeros has not been considered there. In Sections 3.2 and 3.3 we study the second and third cases where the weights are represented via the confluent hypergeometric Kummer function of 2nd kind. These are, to the best of our knowledge, new. The fourth case is treated in Section 3.4, where the orthogonality weights are expressed via hypergeometric functions and depend on two parameters. For special choices of those parameters we recover some known polynomial families. However, the existent literature on the subject has not taken into account the explicit representation of the weights supported on a set containing all the zeros and this has been accomplished here for all the threefold symmetric polynomials satisfying the Hahn property within the context of -orthogonality. It turns out that the orthogonality measures under analysis are solutions to a second order differential equation and, as such, the recurrence coefficients for the whole set of the corresponding multiple orthogonal polynomials of type II can only be obtained algorithmically via the nearest-neighbor algorithm explained in [37] [19]. Some of the -orthogonal polynomials that we found appear in the theory of random matrices, in particular in the investigation of singular values of products of Ginibre matrices, which uses multiple orthogonal polynomials with weight functions expressed in terms of Meijer G-functions [24]. For these Meijer G-functions are hypergeometric or confluent hypergeometric functions.

2 Threefold 2-orthogonal polynomial sequence

A sequence of monic polynomials (with ) is symmetric whenever for all . This means that all even degree polynomials are even functions while odd degree polynomials are odd functions. Hermite and Gegenbauer polynomials are examples of symmetric polynomial sets, which also happen to be the only classical orthogonal polynomials that are symmetric. Many other examples of symmetric orthogonal polynomial sequences are around in the literature.

The notion of symmetry of polynomial sequences has been extended and commonly referred to as -symmetric in works by Maroni [28], [15] and followed by Ben Cheikh and his collaborators [6] [8] [9][25]. The case would correspond to the usual symmetric case. We believe the name is misleading, as will soon become apparent, and therefore we call it differently as it pictures better the nature of the problem.

Definition 2.1.

A polynomial sequence is -fold symmetric if

 Pn(ωkx)=ωnkPn(x),  for any  n⩾0  and  k=1,2,…,m−1, (2.1)

where

So, an -fold symmetric sequence is a -symmetric sequence in [28], [15], [6], [8],[9],[25]. (The symmetric sequences of Hermite and Gegenbauer polynomials are examples of twofold symmetric polynomials.) In particular, a threefold symmetric sequence , is such that

 Pn(ωx)=ωnPn(x)  and  Pn(ω2x)=ω2nPn(x)  with  ω=e2iπ3, n⩾0,

which corresponds to say that there exist three sequences with such that

 P3n+j(x)=xjP[j]n(x3),j=0,1,2. (2.2)

Throughout, we will refer to as the diagonal components of the cubic decomposition of the threefold symmetric sequence , which is line with the terminology adopted in a more general cubic decomposition framework in [33].

In the case of a -orthogonal polynomial sequence  with respect to a vector linear functional , we have, as discussed in Section 1,

 ⟨u0,xmPn⟩={0forn⩾2m+1N0(n)≠0forn=2m (2.3) ⟨u1,xmPn⟩={0forn⩾2m+2N1(n)≠0forn=2m+1, (2.4)

and there exists a set of coefficients such that satisfies a third order recurrence relation (see [28, 39])

 Pn+1(x)=(x−βn)Pn(x)−αnPn−1(x)−γn−1Pn−2(x) (2.5)

with and . Straightforwardly from the definition, one has

 γ2n+1=⟨u0,xn+1P2n+2⟩⟨u0,xnP2n⟩≠0,γ2n+2=⟨u1,xn+1P2n+3⟩⟨u1,xnP2n+1⟩≠0,  n⩾0, (2.6)

or, equivalently,

 ⟨u0,xn+1P2n+2⟩=n∏k=0γ2k+1% and ⟨u1,xn+1P2n+3⟩=n∏k=0γ2k+2,for n⩾0.

Whenever a -orthogonal polynomial sequence is threefold symmetric, the recurrence relation (2.5) reduces to a three-term relation, where the - and -coefficients all vanish. For this type of -orthogonal polynomial sequences more can be said.

Proposition 2.2.

