New q-Laguerre polynomials having factorized permutation interpretations

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

New q-Laguerre polynomials having factorized permutation interpretations ✩ Gi-Sang Cheon a,b , Ji-Hwan Jung b,∗ , Suh-Ryung Kim b,c a

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea Applied Algebra and Optimization Research Center, Sungkyunkwan University, Suwon 16419, Republic of Korea c Department of Mathematics Education, Seoul National University, Seoul 08826, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 9 April 2018 Available online xxxx Submitted by S. Cooper

a b s t r a c t In this paper, we generalize the Laguerre polynomials in terms of q-analogue for Riordan matrices. To be more specific, for α ∈ N0 , we introduce new q-Laguerre (α) (α) polynomials Ln (x; q) by defining the Eulerian generating function for Ln (x; q) as ⎛

Keywords: q-Riordan matrix q-Laguerre polynomials Permutations q-Rook numbers

⎝

α  j=0

⎞   xz 1 ⎠ eq . j+1 1+q z 1+z (α)

Interestingly, it turns out that Ln (x; q) have combinatorial descriptions in the aspect of the inversions of factorized permutations and q-rook numbers. We locate their zeros and develop their algebraic properties as well. © 2018 Elsevier Inc. All rights reserved.

1. Introduction For a real number α, the polynomial solutions denoted by Ln (x) of the differential equation xy  + (α + (α) 1 − x)y  + ny = 0 are called generalized Laguerre polynomials. Throughout this paper, we consider Ln (x) as a monic polynomial of degree n. It is well-known that (α)

L(α) n (x)

=

n k=0

 n+α k x k! k + α

n−k n!

(−1)

(1)

✩ This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2016R1A5A1008055) and the Ministry of Education (NRF-2016R1A6A3A11930452). * Corresponding author. E-mail addresses: [email protected] (G.-S. Cheon), [email protected] (J.-H. Jung), [email protected] (S.-R. Kim).

https://doi.org/10.1016/j.jmaa.2018.09.057 0022-247X/© 2018 Elsevier Inc. All rights reserved.

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which is derived by applying Leibniz’s theorem for differentiation of a product to Rodrigues’ formula [16]. Laguerre polynomials are closely related to the Riemann hypothesis (see [3,18,21]) and have been extensively studied analytically and combinatorially by many authors. They are classical orthogonal polynomials satisfying the three term recurrence relation: (α)

(α)

Ln+1 (x) = (x − (2n + α + 1))L(α) n (x) − n(n + α)Ln−1 (x).

(2)

It is also well-known [10] that the Laguerre polynomials are closely related to rook polynomials. For α ∈ N0 and n ∈ N, the rook polynomial for a board B ⊆ [n + α] × [n] is defined by RB (x) =

n

(−1)k rk xn−k

k=0

where rk denotes the number of all placements of k non-attacking rooks on B i.e., rooks not in the same column or in the same row. If B is an (n + α) × n complete board, then the rook polynomial of B coincides (α) with the Laguerre polynomial Ln (x). Various q-analogues of Laguerre polynomials have been introduced and studied (see [7,15,17,19,20]). Among q-analogues of Laguerre polynomials, the ones given by Garsia and Remmel [7] and by Kasraoui et al. [15] include a combinatorial description. In 1980, Garsia and Remmel [7] introduced a q-analogue of the (α) Laguerre polynomial Ln (x) by generalizing the explicit formula (1) as L(α) n (x|q)

=

n

q

  n+α xk [k]q ! k + α q

k(k+α) [n]q !

k=0

where [k]q ! = [1]q [2]q · · · [k]q for [k]q = 1 +q+· · ·+q k−1 with [0]q = 0 and (α) Ln (x|q)

n k q

=

[n]q ! [k]q ![n−k]q ! .

The polynomials

(α) Ln (−x)

reduce to (−1) when q → 1. Another q-analogue was introduced by Kasraoui, Stanton and Zeng [15] in 2011. They defined q-Laguerre polynomials n

√ Ln (x, y; q) =

y q−1



n Qn

(q − 1)x + y + 1 1 √ ; √ , yq|q √ 2 y y

 (3)

by re-scaling Al-Salam–Chihara polynomials Qn (x) := Qn (x; a, b|q) [1] which form a family of basic hypergeometric orthogonal polynomials defined by the recurrence relation: Qn+1 (x) = (2x − (a + b)q n )Qn (x) − (1 − q n )(1 − abq n−1 )Qn−1 (x), (n ≥ 1)

