On a generalization of Hermite polynomials

Journal of Functional Analysis 266 (2014) 2910–2920

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Journal of Functional Analysis www.elsevier.com/locate/jfa

On a generalization of Hermite polynomials Piotr Krasoń a,∗ , Jan Milewski b a b

Department of Mathematics, Szczecin University, 70-415 Szczecin, Poland Institute of Mathematics, Poznań University of Technology, 60-965 Poznań, Poland

a r t i c l e

i n f o

Article history: Received 5 March 2013 Accepted 28 December 2013 Available online 10 January 2014 Communicated by D. Voiculescu Keywords: Hermite–Jordan polynomials Eigenvalue problem Differential operator

a b s t r a c t We consider a new generalization of Hermite polynomials to the case of several variables. Our construction is based on an analysis of the generalized eigenvalue problem for the operator ∂Ax +D, acting on a linear space of polynomials of N variables, where A is an endomorphism of the Euclidean space RN and D is a second order differential operator. Our main results describe a basis for the space of Hermite–Jordan polynomials. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Classical Hermite polynomials (cf. [2,4]) are the solutions in the space of polynomials in one variable of the following eigenvalue problem: d d2 W + αx W = λW. dx2 dx

(1.1)

For a Hermite polynomial of degree n, with the leading coefficient an the corresponding eigenvalue λ(n, α) = nα. Typically, a value of −2 for the constant α is important for various applications. One can also find the following formula for the coefficient am as a function of an : * Corresponding author. E-mail addresses: [email protected] (P. Krasoń), [email protected] (J. Milewski). 0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jfa.2013.12.027

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am = (2α)(m−n)/2

n! an m![(n − m)/2]!

for α = 0.

2911

(1.2)

In (1.2) we assume that m has the same parity as n, otherwise we put am = 0. The formula (1.2) is obtained by downward recursion, with step −2, on the monomials xk starting from xn . For α = 0 there are two linearly independent eigenpolynomials corresponding to the eigenvalue λ = 0: φ0 = 1 and φ1 = x. Notice that for the case α = 0, the only eigenvalue in the space of polynomials is λ = 0. There are some generalizations of Hermite polynomials to the case of several independent variables (cf. [7,8,3]). In this paper, we study one such new generalization. The novelty of our approach relies on the consideration of a generalized eigenvalue problem. We think that our approach will be useful in the analysis of some problems in mathematical physics and partial differential equations. We consider the following differential operator ∂Ax + D.

(1.3)

 2 In (1.3) D = di,j ∂x∂i ∂xj is a second order differential operator with constant coefficients and ∂Ax denotes differentiation along the vector Ax, where A is an endomorphism of RN .  ∂ . We are interested in a generalized eigenvalue problem [1] Thus ∂Ax = i,j Ai,j xi ∂x j for the operator (1.3): (∂Ax + D − λid)k φ = 0.

(1.4)

Definition 1.1. If φ satisfies (1.4) for k = r and does not satisfy it for k = r − 1, we call it a generalized eigenfunction for the operator (1.3) of order r. In the space of polynomials with N variables we call such eigenfunctions the Hermite–Jordan polynomials of order r. Definition 1.2. Let V be a vector space and A ∈ End(V ) a linear operator. A sequence (un ) of linearly independent vectors in V is called a chain of generalized eigenvectors corresponding to the eigenvalue λ if Au1 = λu1

and Auk = λuk + uk−1

for k > 1.

Remark 1.3. Note that an eigenvalue λ of a linear operator A acting on a finite dimensional vector space V yields in Definition 1.2 a finite sequence (un ) corresponding to a Jordan cell of λ. In the case when the chain of generalized eigenvectors (un ) is maximal it forms the standard basis of a Jordan cell. For an infinite dimensional V we might obtain an infinite chain of generalized eigenvectors for the operator A. n

Example 1.4. un+1 := φn = xn! is a generalized eigenpolynomial of order n + 1 for the operator d/dx for the eigenvalue 0. More precisely, the sequence (φn ) is an infinite chain of generalized eigenpolynomials, i.e. dφn /dx = 0 · φn + φn−1 for n > 0 and φ0 is an eigenpolynomial.

