aa r X i v : . [ m a t h - ph ] F e b Operator orderings and Meixner-Pollaczek polynomials
Genki Shibukawa
Abstract
The aim of this paper is to give identities which are generalizations of the formu-las given by Koornwinder [J. Math. Phys. 30, (1989)] and Hamdi-Zeng [J. Math.Phys. 51, (2010)]. Our proofs are much simpler than and different from the previousinvestigations.
Let W be the Weyl algebra generated by p and q with the relation [ p, q ] := pq − qp = 1. Inthis paper, we prove the following theorems. Theorem 1.1.
We put T := pq + qp . We obtain n m X k =0 (cid:18) mk (cid:19) p k q n p m − k = 2 m n X k =0 (cid:18) nk (cid:19) q k p m q n − k (1.1) = m n ! i − n P ( m − n ) n (cid:16) i ( T + m − n )2 ; π (cid:17) p m − n ( m ≥ n )2 n m ! i − m q n − m P ( n − m ) m (cid:16) i ( T + n − m )2 ; π (cid:17) ( n ≥ m ) . In particular, we have([5]) (1.2) n X k =0 (cid:18) nk (cid:19) p k q n p n − k = n X k =0 (cid:18) nk (cid:19) q k p n q n − k = n ! i − n P ( ) n (cid:18) iT π (cid:19) . Here P ( α ) n ( x ; φ ) is the Meixner-Pollaczek polynomial given by the hypergeometric series P ( α ) n ( x ; φ ) := (2 α ) n n ! e inφ F (cid:20) − n, α + ix α ; 1 − e − iφ (cid:21) . (1.3) Theorem 1.2.
Let T m,n be the sum of all possible terms containing m factors of p and n factors of q . We have T m,n = n !2 n (cid:0) m + nn (cid:1) i − n P ( m − n ) n (cid:16) i ( T + m − n )2 ; π (cid:17) p m − n ( m ≥ n ) m !2 m (cid:0) m + nm (cid:1) i − m q n − m P ( n − m ) m (cid:16) i ( T + n − m )2 ; π (cid:17) ( n ≥ m ) . (1.4) 1 n particular, we have([6],[5],[4]) (1.5) T n := T n,n = n !2 n (cid:18) nn (cid:19) i − n P ( ) n (cid:18) iT π (cid:19) . The formula (1.5) for T n was first observed by Bender, Mead and Pinsky([2]), and provedby Koorwinder([6]). The idea of the proof in [6] is to consider the irreducible unitary rep-resentations of the Heisenberg group and some analysis for special functions. Moreover, acombinatorial proof was given by Hamdi and Zeng([5]). They used the rook placement in-terpretation of the normal ordering of the Weyl algebra and gave also a proof of (1.2), whichwas first observed by [3]. Our results extend these to general m and n .The proofs given in this paper are much simpler than the investigations([6], [3]). Actually,we only use some basic properties of the Weyl algebra and a certain transformation formulaof the hypergeometric function. Our proofs clarify the reason why (1.2) and (1.5) are equalup to constant, which is not explained in [5]. The operations L A , R A ∈ End C ( W ) are respectively left and right multiplications, that is,(2.1) L A .X := AX, R A .X := XA, ( A, X ∈ W ) . We introduce some useful operators([7]).(2.2) ˇad( A ) := L A + R A . We remark that
L, R : W → End C ( W ) are linear, hence ˇad is also linear. In addition, sinceˇad( A ) N . N A N , we obtain the following lemma immediately. Lemma 2.1.
Let t , · · · , t n be indeterminates. For any N ∈ Z ≥ , we obtain (2.3) ( n X k =1 t k ˇad( A k ) ) N . N ( n X k =1 t k A k ) N . In particular, we have (2.4) ( t ˇad( p ) + t ˇad( q )) N . N ( t p + t q ) N . Remark 2.2.
When N = n in Lemma 2.1, comparing the coefficients of t · · · t n on bothsides of the (2.3), we obtain the following formula immediately.(2.5) F ( ˇad( A n )) . n F ( A n ) . Here, A n := ( A , · · · , A n ) , ˇad( A n ) := ( ˇad( A ) , · · · , ˇad( A n )) and F ( A n ) := X σ ∈ S n A σ (1) · · · A σ ( n ) , F ( ˇad( A n )) := X σ ∈ S n ˇad( A σ (1) ) · · · ˇad( A σ ( n ) ) . (2.6) 2 emma 2.3. The operators ˇad( p ) and ˇad( q ) are commutative.Proof. Obviously L A and R B are commutative. Since L is a homomorphism and R is ananti-homomorphism, we have[ ˇad( p ) , ˇad( q )] = [ L p + R p , L q + R q ] = [ L p , L q ] + [ R p , R q ] = L pq − qp − R pq − qp = 0 . Proposition 2.4. (2.7) ˇad( p ) m ˇad( q ) n . n m X k =0 (cid:18) mk (cid:19) p k q n p m − k = 2 m n X k =0 (cid:18) nk (cid:19) q k p m q n − k . Proof.