[14] Let be a 2-orthogonal polynomial sequence with respect to the linear functional satisfying (2.3)-(2.4). The following statements are equivalent:

1. The sequence is threefold symmetric.

2. The linear functional is threefold symmetric, that is,

 (uν)3n+μ=0,  for  ν=0,1  and  μ=1,2  % with  μ≠ν. (2.7)
3. The sequence satisfies the third order recurrence relation

 Pn+1(x)=xPn(x)−γn−1Pn−2(x),n⩾2, (2.8)

with and .

Each of the components of the cubic decomposition are also -orthogonal polynomial sequences.

Lemma 2.3.

[28] Let be a threefold symmetric -OPS. The three polynomial sequences (with ) in the cubic decomposition of described in (2.2) are 2-orthogonal polynomial sequences satisfying:

 P[j]n+1(x)=(x−β[j]n)P[j]n(x)−α[j]nP[j]n−1(x)−γ[j]n−1P[j]n−2(x), (2.9)

where

 β[j]n=γ3n−1+j+γ3n+j+γ3n+1+j,n⩾0,α[j]n=γ3n−2+jγ3n+j+γ3n−1+jγ3n−3+j+γ3n−2+jγ3n−1+j,n⩾1,γ[j]n=γ3n−2+jγ3n+jγ3n+2+j≠0,n⩾2.

Moreover, is 2-orthogonal with respect to the vector functional with

 u[j]ν=σ3(xju3ν+j)  for each  j=0,1,2 %and ν=0,1.

where represents the dual sequence of and is the operator defined by for any and .

The orthogonality measures are supported on a starlike set with three rays.

Theorem 2.4.

[3] If for in (2.8), then there exists a vector of linear functionals such that the polynomials defined by (2.8) satisfy the 2-orthogonal relations (2.3)-(2.4). Moreover, the vector of linear functionals satisfies (2.7) and there exist a vector of two measures such that

 ⟨u0,f(x)⟩=∫Sf(x)dμ0(x) ⟨u1,f(x)⟩=∫Sf(x)dμ1(x)

where represents the starlike set

 S\coloneqq2⋃k=0Γkwith Γk=[0,e2πik/3∞),

and the measures have a common support which is a subset of and are invariant under rotations of .

Regarding the behavior of the zeros of any threefold symmetric 2-orthogonal polynomial sequence, it was shown in [9] that between two nonzero consecutive roots of there is exactly one root of and one root of and all those roots lie on the starlike set .

Proposition 2.5.

Let be a 2-OPS satisfying (2.8). If , then the following statements hold:

1. If is a zero of , then are also zeros of with .

2. is a zero of of multiplicity when .

3. has distinct positive real zeros with .

4. Between two real zeros of there exist only one zero of and only one zero of , that is, .

Proof.

The result is a consequence of [9, Theorem 2.2] for the case . ∎

So, with has zeros on the positive real line and all the other zeros are obtained by rotations of of and is a single zero for and a double zero for . The connection between the asymptotic behavior of the -recurrence coefficients in (2.8) and the upper bound for the largest zero is discussed in [2], but for bounded recurrence coefficients. Here, we extend that discussion, by embracing the cases where the recurrence coefficients are unbounded with different asymptotic behavior for even and odd order indices, which will be instrumental in Section 3.

Theorem 2.6.

If in (2.8) are positive and, additionally, and for large , with and , then largest zero in absolute value behaves as

 |xn,n|⩽322/3c1/3nα/3+o(nα/3),n⩾1, (2.10)

where .

Proof.

Consider the Hessenberg matrix

 Hn=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010⋯000000010⋯0000γ10010⋯0000γ20010⋯00⋱⋱⋱⋱00⋯0γn−40010000⋯0γn−30010000⋯0γn−200⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

so that the recurrence relation (2.8) can be expressed as

 Hn⎛⎜ ⎜ ⎜ ⎜ ⎜⎝P0(x)P1(x)⋮Pn−1(x)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=x⎛⎜ ⎜ ⎜ ⎜ ⎜⎝P0(x)P1(x)⋮Pn−1(x)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠−Pn(x)⎛⎜ ⎜ ⎜ ⎜⎝00⋮1⎞⎟ ⎟ ⎟ ⎟⎠

and each zero of is an eigenvalue of the matrix . The spectral radius of the matrix ,