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In the space of analytic functions, the operator d/dx has a continuous spectrum. The eigenfunctions exp(λx) correspond to the eigenvalue λ. The usual Maclaurin series can be expressed in terms of the generalized eigenpolynomials exp(λx) =



λn φn .

(1.5)

n0 n

Example 1.5. φn = xn! is a generalized eigenpolynomial of order [n/2]+1 for the operator d2 /dx2 . In the space of analytic functions, the operator d2 /dx2 also has a continuous spectrum. The following propositions generalize this situation and are easy to prove. Proposition 1.6. Let A : V → V be a continuous linear operator acting on a linear topological space V . Assume that we have an infinite chain (un ) of generalized eigenvectors corresponding to the eigenvalue λ0 = 0 satisfying Au1 = 0,

Aun = un−1

for n > 1

and the series u(λ) :=

∞ 

λn−1 un

n=1

is convergent for λ in some neighbourhood of zero. Then u(λ) is an eigenvector corresponding to λ. Proposition 1.7. Let A be as in Proposition 1.6 and assume that an interval (a, b) is contained in the spectrum of A. Let u : (a, b) −→ V

(1.6)

Au(λ) = λu(λ).

(1.7)

be a C ∞ -map satisfying

Then the sequence 1 dn u(λ), n! dλn

n ∈ N0

forms an infinite chain of generalized eigenvectors of A corresponding to the eigenvalue λ.

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Proof. Indeed, A

1 dn n! dλn u(λ)

2913

is an eigenvector of A for n = 0 and

 dn−1 1 dn 1 dn  1 dn 1 u(λ) = u(λ) + λ u(λ) λu(λ) = n n n−1 n! dλ n! dλ (n − 1)! dλ n! dλn

for n  1. 2 Remark 1.8. In the hypotheses of Proposition 1.7, we have to assume that V is such that the notion of a C ∞ -map u makes sense. V may be, for example, a Banach space. In Section 3, we suggest one algorithm for finding the leading term of a Hermite– Jordan polynomial, and then a recursive algorithm for finding all of its non-zero coefficients. These, under some mild hypotheses on a spectrum of ∂Ax + D, yield a unique basis for the space spanned by Hermite–Jordan polynomials (cf. Theorem 3.12). Note that, constructed in this way Hermite–Jordan polynomials are not just products of the corresponding polynomials with one variable, even when A is diagonalizable. Therefore we obtain a non-trivial generalization of the case of polynomials with one variable. Classical Hermite polynomials find applications in physics for the analysis of harmonic oscillators [6]. In a future paper, we will address some applications for Hermite– Jordan polynomials. 2. Basic notions We need the following definition: Definition 2.1. We say that a polynomial P (x1 , . . . , xN ) is of parity n if every nonzero monomial which is a summand of P (x1 , . . . , xN ) is of the total degree that has the same parity as n. Observe that the operator ∂Ax + D preserves parity of a polynomial. By H r,n (λ) we will denote a Hermite–Jordan polynomial of degree n, parity n and r,n order r corresponding to the eigenvalue λ. Hm (λ) will be the homogenous part of degree m of H r,n (λ). Thus we have H

r,n

(λ) =

n 

r,n Hm (λ).

(2.8)

m=0

Remark 2.2. Notice that similarly to the one-dimensional case (cf. (1.2)) in (2.8) r,n (λ) = 0 if m is of different parity than n. Hm Remark 2.3. If for every triple {r, n, λ} we choose H r,n (λ) then it follows that span{H r,n (λ): r, n, λ} is the space of all polynomials in N -variables.

