Complex Hermite Polynomials: Their Combinatorics and Integral Operators

Home , Hermite polynomials

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 4, April 2015, Pages 1397–1410 S 0002-9939(2014)12362-8 Article electronically published on December 9, 2014

COMPLEX HERMITE POLYNOMIALS: THEIR COMBINATORICS AND INTEGRAL OPERATORS

MOURAD E. H. ISMAIL AND PLAMEN SIMEONOV

(Communicated by Jim Haglund)

Abstract. We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these polynomials. We use the combinatorial interpretations to give new proofs of several orthogonality relations satisfied by these polynomials with respect to positive exponential weights in the complex plane. We also construct four integral operators of which the first type of complex Hermite polynomials are eigenfunctions and we identify the corresponding eigenvalues. We prove that the products of these complex Hermite polynomials are complete in certain L2-spaces.

1. Introduction We consider two types of complex Hermite polynomials. The ﬁrst type is sim- ply the Hermite polynomials in the complex variable z,thatis,{Hn(z)}.These polynomials have been introduced in the study of coherent states [4], [6]. They are deﬁned by [18], [11],

n/2 n!(−1)k (1.1) H (z)= (2z)n−2k,z= x + iy. n k!(n − 2k)! k=0

The second type are the polynomials {Hm,n(z,z¯)} deﬁned by the generating function ∞ um vn (1.2) H (z,z¯) =exp(uz + vz¯ − uv). m,n m! n! m, n=0 These polynomials were introduced by Itˆo [14] and also appear in [7], [1], [4], [19], and [8]. They are essentially the same as in [5, (2.6.6)]. Equation (1.2) yields the

Received by the editors January 25, 2013 and, in revised form, May 31, 2013 and July 18, 2013. 2010 Mathematics Subject Classiﬁcation. Primary 05A15, 05A18, 33C45, 45P05; Secondary 42A65. Key words and phrases. Complex Hermite polynomials, matchings of multisets, orthogonality, combinatorics of linearization of products, eigenvalues, eigenfunctions, integral operators, completeness. The ﬁrst author’s research was supported by the NPST Program of King Saud University; project number 10-MAT1293-02 and King Saud University in Riyadh and by the Research Grants Council of Hong Kong grant # CityU 1014111.

c 2014 American Mathematical Society 1397

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explicit formula ∧ mn m n (1.3) H (z,z¯)= (−1)kk! zm−kz¯n−k. m,n k k k=0

One simple combinatorial interpretation of the polynomials {Hn(x/2)} is apparent from (1.1): edge(π) fix(π) (1.4) Hn(x/2) = (−1/2) x π∈M(n) where M(n) is the set of all unordered matchings π of n vertices, edge(π)isthe number of edges in π (pairs of matched by π vertices) and fix(π)isthenumberof vertices fixed by π (vertices not matched to other vertices by π). It is clear that the explicit formula (1.1) is equivalent to the generating function ([18], [11]), ∞ H (z) (1.5) n tn =exp(2zt − t2). n! n=0

The complex Hermite polynomials {Hn(z)} satisfy the orthogonality relation ([20], [15]), n −ax2−by2 π n a + b (1.6) Hm(x + iy)Hn(x − iy) e dx dy = √ 2 n! δm,n R2 ab ab where 1 1 (1.7) 0

The polynomials {Hm,n(z,z¯)} satisfy the orthogonality relation ([8]), 1 −x2−y2 (1.8) Hm,n(x+iy, x−iy)Hp,q(x + iy, x − iy) e dx dy = m! n! δm,p δn,q. π R2