 ρ(Hn)=max{|λ|: λ is an eigenvalue of Hn},

is bounded from above by where denotes a matrix norm (see [23, Section 5.6]). We take the matrix norm

 ||Hn||S=||S−1HnS||∞=max1⩽i⩽n{n∑j=1∣∣(S−1HnS)i,j∣∣},

where corresponds to a non-singular matrix and denotes the th row and th column entry of the product matrix . In particular if is an invertible diagonal matrix , then

 ||Hn||S=max{d2d1,d3d2,d4+d1γ1d3,…,dk+dk−3γk−3dk−1,…,dn+dn−3γn−3dn−1,dn−2γn−2dn}.

Setting , for some positive constant , gives

 ||Hn||S⩽2α/3(d+cd2)nα/3+o(nα/3)as  n→+∞.

The choice of gives a minimum to , so that

 ||Hn||S⩽341/3(c nα)1/3+o(nα/3)as  n→+∞,

which implies the result. ∎

To summarize, Proposition 2.5 combined with Lemma 2.3 allow us to conclude that each has exactly real zeros with . Moreover, between two consecutive zeros of there is exactly one zero of and another of : . Taking into consideration the asymptotic behavior for the largest zero of described in Theorem 2.6, we then conclude that

 x[j]n,k⩽274c nα+o(nα).

Therefore, for each , all the zeros of distinct from correspond to the cubic roots of , with , and they all lie on the starlike set and within the disc centred at the origin with radius .

Multiple orthogonal polynomials on starlike sets received attention in recent years, under different frameworks. This includes the asymptotic behavior of polynomial sequences generated by recurrence relations of the type (2.8) when further assumptions are taken regarding specific behaviour for the -coefficient [2] or for certain type of the 2-orthogonality measures in [12] and [27]. Faber polynomials associated with hypocycloidal domains and stars have also been studied in [21].

Here, we describe all the threefold -orthogonal polynomial sequences that are classical in Hahn’s sense, and our study includes the representations for the measures supported on a set containing all the zeros of the polynomial sequence. From Theorem 2.4 and Proposition 2.5, the support lies on a starlike set, that, according to Theorem 2.6 is bounded if the -coefficients are bounded, and unbounded otherwise.

3 Threefold symmetric Hahn-classical 2-orthogonal polynomial sequence

The classical orthogonal polynomial sequences of Hermite, Laguerre, Jacobi and Bessel collectively satisfy the so-called Hahn property: the sequence of its derivatives is again an orthogonal polynomial sequence. In the context of -orthogonality, this algebraic property is portrayed as follows:

Definition 3.1.

A monic 2-orthogonal polynomial sequence  is ”-Hahn-classical” when the sequence of its derivatives , with is also a -orthogonal polynomial sequence.

The study of this type of -orthogonal sequences was initiated in the works by Douak and Maroni [14]-[17]. In those works (as well as in [32]) several properties of these polynomials were given, with the main pillars of the study being the structural properties, including the recurrence relations, satisfied by the polynomials. All in all, those studies encompassed the analysis of a nonlinear system of equations fulfilled by the recurrence coefficients. Douak and Maroni treated some special solutions to that system of equations, bringing to light several examples of these threefold symmetric ”-Hahn-classical” polynomials: see [13, 16, 17]. However, for those cases the support of the corresponding orthogonality measures that they found consisted of the positive real axis, which does not contain all the zeros. Here, we base our analysis on the properties of the orthogonality measures and deduce the properties of the recurrence coefficients. We incorporate the works by Douak and Maroni and go beyond that by fully describing all the threefold ”-Hahn-classical” polynomials and bringing up explicitly the orthogonality measures along with the asymptotic behavior of the largest zero in absolute value as well as a Bochner type result for the polynomials (i.e., characterizing these polynomials via a third order differential equation).

We start by observing that the threefold symmetry of readily implies the threefold symmetry of . This is a straightforward consequence of Definition 2.1, as it suffices to take single differentiation of relation (2.1). Such property is valid for any polynomial sequence, regardless any orthogonality properties.