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The following proposition is a consequence of the fact that operator D lowers the degree of a polynomial by two. Proposition 2.4. For a given operator ∂Ax + D, Hnr,n (λ) is a generalized eigenpolynomial of order r  r, of the operator ∂Ax corresponding to the same eigenvalue λ. Proof. Indeed since (∂Ax + D − λid)r H r,n (λ) = 0 and D lowers the degree of a homogenous polynomial by 2 we see that (∂Ax − λid)r Hnr,n (λ) = 0. 2 We also have the following: Proposition 2.5. Let Pn (x1 , . . . , xN ) be a homogenous generalized eigenpolynomial of the operator ∂Ax corresponding to the eigenvalue λ of order r . There exists a Hermite– Jordan polynomial Gr,n (λ) with Gr,n n (λ) = Pn (x1 , . . . , xN )

(2.9)

and for any Gr,n (λ) satisfying (2.9) we have r  r . Proof. Let Vn be a linear space of polynomials Wn (x1 , . . . , xN ) of degree n and parity n. The space Vn is invariant with respect to the operators ∂Ax and ∂Ax + D. Consider the subspace Vn,λ ⊂ Vn of Hermite–Jordan polynomials of eigenvalue λ included in Vn . Of course Vn,λ = Vn ∩ span{H r,k (λ): k  n}. Let V˜n,λ be the subspace of Vn spanned by the homogenous components of degree n of the polynomials in Vn,λ . By Remark 2.3  and Proposition 2.4 we see that Pn (x1 , . . . , xN ) ∈ V˜n,λ . Thus Pn = k Wnrk ,n (λ), where  r,n (λ) W rk ,n (λ) are Hermite–Jordan polynomials. But k Wnrk ,n (λ) = Gr,n n (λ), where G is a Hermite–Jordan polynomial with r  max{rk }. By Proposition 2.4 we see that, r  r . 2 The following lemmas will be useful in the proof of Theorem 3.10. Lemma 2.6. Let ψ be a polynomial of degree n and parity n. There exist polynomials Ψ and a Hermite–Jordan polynomial H r,k (λ) both of parity n and degrees not exceeding n such that: (∂Ax + D − λid)Ψ = ψ + H r,k (λ).

(2.10)

Proof. Let V be a finite dimensional vector space and T : V → V a linear map. Using Jordan basis for T one easily sees that one can choose the smallest r ∈ N such that: V = im T + ker T r .

(2.11)

Hence for any v ∈ V there exist u ∈ V and w ∈ ker T r such that: v = T u + w.

(2.12)

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Let V be the linear space of polynomials of at most degree n and of parity n, T = (∂Ax + D − λid). Taking in (2.12) v = ψ, Ψ = u and H r,k (λ) = −w we obtain the assertion. 2 Lemma 2.7. Let Ψn be a generalized eigenpolynomial of degree n of the operator ∂Ax satisfying (∂Ax − λid)Ψn = Hns,n (λ)

(2.13)

for a Hermite–Jordan polynomial H s,n (λ). There exist Hermite–Jordan polynomials Gr,n (λ) and F t,p (λ) with p  n − 2 and p of parity n such that: Gr,n n (λ) = Ψn

(2.14)

(∂Ax + D − λid)Gr,n (λ) = H s,n (λ) + F t,p (λ).

(2.15)

and

Proof. By Lemma 2.6 there exist a polynomial Q and a Hermite–Jordan polynomial F t,p (λ) both of degree p  n − 2 and parity n such that   (∂Ax + D − λid)Q = H s,n (λ) − Hns,n (λ) − DΨn + F t,p (λ).

(2.16)

(∂Ax + D − λid)(Q + Ψn ) = H s,n (λ) + F t,p (λ).

(2.17)

Thus

Put Gr,n (λ) = Q + Ψn .

2

Example 2.8. Consider the generalized Hermite problem in R2 with the following operators: ∂Ax = x

∂ ∂ −y , ∂x ∂y

D=

∂2 . ∂x∂y

The monomials xm y n are the eigenpolynomials of the operator ∂Ax corresponding to the eigenvalue λm,n = m − n and since   (∂Ax + D) xm y n = λm,n xm y n + nmxm−1 y n−1 , we see that they are also generalized eigenpolynomials of order r(n, m) := min(m + 1, n + 1) of the operator ∂Ax + D. Therefore on the space Vn+m the polynomials Dk xm y n generate the basis for the generalized eigenspace of dimension r(n, m) of the operator ∂Ax + D corresponding to the Jordan cell of dimension r(n, m). On the other hand the corresponding Jordan cells of the operator ∂Ax are one-dimensional.

