aa r X i v : . [ m a t h . C O ] J un ORTHOGONAL POLYNOMIALS AND OPERATOR ORDERINGS
ADEL HAMDI AND JIANG ZENG
Abstract.
An alternative and combinatorial proof is given for a connection between asystem of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra,which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math.Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the samevein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connectinga special one-parameter class of Hermitian operator orderings and the continuous Hahnpolynomials are also proved. Introduction
The Meixner-Pollaczek polynomials are defined by P ( a ) n ( x ; φ ) = (2 a ) n n ! e inφ F ( − n, a + ix ; 2 a ; 1 − e − iφ ) . Consider the following special Meixner-Pollaczek polynomial: S n ( x ) = n ! P (1 / n (cid:18) x ; 12 π (cid:19) = i n n ! n X k =0 ( − k k ! (cid:18) nk (cid:19) k − Y j =0 ( ix + 1 + 2 j ) , (1)which turns out to be the orthogonal polynomial of degree n on R with respect to the weightfunction x / cosh( πx/ S n +1 ( x ) = xS n ( x ) − n S n − ( x )with S ( x ) = 1. The first values of these polynomials are as follows: S ( x ) = x, S ( x ) = x − , S ( x ) = x − x, S ( x ) = x − x + 9 . Let S n be the set of permutations on { , , . . . , n } . For any σ ∈ S n let cyco σ be the numberof cycles in σ of odd length. Then it is easy to see that the polynomial S n ( x ) has the followingcombinatorial interpretation: S n ( x ) = ( − i ) n X σ ∈ S n ( ix ) cyco σ . It is interesting to note that the corresponding moment is the secant number E n defined by X n ≥ E n x n (2 n )! = 1cos x . If [ q, p ] := qp − pq = i, (2) Date : November 11, 2018. and T n is the sum of all possible terms containing n factors of p and n factors of q , thenBender, Mead and Pinsky [1, 2] first observed and Koornwinder [6] proved the followingresult T n = (2 n − n ! S n ( T ) . (3)For example, we have T = pq + qp and T = T + p q + q p . It follows from (2) that[ q, p ] = ( qp ) − qp q − pq p + ( pq ) = −
1. Hence T − qp ) + ( pq ) ) and p q + q p + 1 = p ( pq + i ) q + q ( qp − i ) p = ( pq ) + ( qp ) = 12 ( T − . (4)Finally we get T = ( T −
1) + ( p q + q p + 1) = S ( T ).To prove (3), Koornwinder [6] made use of the connection between Laguerre polynomialsand Meixner-Pollaczek polynomials, the Rodrigues formula for Laguerre polynomials, anoperational formula involving Meixner-Pollaczek polynomials, and the Schr¨odinger modelfor the irreducible unitary representations of the three-dimensional Heisenberg group. Tothe best knowledge of the authors Koornwinder’s proof is the only published one for (3). Inthis paper we shall give an elementary proof of (3) using the rook placement interpretationof the normal ordering of two non commutative operators. See [7], and also [8, 4] for tworecent papers on this theory.On the other hand, Bender and Dunne [3] discussed a correspondence between polynomialsand rules for operator orderings. More precisely, given two operators q and p satisfying (2),they consider the possible operator orderings O as a sum O ( q n p n ) = P nk =0 a ( n ) k q k p n q n − k P nk =0 a ( n ) k , where the coefficients a ( n ) k may be chosen arbitrarily with a (0)0 = a (1)0 = a (1)1 = 1. Hence O ( q p ) = 1 and O ( q p ) = ( qp + pq ). They pointed out that with every operator corre-spondence rule O one can associate a class of polynomials p n ( x ) defined by O ( q n p n ) = p n [ O ( qp )] . (5)If a ( n ) k = (cid:0) nk (cid:1) , their result is equivalent to O ( q n p n ) := n X k =0 (cid:18) nk (cid:19) q k p n q n − k = S n ( T ) . (6)Their polynomial is actually equal to 2 − n S n (2 x ). For example, we have O ( q p ) = q p + 2 qp q + p q = T −