Regarding threefold symmetric 2-orthogonal polynomial sequences possessing Hahn’s property, more can be said. The next result summarizes a characterization of the orthogonality measures along side with the recurrence relations of the original sequence and the sequence of derivatives . This characterization can be found in the works [14, 15, 32]. Nonetheless, we revisit these results for a matter of completion while bringing different approaches to the original proofs, highlighting that all the structural properties for the polynomials can be derived from the corresponding 2-orthogonality measures.

Theorem 3.2.

Let be a threefold symmetric 2-orthogonal polynomial sequence for satisfying the recurrence relation (2.8). The following statements are equivalent:

1. The polynomial sequence is a threefold symmetric 2-orthogonal polynomial sequence, satisfying the third-order recurrence relation

 Qn+1(x)=xQn(x)−˜γn−1Qn−2(x), (3.1)

with initial conditions for .

2. The vector functional satisfies the matrix differential equation

 (3.2a) where (3.2b) for some constants ϑ1 and ϑ2 such that ϑ1,ϑ2≠n−1n,for all  n⩾1. (3.2c)
3. there are coefficients such that satisfies

 (ϕ(x)u0)′′+(2γ1(ϑ2+ϑ1−2)x2u0)′+2γ1(ϑ1−2)xu0=0 (3.3)

and

 (ϑ1−2)(2ϑ2−1)u1=ϕ(x)u′0−2γ1(ϑ1−1)(2ϑ2−3)x2u0, if ϑ1≠2, (3.4a) xu′1=2u′0, if ϑ1=2, (3.4b)

where

 ϕ(x)=(ϑ1(2ϑ2−1)−2γ1(ϑ1−1)(ϑ2−1)x3). (3.5)
4. There exists a sequence of numbers such that

 Pn+3(x)= Qn+3(x)+((n+1)γn+2−(n+3)˜γn+1)Qn(x), (3.6)

with initial conditions

Proof.

We prove that (a) (d) (b) (c) (a).

In order to see (a) implies (d), we differentiate the recurrence relation satisfied by to then replace the differentiated terms by its definition of . A substitution of the term by the expression provided in (3.1) finally gives (3.6).

Now we prove that (d) implies (b). Any linear functional in can be written as

 w=∞∑k=0⟨w,Pk⟩uk.

Based on this, the relation implies

 v′n=−(n+1)un+1,n⩾0, (3.7)

 vn=un+((n+1)γn+2−(n+3)˜γn+1)un+3,n⩾0. (3.8)

The choice of and in both (3.7) and (3.8) respectively gives

 [v′0v′1]=−[u12u2]and[v0v1]=[u0+(γ2−3˜γ1)u3u1+(2γ3−4˜γ2)u4]. (3.9)

As explained in [32] (and, alternatively, in [14], [15] and [28]), the elements of the dual sequence of the 2-orthogonal polynomial sequence  can be written as

 u2n=En(x)u0+an−1(x)u1,u2n+1=bn(x)u0+Fn(x)u1,

where and are polynomials of degree , while and are polynomials of degree less than or equal to under the assumption that , , and . The recurrence relations (2.8) fulfilled by yield (see [32, Lemma 6.1])

 γ2n+2Fn+1(x)−xFn(x)=−an−1(x), γ2n+3an+1(x)−xan(x)=−Fn(x), γ2n+2bn+1(x)−xbn(x)=−En(x)and γ2n+3En+2(x)−xEn+1(x)=−bn(x),

with initial conditions and . In particular, we obtain:

 b1(x)=−1γ2,F1(x)=1γ2x,E2(x)=1γ1γ3x2,anda1(x)=−1γ3,

so that

 (3.10)

Consequently, the first identity in (3.9) reads as

 (3.11a) Using (3.10) in the second identity in (3.9) leads to (3.11b)

where (c.f. [32, Eq. (6.17)])

 ϕ00(x)=3˜γ1γ2,ϕ01(x)=(1−3˜γ1γ2)x, ϕ10(x)=2(1γ1−2˜γ2γ3)x2andϕ11(x)=−1+4˜γ2γ3.

We set and , and now (3.2a)-(3.2b) follows after differentiating both sides of the equation in (3.11b) and then compare with (3.11a), which is a system of two functional equations in :

 {ϑ1u′0+(1−ϑ1)xu′1+(2−ϑ