Let k be a ﬁxed positive integer and let n=(n1,...,nk)beak-tuple of non- ∈ Nk | | k negative integers, that is, n 0.Set n = j=1 nj . Azor, Gillis, and Victor [3], and Godsil [9], independently, found a combinatorial interpretation of the integrals k 1 − 1 | | − 2 √ 2 n x (1.9) A(n)= 2 e Hnj (x) dx π R j=1 as the number of inhomogeneous perfect matchings of a multiset with k sets (components) of sizes (number of elements) n1,...,nk. By multiset we mean a collection of sets or components. The elements of each set are labeled (distinguishable). A matching of one or more multisets is perfect if each element is matched to another element, and inhomogeneous if each element is matched to an element from a diﬀerent set. The pairs of matched elements (edges) are unordered (unoriented). The matching is weighted if a weight is assigned to each pair of matched elements in the matching, and the weight of the matching is the product of the weights of its edges. The total weight of a collection of matchings is the sum of their weights. In what follows we shall often skip the adjective “inhomogeneous” when it is clear from the context that the matchings considered ∈ Nk are inhomogeneous. We say that a multiset has size n =(n1,...,nk) 0 if it

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has k components of respective sizes n1,...,nk. We allow some but not all of the multiset components to be empty. The numbers in (1.9) are also the coefficients in the linearization of products of Hermite polynomials when expanded in Hermite polynomials; see [2] and [11, Chapter 9] for references and motivation. Large parameter asymptotics of such coefficients were studied in [13] from a combinatorial point of view. The Hermite polynomials of a real variable have combinatorial properties. The combinatorics of their moments, explicit representation, and three term recurrence relation is available in Viennot’s Lecture Notes [21]. The combinatorics of the general case of Sheffer orthogonal polynomials is in [16]; see also [22]. A general approach to the combinatorics of linearization coefficients is in [16]. The first combinatorial interpretation of complex Hermite polynomials that we will discuss involves two colored multisets. The following theorem will be proved in Section 2. Theorem 1.1. Suppose that we have two colored multisets. The first, of color I, ∈ Nk ∈ Nk has size m 0 and the second, of color II, has size n 0 . We match elements from different sets, and to each pair of matched elements we assign weight 1 if the elements have the same color. We assign weight 1/a +1/b if the elements have different colors. Then the total weight of all perfect matchings of this type is the number B(m, n) defined by (1.10) √ k ab − 1 | | | | − 2− 2 2 ( m + n ) − ax by B(m, n)= 2 [Hmj (x + iy)Hnj (x iy)] e dx dy. π R2 j=1 Observe that the orthogonality relation (1.6) follows immediately from Theorem 1.1. The following theorem was first proved by K. Gorska [10]. Theorem 1.2. Let a and b satisfy conditions (1.7). Then the complex Hermite polynomials {Hn(z)} satisfy the orthogonality relations −ax2−by2 Hm(x + iy)Hn(x + iy)e dx dy R2 (1.11) −ax2−by2 π n = Hm(x − iy)Hn(x − iy)e dx dy = √ 2 n! δm,n. R2 ab The orthogonality relations in Theorem 1.2 play a fundamental role in the con- struction of coherent states (for example see [20]). A generating function for the numbers {B(m, n)} isderivedinSection2.Us- ing this generating function, we establish the combinatorial interpretation of these numbers given by Theorem 1.1 and then we show that the orthogonality relations (1.6) and (1.11) are simple consequences of this combinatorial interpretation and the generating function (2.1). In Section 2 we also give a generalization of the two-color multiset combinatorics to arbitrary number of colored multisets. At the end of Section 2 we treat a special case of this multiset combinatorics when each multiset consists of a single set. Concerning the polynomials {Hm,n(z,z¯)}, the following combinatorial result is proved in Section 3.