32 + 2 T − i · T + i T − . If a ( n ) k = (cid:18) n + lk (cid:19)(cid:18) n + lk + l (cid:19)(cid:18) n + ll (cid:19) − , (7) RTHOGONAL POLYNOMIALS AND OPERATOR ORDERINGS 3 where l is an arbitrary parameter, Bender and Dunne [3] observed that the correspondingpolynomials belong to the large class of continuous Hahn polynomials. An explicit formulafor the n th polynomial is P n ( x ) = i n (cid:18) n + 2 ln (cid:19) − Γ( n + l + 1)Γ( l + 1) F ( − n, n + 2 l + 1 ,
12 + ix ; 1 , l + 1; 1) . Bender and Dunne [3] announced the following result O ( q n p n ) := P nk =0 a ( n ) k q k p n q n − k P nk =0 a ( n ) k = P n ( T / . (8)Note that the denominator has a closed formula D n := n X k =0 a ( n ) k = (cid:18) n + ll (cid:19) − n X k =0 (cid:18) n + lk (cid:19)(cid:18) n + ln − k (cid:19) = (cid:18) n + ll (cid:19) − (cid:18) n + 2 ln (cid:19) . For example, by (4) we have O ( q p ) = 12 1 + l l ( p q + 2 2 + l l qp q + q p )= 12 1 + l l (cid:18) T −
32 + 2 + l l ( T + i )( T − i )2 (cid:19) = T −
14 1 + 2 l l = P ( T / . Since (6) and (8) were announced without proof, we shall provide a proof similar to that of(3).We shall first recall briefly the rook theory of normal ordering in Section 2 and thenprove (3), (6) and (8) in Sections 3, 4 and 5, respectively.2.
Rook placements and the normal ordering problem
Let D and U be two operators satisfying the commutation relation [ D, U ] = 1. Then thealgebra generated by D and U is the Weyl algebra. Each element of this algebra, identifiedas a word w on the alphabet { D, U } , can be uniquely written in the normally ordered formas w = X r,s c r,s U r D s . The coefficients c r,s can be computed using the rook theory. The reader is referred to [8, 4]for more details. Given a word w with n letters U ’s and m letters D ’s, we draw a latticepath from ( n,
0) to (0 , m ) as follows: read the word w from left to right and draw a unit lineto the right (resp. down) if the letter is D (resp. U ). This lattice path outlines a Ferrersdiagram B w as illustrated in Figure 1. ADEL HAMDI AND JIANG ZENG w = DU DDU DU = ⇒ Figure 1. Correspondence between words and Ferrers diagramsThe commutation rule DU = U D + 1 implies that the normal writing of w amounts toreplacing successively each DU by U D or 1, this procedure amounts to deleting each up-right-most corner of the Ferrers board or deleting it with its row and column. Let r k ( B ) bethe number of placing k ( k ≥
0) non-attaking rooks on the Ferrers board B w . It is known(see [7, 8]) that w = X k r k ( B w ) U n − k D m − k . (9)Now, it follows from [8, Theorem 5.1] that X B ⊆ [ n ] × [ n ] r k ( B ) = (2 n )!2 k k !( n − k )!( n − k )! , (10)where the sum is over all the Ferrers diagrams contained in the square [ n ] × [ n ] and outlinedby the lattice paths starting from (0 , n ) and ending at ( n, T n ( D, U ) be the sum ofall the words with n letters D and n letters U . We derive from (9) and (10) that T n ( D, U ) = n X k =0 (2 n )!2 k k !( n − k )!( n − k )! U n − k D n − k . (11)On the other hand, let x := DU + U D = 2