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Fig. 1. Young diagram corresponding to the eigenvalue λi .

3. Construction of a basis of Hermite–Jordan polynomials In this section we suggest a choice and an effective construction of a basis of Hermite– Jordan polynomials. In the finite dimensional space WN of 1-forms on RN we choose a Jordan basis for the operator ∂Ax . Assume that ∂Ax has r-distinct eigenvalues λ1 , . . . , λr . Assume that λi gives the li Jordan blocks of sizes ki,1  · · ·  ki,li . For every λi we associate a Young diagram [5] (cf. Fig. 1). Let φki,j : 1  k  ki,j , 1  j  li be the λi -part of the Jordan basis i.e. ∂Ax φki,j = λi φki,j + φk−1 i,j ,

k > 1,

∂Ax φ1i,j = λi φ1i,j .

(3.18)

The basis of 1-forms corresponding to the Jordan canonical form consists of all φki,j : 1  k  ki,j , 1  j  li , 1  i  r. In Fig. 1 i = const, the horizontal direction is the k-direction, the vertical direction is the j-direction and the (j, k)-cell corresponds to the eigenform φki,j . Definition 3.1. By concatenating Young diagrams (resp. Young tableaux) corresponding to various eigenvalues we obtain a diagram [(i, j, k)] (resp. tableau [nki,j ]) of the shape shown in Fig. 2. Slightly abusing notation we will call it a Young diagram (resp. Young tableau) associated to the Jordan canonical form of the operator ∂Ax acting on the space of 1-forms. Notice that we do not require that our Young tableaux have different entries. Remark 3.2. Since the Jordan canonical form is not unique and depends on the order of Jordan blocks we assume that both the order of eigenvalues and the order of blocks corresponding to a given eigenvalue λi are chosen so that we obtain a unique Young diagram (resp. Young tableau) associated with the Jordan canonical form of the operator ∂Ax acting on the space of 1-forms (cf. Fig. 2). Let further:    nk Φ nki,j := φki,j i,j . i,j,k

(3.19)

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Fig. 2. Young tableau corresponding to the product of powers of generalized eigenforms. k

In (3.19) the expression (φki,j )ni,j denotes the nki,j -power of the form φki,j . Remark 3.3. We are interested in the case when all nki,j  0. Notice that in all following formulas involving differentiation if one of the entries of a resulting Young tableau is negative then the corresponding coefficient is zero. We also need the following: Definition 3.4. For a Young diagram [(i, j, k)] a triple (m, s, t) satisfying 1 < m  ks,t , m 1  t  ls , 1  s  r we define an operator Bs,t acting on a Young tableau [nki,j ] in m k the following manner: Bs,t ([ni,j ]) is the Young tableau with the same entries except that (m, s, t)-entry is decreased by 1 and (m − 1, s, t)-entry is increased by 1. Now we formulate several lemmas useful in further computations. Lemma 3.5 (Generalized Newton–Leibniz formula). For any vector Y ∈ RN , any functions f1 , . . . , fl of class C m , and constants μ, μ1 , . . . , μl such that μ = μ1 + · · · + μl one has (∂Y + μid)m (f1 · · · · · fl )      m! (∂Y + μ1 id)m1 f1 · · · (∂Y + μl id)ml fl . = m1 ! · · · ml ! m +···+m =m 1

l

Proof. For m = 1 and l = 1 the lemma is obvious. Using Leibniz rule we induct first with respect to l and then m. 2 Lemma 3.6. For the 1-forms φki,j which form the Jordan basis of ∂Ax acting on the space WN one has  n   w1  wl 1 l (∂Ax − λi id)m φki,j = S(m1 , w1 ; . . . ; ml , wl ) φk−m · · · φk−m , i,j i,j

(3.20)

P. Krasoń, J. Milewski / Journal of Functional Analysis 266 (2014) 2910–2920

2918

where S(m1 , w1 ; . . . ; ml , wl ) =

m! n! · (m1 !)w1 · · · (ml !)wl w1 ! · · · wl !