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Theorem 1.3. Suppose that we have two color I multisets B1 and B2 and two color II multisets R1 and R2, each having k components. Multisets B1 and B2 have sizes ∈ Nk ∈ Nk m and p 0 and multisets R1 and R2 have sizes n and q 0, respectively. Consider all inhomogeneous perfect matching of these four multisets where each element is matched to an element of different color. Elements from the same index components of B1 and R1 cannot match each other, and the same restriction holds for B2 and R2. Then the number of perfect matchings of this type is (1.12) k 1 − − −x2−y2 I(m, n, p, q)= Hmj ,nj (x + iy, x iy)Hpj ,qj (x + iy, x iy) e dx dy. π R2 j=1 In Section 3 we first derive an exponential generating function for integrals involving products of polynomials Hm,n(z,z¯), and as a particular case, a generating function for the numbers {I(m, n, p, q)}. The former generating function is used to prove a more general version of Theorem 1.3 for even number of colored multisets which is then specialized to the case of four colored multisets in Theorem 1.3. In Section 4 we construct two integral operators and two operators of Fourier −ax2−by2 type acting on weighted L2-spaces with weights e and constants a and b satisfying (1.7). We show that the polynomials {Hn(x + iy)} are eigenfunctions for these operators and we find the corresponding eigenvalues. In Section 4 we also 2 −ax2−by2 prove that the set {Hm(x + iy)Hn(x − iy)} is complete in L2(R ,e ), and 2 2 we compute the moments of e−ax −by . In our proofs we shall use the following version of the exponential formula for exponential generating functions [17].

Theorem 1.4. Let S1,...,Sk be sets of sizes |Sj | = nj , j =1,...,k,andletM(n) be the collection of all inhomogeneous perfect matchings of the elements of these k { } { ≤ ≤ }⊂C sets, where n =(n1,...,nk).Letalso w = wj1,j2 , 1 j1

2. Combinatorial interpretations of the polynomials {Hn(z)} We ﬁrst derive an exponential generating function of the numbers {B(m, n)}.

Theorem 2.1. Suppose that a and b satisfy (1.7).Then

∞ k smj tnj B(m, n) j j m !n ! m ,...,m ,n ,...,n =0 j=1 j j (2.1) 1 k 1 k a + b =exp (s s + t t )+ s t . i j i j ab i j 1≤i

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Proof. Using (1.5) and (1.10) we write the left hand side of (2.1) as (2.2) √ k √ √ ab −ax2−by2 − 2 − 2 − e exp sj /2+ 2(x + iy)sj tj /2+ 2(x iy)tj dx dy π R2 j=1 √ ab 1 k 1 k 2 1 k 2 = exp − (s2 + t2)+ (s + t ) − (s − t ) π 2 j j 2a j j 2b j j j=1 j=1 j=1 1 k 2 i k 2 × exp −a x − √ (sj + tj) − b y − √ (sj − tj ) dx dy. R2 a 2 j=1 b 2 j=1 Notice that by (1.7) the exponential expression in the second line of (2.2) reduces to the right hand side of (2.1). Next, for α>0andβ ∈ C we have the integral evaluation 2 (2.3) e−α(y−β) dy = π/α. R √ ∈ R − If β , (2.3) follows by setting x = 2α(y β) and using the Gaussian integral √1 −x2/2 ∈ R →∞ R e dx =1.Ifβ/ , (2.3) follows by taking the limit M in 2π 2 e−αz dz =0 γ(M) where M>0andγ(M) is the simple, closed, positively oriented contour formed by the boundary of the parallelogram with vertices√ ±M and ±M − β. By (2.3), the last integral in (2.2) equals π/ ab. This completes the veriﬁcation of (2.1). Proof of Theorem 1.1. The theorem follows from the generating function (2.1) and Theorem 1.4. Orthogonality relation (1.6) follows from the special case k =1ofTheorem1.1 and Theorem 2.1. Proof of Theorem 1.2. Orthogonality relations (1.11) follow from Theorems 1.1 and 2.1 with k = 2 by taking either m =(m, n)andn =(0, 0) or m =(0, 0) and n =(m, n).

We now come to the combinatorics of the polynomials {Hn(x + y)}.

Theorem 2.2. The Hermite polynomials {Hn(x + y)} have the explicit form edge(π) fix(π) Hn((x + y)/2) = (−1/2) (x + y) π∈M(n) (2.4) fix(π) = (−1/2)edge(π) xfixRed(π)yfixBlue(π) fixRed(π) π∈M(n) where fixRed(π) is the number of fixed points which we color red and fixBlue(π) is the number of fixed points which we color blue. Proof. Equation (2.4) follows from equation (1.4) in which we arbitrarily color the fixed points of π in red or blue colors.