(4)

with initial values Q0 (x) = 1 and Q1 (x) = 2x − (a + b). Then they derived a combinatorial interpretation from the Simion and Stanton’s combinatorial model for octabasic Laguerre polynomials [20], and showed that Ln (x, y; q) satisfies the three-term recurrence relation: Ln+1 (x, y; q) = (x − y[n + 1]q − [n]q )Ln (x; q) − y[n]2q Ln−1 (x; q),

(n ≥ 1)

(5)

where L0 (x, y; q) = 1 and L1 (x, y; q) = x −y(1 +q) −1. We note that (5) reduces to the three-term recurrence relation (2) only when α = 0, y = 1 and q → 1. (α) (α) In this paper, we introduce a new q-analogue, denoted by Ln (x; q), of the polynomials Ln (x) defined as the q-Sheffer [5] for the pair ⎛

α 

⎞

z ⎠ 1 , . j+1 z 1 + z 1 + q j=0

⎝

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(α)

The Eulerian generating function for Ln (x; q) tends to the exponential generating function for the poly(α) (α) nomials Ln (x) as q → 1 and thus Ln (x; q) may be regarded as a natural extension of the generalized (α) (α) Laguerre polynomials Ln (x). The coefficients of the polynomials Ln (x; q) have a nice combinatorial interpretation in terms of inversions of factorized permutations (Theorem 3.4), which leads to a combinatorial interpretation involving q-rook numbers introduced by Garsia and Remmel [8] (Theorem 4.1). Furthermore, (α) Ln (x; q) are orthogonal and satisfy the three-term recurrence relation which is reduced to the three-term recurrence relation (2) for any α ∈ N0 (Theorem 5.2). The identities (Theorem 5.3 and Corollary 5.4) and (α) the fact that the polynomials Ln (x; q) with n ≥ 1 have only positive real zeros (Theorem 5.5) also support (α) the validity of Ln (x; q) as the desired q-analogues of generalized Laguerre polynomials. As a matter of fact, Gessel [10, p. 174] suggested finding a rook polynomial interpretation for the orthogonality, in the usual sense, of q-Laguerre polynomials. We hope that our q-version of Laguerre polynomials shall provide clues to this problem. 2. The new q-Laguerre polynomials Let us begin with some further terminology used in [6,9,13]. The kth symbolic power f [k] of f is inductively defined by Dq f [k] (z) = [k]q f [k−1] (z)Dq f (z) for k ≥ 1

and f [0] (z) = 1.

Here, Dq f denotes the q-derivative of f defined by f (z) − f (qz) . z − qz

Dq f (z) =

For n ∈ N0 , let Eq (n) be the set of Eulerian generating functions of the form an

zn z n+1 z n+2 + an+1 + an+2 + ··· , [n]q ! [n + 1]q ! [n + 2]q !

(an = 1).

A polynomial sequence {sn (x; q)}n∈N0 is said to be a q-Sheffer [5] for a pair (g, f ) if there exist g ∈ Eq (0) and f ∈ Eq (1) such that

sn (x; q)

n≥0

zn = geq [xf ] [n]q !

  zn where eq (z) = n≥0 z n /[n]q ! is the q-analogue of the exponential function ez and g[f ] for g = n≥0 gn [n] q! is the q-composition of g with f defined by g[f ] =

n≥0

gn

f [n] . [n]q !

n We call sn (x; q) the q-Sheffer polynomial of degree n. Let sn (x; q) = k=0 dn,k xk . It is also known [5] that the sequence {sn (x; q)}n∈N0 is a q-Sheffer for a pair (g, f ) if and only if the coefficient matrix [dn,k ]n,k∈N0 representing the polynomials sn (x; q) forms the q-Riordan matrix denoted (g, f )q whose kth column generating function is n≥k

dn,k

f [k] zn =g . [n]q ! [k]q !

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  (α) For α ∈ N0 , we now define the q-Laguerre polynomial sequence Ln (x; q) to be a q-Sheffer for the n∈N0   α 1 z pair j=0 1+q j+1 z , 1+z . Equivalently, the Eulerian generating function is ⎛ ⎞   α n  xz z 1 (α) ⎝ ⎠ = . Ln (x; q) eq [n]q ! 1 + q j+1 z 1+z j=0

n≥0

(6)

Note that if q → 1 then the equation (6) gives the exponential generating function for the polynomials (α) Ln (x) as follows: ∞

L(α) n (x)

n=0 (α)

If we denote Ln (x; q) =

n

(α) k k=0 n,k x



(α)

n,k

xz 1 zn = e 1+z . α+1 n! (1 + z)