U D + 1, then
U D = x − . (12)It is also folklore (see [5, p. 310]) that U n D n = n X k =0 s ( n, k )( U D ) k , (13)where s ( n, k ) is the Stirling number of the first kind . Since n X j =0 s ( n, j ) t j = t ( t − · · · ( t − n + 1) , we can rewrite (13) as U n D n = n Y j =1 ( U D − j + 1) . (14) RTHOGONAL POLYNOMIALS AND OPERATOR ORDERINGS 5 Proof of equation (3)If we set D = − iq and U = p , then equation (2) becomes [ D, U ] = 1 and the algebragenerated by D and U is the Weyl algebra. Substituting (14) and (12) into (11), we obtain,by replacing k by n − k , T n ( D, U ) = (2 n − n X k =0 ( − k k ! (cid:18) nk (cid:19) k − Y j =0 ( − x + 2 j + 1) = ( − i ) n (2 n − n ! S n ( ix ) . Since T n ( D, U ) = ( − i ) n T n ( q, p ), letting T = pq + qp = ix , we derive T n ( q, p ) = (2 n − n ! S n ( T ) , which is exactly (3). 4. Proof of equation (6)For the commutation relation DU − U D = 1 and for n ≥ k ≥
0, it is readily seen, byinduction on k , that D k U n = k X j =0 (cid:18) kj (cid:19) n j U n − j D k − j , (15)where n j = n ( n − ... ( n − j + 1) = n ! / ( n − j )!, then O ( D n U n ) = n X k =0 (cid:18) nk (cid:19) k X j =0 (cid:18) kj (cid:19) n j U n − j D n − j = n X j =0 (cid:18) nj (cid:19) n − j n j U n − j D n − j . Substituting (14) and (12) in the last sum, we obtain, by replacing n − j by k , O ( D n U n ) = n X k =0 ( − k (cid:18) nk (cid:19) n n − k k − Y j =0 ( − x + 2 j + 1) = n X k =0 ( − k (cid:18) nk (cid:19) n ! k ! k − Y j =0 ( − x + 2 j + 1) . As T = qp + pq = ix we derive O ( q n p n ) = i n O ( D n U n ) = S n ( ix ). ADEL HAMDI AND JIANG ZENG Proof of equation (8)By definition and (15), we have O ( D n U n ) = (cid:18) n + 2 ln (cid:19) − n X k =0 (cid:18) n + lk (cid:19)(cid:18) n + lk + l (cid:19) D k U n D n − k = (cid:18) n + 2 ln (cid:19) − n X k =0 (cid:18) n + lk (cid:19)(cid:18) n + lk + l (cid:19) k X j =0 (cid:18) kj (cid:19) n j U n − j D n − j = (cid:18) n + 2 ln (cid:19) − n X j =0 (cid:18) nj (cid:19) U n − j D n − j ( n + l )!( n + l − j )! (cid:18) n + 2 l − jn − j (cid:19) . Substituting (14) and (12) and replacing j by n − k we obtain O ( D n U n ) = (cid:18) n + 2 ln (cid:19) − ( n + l )! l ! n X k =0 (cid:18) nk (cid:19) ( n + 2 l + 1) k ( l + 1) k ( − k k k ! k − Y j =0 ( − x/ / j ) . Since O ( q n p n ) = i n O ( D n U n ) and T = ix , we derive Eq. (8). Acknowledgments : This work was partially supported by the French National ResearchAgency through the grant ANR-08-BLAN-0243-03, and initiated during the first author’svisit to Institut Camille Jordan, Universit´e Lyon 1 in the summer of 2009.
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Concrete Mathematics , Addison-Wesley Publishing Co.1989.[6] T. Koornwinder, Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (1989),no. 4, 767–769.[7] A. M. Navon, Combinatorics and fermion algebra, Il Nuovo Cimento B (1971-1996), 16 (1973), 324-330.[8] A. Varvak, Rook numbers and the normal ordering problem, J. Combin. Theory Ser. A 112 (2005), no.2, 292–307.
Faculty of Science of Gabes, Department of Mathematics, Cit´e Erriadh 6072, Zrig, Gabes,TUNISIA
E-mail address : aadel [email protected] Universit´e de Lyon, Universit´e Lyon 1, Institute Camille Jordan, UMR 5208 du CNRS,43, Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, FRANCE
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