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2.1. Combinatorics involving colored multisets. The generating function in Theorem 2.1 and the combinatorial interpretation in Theorem 1.1 can be extended to any number of multisets. As a multivariable analog of the numbers in (1.10) we define the numbers (2.5) B(n1,...,ns) − s | | s 1/2 s k αs ν=1 nν 2 l=1 al − s a x2 = H (L (x)) e l=1 l l dx ···dx s/2 nν,j ν 1 s π Rs ν=1 j=1 ∈ Nk ≥ where nν =(nν,1,...,nν,k) 0 , ν =1,...,s, αs 0, al > 0, l =1,...,s, x =(x1,...,xs), and s (2.6) Lν (x)= ων,lxl,ν=1,...,s l=1 are linear functions with complex coefficients satisfying the conditions s ω2 (2.7) ν,l =1,ν=1,...,s. al l=1 Theorem 2.3. The numbers defined by (2.5)–(2.7) have the generating function

∞ s k nν,j tν,j B(n1,...,ns) nν,j! nν,j =0 ν=1 j=1 1≤ν≤s, 1≤j≤k (2.8) s ω ω −2αs ν1,l ν2,l =exp 2 tν1,j1 tν2,j2 . al (ν1,j1)=( ν2,j2) l=1 1≤ν1,ν2≤s, 1≤j1,j2≤k s/2 s 1/2 Proof. By (2.5) and (1.5) the left hand side of (2.8) times π / l=1 al equals s k ∞ −αs nν,j − s 2 [2 tν,j] l=1 alxl ··· e Hnν,j (Lν (x)) dx1 dxs Rs nν,j! ν=1 j=1 nν,j =0 s k − s 2 − − l=1 alxl 1 αs − 2αs 2 ··· = e exp 2 Lν (x)tν,j 2 tν,j dx1 dxs Rs ν=1 j=1 s k − − 2αs 2 =exp 2 tν,j ν=1 j=1 (2.9) s s s k − × − 2 1 αs ··· exp alxl +2 ων,ltν,j xl dx1 dxs Rs l=1 l=1 ν=1 j=1 s k s s k 2 − − 1 − 2αs 2 2αs =exp 2 tν,j +2 ων,ltν,j al ν=1 j=1 l=1 ν=1 j=1 s − s k 2 2 αs × exp − al xl − ων,ltν,j dx1 ···dxs. Rs al l=1 ν=1 j=1

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By (2.7) the exponential expression in line ﬁve of (2.9) reduces to the right hand s side of (2.8), and by (2.3) the integral in line six of (2.9) reduces to l=1 π/al. This completes the veriﬁcation of (2.8).

We can now state and prove the following generalization of the two-color multiset combinatorics. Theorem 2.4. Suppose that we have s multisets colored in s diﬀerent colors and ∈ Nk having k components each. Let nν =(nν,1,...,nν,k) 0 bethesizeofthecolorν multiset, ν =1,...,s. We match elements from diﬀerent sets and assign a weight to each pair of matched elements in the following way: When an element from component j1 of the color ν1 multiset is matched with an element from component j2 of the color ν2 multiset (with (ν1,j1) =( ν2,j2)), the weight assigned is s − ων ,lων ,l (2.10) 21 2αs 1 2 al l=1

where al > 0, l =1,...,s and {ων,l, 1 ≤ ν ≤ s, 1 ≤ l ≤ s}⊂C satisfy conditions (2.7),andαs ≥ 0. Then the total weight of all such weighted perfect matchings is B(n1,...,ns). Proof. This result follows from the generating function (2.8) and Theorem 1.4.

The case s =2andk = 1 is worth recording separately. In this case from (2.5), (2.7), and (2.8) with αs = α2 =1/2 we obtain the orthogonality relation (2.11) n −ax2−by2 π n αγ βδ Hm(αx + βy)Hn(γx + δy)e dx dy = √ 2 n! + δm,n R2 ab a b where α2 β2 γ2 δ2 (2.12) a>0,b>0, + = + =1. a b a b The orthogonality relations (1.6) and (1.11) are special cases of (2.11).