Since L A and R B are commutative, L is a homomorphism and R is an anti-homomorphism,we obtainˇad( p ) m ˇad( q ) n . L p + R p ) m . n q n = 2 n m X k =0 (cid:18) mk (cid:19) L p k R p m − k .q n = 2 n m X k =0 (cid:18) mk (cid:19) p k q n q m − k . On the other hand, since ˇad( p ) and ˇad( q ) are commutative, we haveˇad( p ) m ˇad( q ) n = ˇad( q ) n ˇad( p ) m . Hence, the second equality of (2.7) can be proved in the same way.
Remark 2.5.
Wakayama([7]) has constructed the oscillator representation of the simple Liealgebra sl by ˇad and ad in End C ( W ) and then, proves that ˇad( p ) n ˇad( q ) n . T = pq + qp and pq − qp = 1, we have(2.8) pq = T + 12 , qp = T − . The proof of the following lemma is straightforward.
Lemma 2.6. (1)
Let f ( T ) ∈ C [ T ] , l ∈ Z ≥ . We have (2.9) p l f ( T ) = f ( T + 2 l ) p l , q l f ( T ) = f ( T − l ) q l . (2) For any l ∈ Z ≥ , we have (2.10) p l q l = (cid:18) T (cid:19) l , q l p l = ( − l (cid:18) − T (cid:19) l . Here, ( x ) l := x ( x + 1) · · · ( x + l − , ( x ) := 1 . roposition 2.7. (2.11) n ! i − n P ( α ) n (cid:18) ix π (cid:19) = n X k =0 (cid:18) nk (cid:19) ( − k (cid:16) α − x (cid:17) k (cid:16) α + x (cid:17) n − k . Proof.
It follows from the formula (2.3.14) in [1] that(
LHS ) = (2 α ) n F (cid:20) − n, α − x α ; 2 (cid:21) = (cid:16) α + x (cid:17) n F (cid:20) − n, α − x − n − α − x + 1; − (cid:21) = ( RHS ) . Remark 2.8.
One may also prove this proposition using the generating function for Meixner-Pollaczek polynomials.We now prove Theorem 1.1 as follows. If m ≥ n ,2 m n X k =0 (cid:18) nk (cid:19) q k p m q n − k = 2 m n X k =0 (cid:18) nk (cid:19) q k p k p m − n p n − k q n − k = 2 m n X k =0 (cid:18) nk (cid:19) ( − k (cid:18) − T (cid:19) k p m − n (cid:18) T (cid:19) n − k = 2 m n X k =0 (cid:18) nk (cid:19) ( − k (cid:18) − T (cid:19) k (cid:18) T m − n (cid:19) n − k p m − n = 2 m n ! i − n P ( m − n ) n (cid:18) i ( T + m − n )2 ; π (cid:19) p m − n . The second equality follows from (2.10), the third from (2.9) and the fourth from (2.11). ByProposition 2.4, the case of n ≥ m can be proved in the same way. Comparing the coefficients of t m t n on both sides in (2.4) for N = m + n , one obtain the keyProposition. Proposition 3.1.
For any m, n ∈ N , we have (3.1) T m,n = 12 m + n ( m + n )! m ! n ! ˇad( p ) m ˇad( q ) n . . Theorem 1.2 follows immediately from (3.1), (2.7) and (1.1).
Remark 3.2. (1) If m ≥ n , then we have the following result immediately by Theorem 1.2and (2.10).(3.2) T m,n q m − n = n !2 n (cid:18) m + nn (cid:19) i − n (cid:18) T (cid:19) m − n P ( m − n ) n (cid:18) i ( T + m − n )2 ; π (cid:19) . n ≥ m is similar.(2) If m ≥ n , then a explicit expression of the Poincare-Birkhoff-Witt theorem for T m,n follows from (1.4), (1.3) and (2.10). T m,n = 12 n m !( m − n )! (cid:18) m + nn (cid:19) X k ≥ (cid:18) nk (cid:19) k (1 + m − n ) k q k p k + m − n . (3.3)The case of n ≥ m is similar.Recently, a generalization of Theorem 1.2 using the multivariate Meixner-Pollaczek poly-nomials in the framework of the Gelfand pair has been established in [4]. Another proof of[4] in our current approach would be desirable. Acknowledgment
The author would like to thank Professors Masato Wakayama and Hiroyuki Ochiai for manyhelpful comments. This work has been supported by the JSPS Research Fellowship.
References [1] G. E. Andrews, R. Askey and R. Roy:
Special Functions , Encyclopedia of Mathe-matics and its Applications, 71. Cambridge University Press, 1999.[2] C. M. Bender, L. R. Mead and S. S. Pinsky: Resolution of the Operator-OrderingProblem by the Method of Finite Elements, Phys. Rev. Lett. (1986), 2445-2448.[3] C. M. Bender and G. V. Dunne: Polynomials and operator orderings, J. Math. Phys. (1988), 1727-1731.[4] J. Faraut and M. Wakayama: Invariant differential operators on the Heisenberggroup and Meixner-Pollaczek polynomials, Adv. Pure Appl. Math to appear (2013).[5] A. Hamdi and J. Zeng: Orthogonal polynomials and operator orderings, J. Math.Phys. , 043506 (2010).[6] T. H. Koornwinder: Meixner-Pollaczek polynomials and the Heisenberg algebra, J.Math. Phys.30