(3.21)

and the summation is over all indices satisfying the following conditions: m1 w1 + · · · + ml wl = m, w1 + · · · + wl = n,

0  m1 < m2 < · · · < ml < k, l > 0.

(3.22)

Proof. Follows from (3.18) and Lemma 3.5. 2 Example 3.7. To illustrate how the conditions (3.22) in the formula (3.20) work put m = 1. Then the only possibility for the indices is the following: l = 1, m1 = 0, m2 = 1, w1 = n − 1, w2 = 1 and (3.20) specializes to the following equality:  n  n−1 k−1 (∂Ax − λi id) φki,j = n φki,j φi,j .

(3.23)

Lemma 3.8. If in the sum:  l1 +···+ls =m

 n1  ns m! (∂Ax − λi1 id)l1 φki11,j1 · · · (∂Ax − λis id)ls φkiss,js l 1 ! · · · ls !

(3.24)

there is a non-zero summand then the whole sum is non-zero. Proof. Follows from Lemmas 3.5 and 3.6 since all the coefficients in the expansion of the right-hand side in the monomial basis are non-negative. 2 Lemma 3.9. The monomial of the form Φ([nki,j ]) is a generalized eigenvector of the oper  ator ∂Ax corresponding to the eigenvalue λ = i,j,k nki,j λi of order O = 1 + i,j,k (j − 1)nki,j . The following formula holds:     m  k  (∂Ax − λid)Φ nki,j = nm . s,t Φ Bs,t ni,j

(3.25)

m>1,s,t m Proof. Observe that Φ([nki,j ]) and Φ(Bs,t ([nki,j ])) have the same eigenvalue λ. Equality (3.25) follows from Lemma 3.5 for m = 1 and (3.23). Applying (3.25) recursively we see that (∂Ax − λid)O Φ([nki,j ]) = 0, since on the right-hand side all the summands will be equal to 0 (cf. Remark 3.3). Lemma 3.8 guarantees that O is the minimal natural number v such that (∂Ax − λid)v Φ([nki,j ]) = 0. 2

Now we are ready to formulate our main result: Theorem 3.10. Let [(i, j, k)] be a Young diagram as in Definition 3.1. For any Young tableau [nki,j ] corresponding to [(i, j, k)] with non-negative entries there exist Hermite– Jordan polynomials H([nki,j ]) and F t,p (λ) such that:

P. Krasoń, J. Milewski / Journal of Functional Analysis 266 (2014) 2910–2920

(1) H([nki,j ]) corresponds to the eigenvalue λ =  k i,j,k (j − 1)ni,j , (2) the following equality is satisfied:

 i,j,k

2919

nki,j λi of order at least O = 1 +

   m  k   (∂Ax + D − λid)H nki,j = nm + F t,p (λ), s,t H Bs,t ni,j

(3.26)

m>1,s,t

(3) the highest degree homogenous component of H([nki,j ]) is equal to Φ([nki,j ]), (4) p = deg F t,p (λ)  deg H([nki,j ]) − 2 and both F t,p (λ) and H([nki,j ]) are of the same parity. Proof. We start with Φ([nki,j ]). By Lemma 3.9, Φ([nki,j ]) is a generalized eigenvector  k of the operator ∂Ax corresponding to the eigenvalue λ = i,j,k ni,j λi of order O =  k 1 + i,j,k (j − 1)ni,j . By Proposition 2.4 we see that there exists H([nki,j ]) satisfying (1) and (3). By Proposition 2.5 we may apply Lemma 2.7 for a construction of H([nki,j ]). Then we see that (4) is satisfied. Equality (2.15) of Lemma 2.7 and equality (3.25) of Lemma 3.9 yield (2). 2 Remark 3.11. Condition (4) from Theorem 3.10 can be strengthened as follows. Let ˜ ([nk ]) be the space of polynomials generated by monomials of the form Φ([˜ W nki,j ]) for i,j  k  k ˜ i,j  k ni,j for all pairs (i, j). Define which k n W

 k    ˜ nk ni,j = W ∩ Vn−2 i,j

where Vn is as defined in Section 2 and n = Φ([nki,j ]). It is clear that F t,p (λ) ∈ W ([nki,j ]).