2.2. Combinatorics involving colored sets. As an application of Theorem 2.4, we consider the following situation. Suppose that we have a collection of s sets of different colors and sizes n1,...,ns. We match elements from different sets and assign weight to each pair of matched elements. When an element from set ν1 is ≤ ≤ matched with an element from set ν2, the weight assigned is wν1,ν2 ,1 ν1 = ν2 s. The weight of such perfect matching is again the product of the weights of its edges. − s Theorem 2.5. Assume that the matrix W =[δj,l +(1 δj,l)wj,l]j,l=1 is real, symmetric, and positive-definite. Then the total weight of all weighted perfect matchings of s colored sets is the number − 1 |n| s s 2 2 − s x2 (2.13) B(n,W)= H ω x e l=1 l dx ···dx s/2 nν ν,l l 1 s π Rs ν=1 l=1 ∈ Ns s where n =(n1,...,ns) and the matrix Ω=[ων,l]ν,l=1 satisfies the matrix equation (2.14) Ω ΩT = W.

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Proof. Since W is positive-deﬁnite, equation (2.14) has a solution of the form Ω = P T D1/2,whereD is a diagonal matrix whose diagonal entries are the eigenvalues of W , P is a unitary matrix whose rows are the corresponding eigenvectors of W ,and W = P T DP. Then Theorem 2.5 follows from Theorems 2.4 and 2.3 with k =1, αs =1/2, al =1,l =1,...,s, and formulas (2.10) and (2.14).

By Sylvester’s criterion, the matrix W is positive-deﬁnite if and only if its prin- cipal minors are positive. Below we consider the special case when wj,l = wj wl, 1 ≤ j = l ≤ s.

Lemma 2.6. s − s − ν−1 − 2 2 (2.15) det[δj,l +(1 δj,l)wjwl]j,l=1 =1+ ( 1) (ν 1)σs,ν (w1,...,ws ) ν=2

where σs,ν denotes the ν-th elementary symmetric function of s variables.

Proof. Let Ds(w1,...,ws) denote the determinant in (2.15). We factor out wj from row j, j =1,...,s. Then we multiply column l by wl, l =1,...,s.Weget − 2 s Ds(w1,...,ws)=det δj,l +(1 δj,l)wl j,l=1 .

Therefore Ds(w1,...,ws) has a unique representation of the form

s 2 2 Ds(w1,...,ws)=1+ cs,ν σs,ν (w1,...,ws ). ν=2

By (2.15) we have Ds(w1,...,ws−1, 0) = Ds−1(w1,...,ws−1), that is,

s−1 s−1 2 2 2 2 1+ cs,ν σs−1,ν (w1,...,ws−1)=1+ cs−1,ν σs−1,ν (w1,...,ws−1). ν=2 ν=2

Hence cs,ν = cs−1,ν , ν =2,...,s− 1. This implies cs,ν = cν,ν, s ≥ ν ≥ 2. So it ν−1 suﬃces to show that cν,ν =(−1) (ν−1). Notice that cs,s is the leading coeﬃcient of the polynomial √ √ − s (2.16) Ps(t)=Ds t,..., t =det[δj,l +(1 δj,l)t]j,l=1.

(r) It is clear from (2.16) that Ps (1) = 0, r =0,...,s− 2. Adding the top s − 1rows to the last row in the determinant in (2.16) shows that (1 + (s − 1)t) divides Ps(t). Since the constant term of Ps(t)is1,weget

s−1 Ps(t)=(1− t) (1+(s − 1)t),

s−1 hence cs,s =(−1) (s − 1).

Corollary 2.7. A suﬃcient condition for the positive deﬁniteness of the determi- − s−1 1/2 nant in (2.15) is |wj | < (s − 2)2 +1 , j =1,...,s.

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3. The combinatorics of {Hm,n(z,z¯)} ∈ Nk ∈ Nk Let mν =(mν,1,...,mν,k) 0, nν =(nν,1,...,nν,k) 0 , ν =1,...,s. Consider the integrals s k −s/2 I(m1, n1,...,ms, ns)=π Hmν,j ,nν,j Lν (x), Lν (x) Rs (3.1) ν=1 j=1 − s x2 ×e l=1 l dx1 ···dxs, s where x =(x1,...,xs)andLν (x)= l=1 ων,lxl, ν =1,...,s, with complex coeﬃ- cients satisfying s 2 (3.2) ων,l =0,ν=1,...,s. l=1 ∈ Cs ∈ Cs s For vectors ω =(ω1,...,ωs) and η =(η1,...,ηs) we set ω η = l=1 ωlηl. Theorem 3.1. (3.3) ∞ s k mν,j nν,j tν,j wν,j I(m1, n1,...,ms, ns) mν,j! nν,j! mν,j ,nν,j =0 ν=1 j=1 ≤ ≤ ≤ ≤ 1 ν s, 1 j k 1 =exp ω ω t t + ω ω w w +2ω ω t w 4 ν1 ν2 ν1,j1 ν2,j2 ν1 ν2 ν1,j1 ν2,j2 ν1 ν2 ν1,j1 ν2,j2 (ν1,j1)=( ν2,j2) 1≤ν1,ν2≤s 1≤j1,j2≤k s k 1 × exp ω ω − 1 t w . 2 ν ν ν,j ν,j ν=1 j=1 Proof. By (3.1) and (1.2) the left hand side of (3.3) times πs/2 equals (3.4) s k ∞ ∞ mν,j nν,j tν,j wν,j − s 2 l=1 xl ··· Hmν,j nν,j Lν (x), Lν (x) e dx1 dxs Rs mν,j! nν,j! ν=1 j=1 mν,j =0 nν,j =0 s k k k s − − 2 ··· = exp Lν (x) tν,j + Lν (x) wν,j tν,jwν,j xl dx1 dxs Rs ν=1 j=1 j=1 j=1 l=1 s s k s k − 2 − ··· = exp xl + (ων,ltν,j + ων,lwν,j)xl tν,jwν,j dx1 dxs Rs l=1 ν=1 j=1 ν=1 j=1 1 s s k 2 s k =exp (ω t + ω w ) − t w 4 ν,l ν,j ν,l ν,j ν,j ν,j l=1 ν=1 j=1 ν=1 j=1 s 1 s k 2 × exp − xl − (ων,ltν,j + ων,lwν,j) dxl. R 2 l=1 ν=1 j=1 By (3.2) the exponential expression in line four of (3.4) reduces to the exponential expression on the√ right hand side of (3.3), while by (2.3) each integral in line ﬁve of (3.4) equals π.

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The combinatorial interpretation of the polynomials {Hm,n(z,z¯)} follows.

Theorem 3.2. Suppose that we have s color I multisets B1,...,Bs and s color II multisets R1,...,Rs, each having k components. Multisets B1,...,Bs have sizes ∈ Nk ∈ Nk m1,...,ms 0 and multisets R1,...,Rs have sizes n1,...,ns 0, respectively. Consider all inhomogeneous perfect matchings of these 2s multisets with weights assigned in the following way: If the matched elements are from components j1 of 1 Bν1 and j2 of Bν2 , the weight is 2 ων1 ων2 ; if they are from components j1 of Rν1 1 and j2 of Rν2 , the weight is 2 ων1 ων2 ; if they are from components j1 of Bν1 and 1 − j2 of Rν2 , the weight is 2 ων1 ων2 δν1,ν2 δj1,j2 . The total weight of all such weighted perfect matchings is I(m1, n1,...,ms, ns). Proof. The theorem follows from the generating function (3.3) and Theorem 1.4. ∈ Nk Corollary 3.3. Let m, n, p, q 0 . The integrals in (1.12) have the exponential generating function ∞ k umj vnj wpj tqj I(m, n, p, q) j j j j mj ! nj ! pj! qj ! mj ,nj ,pj ,qj =0 j=1 (3.5) 1≤j≤k k

=exp (uj1 + wj1 )(vj2 + tj2 )+ (uj tj + vj wj ) .