k i,j

(3.27)

nki,j is the degree of the polynomial

In the next theorem we address the question of uniqueness of the polynomials H([nki,j ]) constructed in Theorem 3.10. By uniqueness we mean, that for a given Young tableau [nki,j ] corresponding to the Young diagram [(i, j, k)] there exists a unique Hermite–Jordan polynomial satisfying assertions (1)–(4) of Theorem 3.10. Theorem 3.12. If for any non-negative integers ci , i = 1, . . . , r satisfying 0  ci 

 j,k

nki,j

and



ci = 2M,

M >0

(3.28)

i

the following condition holds: c1 λ1 + · · · + cr λr = 0

(3.29)

then the Hermite–Jordan polynomial H([nki,j ]) is defined uniquely. It is a generalized  eigenvector of the operator ∂Ax + D corresponding to the eigenvalue λ = i,j,k nki,j λi of  order O = 1 + i,j,k (j − 1)nki,j and the following equality:

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  (∂Ax + D − λid)H nki,j =



 m  k  nm s,t H Bs,t ni,j

(3.30)

m>1,s,t

is satisfied. Proof. We consider W ([nki,j ]) defined in Remark 3.11. From condition (3.29) we see that ker(∂Ax + D − λid)q ∩ W ([nki,j ]) = 0. It follows that the polynomial F t,p (λ) ≡ 0. 2 Corollary 3.13. If the eigenvalues λ1 , . . . , λr are linearly independent over non-negative integers then the basis {H([nki,j ]): [nki,j ] associated with [(i, j, k)]} of the space polynomials in N -variables constructed in Theorem 3.10 is unique. Moreover, the operators ∂Ax and ∂Ax + D acting on the space of polynomials of N -variables are similar. Proof. Observe that the condition (3.29) is satisfied for any Young tableau associated with the Young diagram [(i, j, k)]. The operators ∂Ax and ∂Ax + D are similar since there exist bases in which they have the same form (cf. (3.25) and (3.30)). 2 Remark 3.14. Notice that to decide whether for a given Young tableau [nki,j ], H([nki,j ]) is unique or not, it is enough to verify condition (3.29) for a finite set of r-tuples (c1 , . . . , cr ) satisfying (3.28). Acknowledgments The first author would like to thank V. Berkovich and the Weizmann Institute in Rehovot, Israel for their hospitality during his visit in the summer of 2012. The research was partially sponsored by the grants: N N 201 607 440 and NCN 2013/09/B/ST1/04416 of the Polish National Center of Science (NCN). References [1] N. Bourbaki, Algebra. Chapters 1–3 (Algèbre. Chapitres 1 à 3), Elements of Mathematics (Éléments de mathématique), Springer, 2007, reprint of the 1970 original. [2] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [3] G. Dattoli, S. Lorenzutta, G. Maino, A. Torre, C. Cesarano, Generalized Hermite polynomials and supergaussian forms, J. Math. Anal. Appl. 203 (1996) 597–609. [4] M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005. [5] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl., vol. 16, Cambridge University Press, 2009, xxviii. [6] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Pergamon Press, 1965. [7] M. Lassalle, Polynômes de Hermite généralisés, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991) 579–582. [8] Y. Xu, Lecture notes on orthogonal polynomials of several variables, in: W. zu Castell, F. Filbir, B. Forster (Eds.), Inzell Lectures on Orthogonal Polynomials, in: Adv. Theory Spec. Funct. Orthogonal Polynomials, vol. 2, Nova Science Publishers, Hauppauge, NY, 2005, pp. 141–196.