1≤j1= j2≤k j=1

This result follows immediately from Theorem 3.1 with s =2,m1 = m, n1 = n, m2 = p, n2 = q,variablest1,j = uj , w1,j = vj , t2,j = wj , w2,j = tj , j =1,...,k, x1 = x, x2 = y, and linear functions L1(x)=L2(x)=x + iy. Proof of Theorem 1.3. This theorem is a special case of Theorem 3.2 with s = k =2 and ω1 = ω2 =(1,i). It also follows from the generating function (3.5) and Theorem 1.4. Corollary 3.4. The polynomials {H (z,z¯)} satisfy the orthogonality relation m,n 1 −x2−y2 (3.6) Hm,n(x+iy, x−iy)Hp,q(x+iy, x−iy) e dx dy = m! n! δm,q δn,p. π R2 This corollary is an immediate consequence of the case k = 1 of equations (1.12) and (3.5). It is important to note that (1.3) implies the symmetry relation

(3.7) Hm,n(z,z¯)=Hn,m(z,z¯). Then orthogonality relation (1.8) follows from equations (3.6) and (3.7).

4. Integral operators In this section we discuss integral operators related to the Poisson kernel for {Hn(z)}. Integral operators associated with the polynomials {Hm,n(z,z¯)} are studied in [12]. The Poisson kernel for the polynomials {Hn(z)} is ([11, Section 4.7]), (4.1) ∞ rn 2zwr − (z2 + w2)r2 K (z,w)= H (z)H (w) =(1− r2)−1/2 exp . r n n 2nn! 1 − r2 n=0

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Let a and b satisfy conditions (1.7). Deﬁne the integral operator T1 by

√ ab −aξ2−bη2 (4.2) (T1f)(x + iy)= Kr(x + iy, ξ − iη)f(ξ + iη) e dξ dη. π R2

It is clear from (4.1) and (1.6) that

a + b n (4.3) (T H )(z)= rnH (z),n∈ N . 1 n ab n 0

Another natural integral operator to consider is

√ ab −aξ2−bη2 (4.4) (T2f)(x + iy)= Kr(x + iy, ξ − iη)f(ξ − iη) e dξ dη. π R2

Similarly, from (4.1) and (1.11) it follows that

n (4.5) (T2Hn)(z)=r Hn(z),n∈ N0.

In order to justify the analysis leading to (4.3) and (4.5), we assume |r| < 1. On the other hand it is known ([11, Section 4.6]) that the Fourier transform has n −x2/2 eigenvalues i and eigenfunctions e Hn(x). Thus one is tempted to let r = i in (4.2) and (4.4). This is indeed possible and we state and prove this fact in Theorem 4.1 below. The proof of Theorem 4.1 uses the representation of the monomials in 2 terms of the Hermite polynomials. Multiplying both sides of equation (1.5) by et and equating the coeﬃcients of tn on both sides of the resulting equation, we obtain

n/2 n! H − (z) (4.6) zn = n 2k ,n∈ N . 2n k!(n − 2k)! 0 k=0

Theorem 4.1.

(4.7) √ n ab iz(ξ−iη)+(z2+(ξ−iη)2)/2 −aξ2−bη2 a + b √ e Hn(ξ + iη) e dξ dη = i Hn(z), π 2 R2 ab (4.8) √ ab iz(ξ−iη)+(z2+(ξ−iη)2)/2 −aξ2−bη2 n √ e Hn(ξ − iη) e dξ dη = i Hn(z), π 2 R2

n ∈ N0,wherea and b satisfy conditions (1.7).

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Proof. The left hand sides of (4.7)–(4.8) are √ √ ∞ − m ab z2/2 Hm(z/ 2) i(ξ iη) −aξ2−bη2 √ e √ Hn(ξ ± iη)e dξ dη π 2 R2 m! 2 m=0 √ √ ∞ m ab 2 H (z/ 2) i = √ ez /2 m √ π 2 m! 2 ⎛ m=0 ⎞ m/2 − × ⎝ m! Hm−2k(ξ iη)⎠ ± −aξ2−bη2 (4.9) m Hn(ξ iη)e dξ dη R2 2 k!(m − 2k)! k=0 √ ∞ √ n+2j n ab 2 i 1 π b ± a = √ ez /2 H (z/ 2) √ √ 2nn! n+2j j!n! ab π 2 j=0 2 2 ab √ n ∞ 1 b ± a 2 H (z/ 2) = √ √ i ez /2 n+2j (−1/8)j j! 2 ab 2 j=0 where we applied (1.5) to get the ﬁrst line, (4.6) to get the third line, and the orthogonality relations (1.6) and (1.11) to get the fourth line. To evaluate the j-sum in the last line of (4.9) we use the integral representation ([11, Equation (4.6.41)]), n (2i) −(u−z)2 n (4.10) Hn(iz)= √ e u du. π R Formula (4.10) leads to the evaluation of the j-sum in the following way: ∞ √ ∞ √ n+2j j H (z/ 2) 1 2 (2iu) (−1/8) n+2j (−1/8)j = √ e−(u+iz/ 2) du j! π R j! j=0 j=0 n √ n √ (2i) 2 2 (2i) 2 2 (4.11) = √ e−(u+iz/ 2) +u /2 un du = √ e−z /2 e−(u/ 2+iz) un du π R π R n √ √ (2i) −z2/2 n+1 −(v+iz)2 n n+1 −z2/2 = √ e ( 2) e v dv =( 2) e Hn(z). π R Formulas (4.7)–(4.8) now follow from (4.9) and (4.11). Corollary 4.2. Consider the integral operators √ ab iz(ξ−iη)+(z2+(ξ−iη)2)/2 −aξ2−bη2 (4.12) (F1,2f)(z)= √ e f(ξ ± iη)e dξ dη π 2 R2

where a>1/2 and b>1/2 satisfy conditions (1.7). Then for every n ∈ N0 the polynomial Hn(z) is an eigenfunction of the operators F1 and F2 with eigenvalues a+b n n ab i and i , respectively. One important problem is the expansion of functions in suitable bases for Hilbert spaces.

Theorem 4.3. The polynomials {Hm(x + iy)Hn(x − iy),m,n∈ N0} form a basis 2 −ax2−by2 for the Hilbert space L2(R ,e ) with a and b related via (1.7).

Proof. Clearly the polynomials {Hm(x + iy)Hn(x − iy),m,n∈ N0} span the space 2 −ax2−by2 of all polynomials of x and y.Letf ∈ L2(R ,e ). We may assume that f

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has compact support in which case it is easy to see that f can be uniformly approximated on R2 by linear combinations of functions of the form g(x)h(y)whereg and h are continuous functions of compact support. But each of these latter functions −ax2 can be uniformly approximated on R by weighted polynomials e pm(x)and −by2 e qn(y). Then, the linear combinations of g(x)h(y) are uniformly approximated on R2 by the corresponding linear combination of the products of these weighted polynomials.

2 2 Finally, we record the formulas for the moments of e−ax −by with a and b related via (1.7). Proposition 4.4. 2 2 (x ± iy)2n+1 e−ax −by dx dy =0, R2 (4.13) 2n −ax2−by2 π (x ± iy) e dx dy = √ (1/2)n. R2 ab

Proof. By orthogonality relation (1.11) and the fact that H0(z) = 1, the moment n of (x ± y) is the integral of the coeﬃcient of H0(x ± iy)ontherighthandsideof (4.6). Acknowledgments This work started at the XXIX International Colloquium on Group-Theoretic Methods in Physics where Andrzej Horzela spoke about the 2D-Hermite polynomials and raised the question of studying their combinatorics. The authors thank S. Twareque Ali, Andrzej Horzela, Karol Penson, and Franciszek Szafraniec for references and discussions. The authors thank Vilmos Totik for the argument in Theorem 4.3. References

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Department of Mathematics, University of Central Florida, Orlando, Florida 32816 – and – King Saud University, Riyadh, Saudi Arabia E-mail address: [email protected] Department of Mathematics and Statistics, University of Houston-Downtown, Hous- ton, Texas 77002 E-mail address: [email protected]

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