(α)

then the matrix [n,k ]n,k∈N0 is the q-Riordan matrix given by ⎛⎛

 n,k∈N0

⎞ ⎞ 1 z ⎠, ⎠ . = ⎝⎝ 1 + q j+1 z 1+z j=0 α 

(7)

q

3. A combinatorial interpretation of the q-Laguerre polynomials (α)

In this section, we give a combinatorial interpretation for Ln (x; q) by using the concept of inversions of a permutation. Let [n] denote the n-set {1, 2, . . . , n} and Πn,k be the set of k-partitions {B1 , . . . , Bk } of [n] where min B1 < · · · < min Bk and the elements of each block Bi = {bi1 , . . . , bik } are arranged in increasing order. An inversion of a permutation ω = w1 w2 · · · wn is a pair (wi , wj ) such that i < j and wi > wj , and the number of inversions of ω is denoted by inv(ω). To find the inversions of a partition π ∈ Πn,k , we arrange it appropriately, remove the braces to obtain a permutation, and then take the inversions of this permutation. The number of inversions of π is denoted by inv(π). The following theorem is useful for giving a combinatorial interpretation of q-Riordan matrices.  zn Proposition 3.1. [6] Let F (z) = n≥0 f (n) [n] ∈ Eq (1) be a function associated to the counting function q! f : N0 → C[[q]] where f (0) = 0. Then hn,k is the (n, k)-entry of the q-Riordan matrix given by (1, F )q if and only if hn,k may be combinatorially interpreted as hn,k =



f (|B1 |) · · · f (|Bk |)q inv(π) (n ≥ k ≥ 1)

π={B1 ,...,Bk }∈Πn,k

with h0,0 = 1 and hn,0 = 0 for n ≥ 1. Now we denote by Fn,k the collection of k-factorized permutations ρ1 /ρ2 / · · · /ρk of [n] where the ρi are nonempty permutations satisfying min ρ1 < · · · < min ρk . In particular, Fn,1 is the collection of permutations of [n]. We also denote by Pn,k the collection of permutations of the set [n] with k disjoint cycles. Let σ = C1 C2 · · · Ck ∈ Pn,k . In this paper, the least element of each cycle in σ is written first and min C1 < · · · < min Ck . For instance, F3,2 and P3,2 are respectively given as follows: F3,2 = {1/23, 1/32, 12/3, 21/3, 13/2, 31/2} , P3,2 = {(1)(23), (12)(3), (13)(2)} .

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An entry of a permutation is called a right-to-left minimum if it is smaller than all the entries on its right. Given a k-factorized permutation ρ = ρ1 /ρ2 / · · · /ρk , we call ρi the ith factor of ρ. In addition, for a k-cyclic permutation σ = C1 C2 · · · Ck , we call Ci the ith cycle of σ. For a nonnegative integer α, we define (α) the set Fn,k as follows: (α)

(i) if α ≥ 1 then Fn,k is the set of elements in Fn+α,k such that 1, 2, . . . , α are in the first factor of ρ and they are right-to-left minima; (α) (ii) if α = 0 then Fn,k is the set of elements in Fn+1,k such that the entry 1 is written first in the first factor of ρ, (α)

and the set Pn,k as (iii) the set of elements in Pn+α,k such that the entry i belongs to the ith cycle of σ for 1 ≤ i ≤ α ≤ k. (0)

(1)

(α)

(α)

In particular, Fn,1 = Pn,1 . For ρ ∈ Fn,k and σ ∈ Pn,k , we define inv(ρ) and inv(σ) in a similar way as the (2)

(2)

number of inversions of a k-partition π ∈ Πn,k . For instance, for 142/73/65 ∈ F5,3 and (143)(27)(56) ∈ P5,3 , we mean inv(142/73/65) = inv(1427365) and inv((143)(27)(56)) = inv(1432756). (α)

(α−1)

Lemma 3.2. For an integer α ≥ 1, there is a bijection ψ : Pn,α → Fn,1 all σ ∈

satisfying inv(σ) = inv (ψ(σ)) for

(α) Pn,α . (α)

(α−1)

(α)

as follows. For σ ∈ Pn,α , let σ = (c1,1 · · · c1,a1 )(c2,1 · · · Proof. Let us define the map ψ : Pn,α → Fn,1 α c2,a2 ) · · · (cα,1 · · · cα,aα ) where ci,1 = i and k=1 ak = n + α. If α = 1 then let ψ be the identity map and if α ≥ 2 then let ψ (σ) =  c1,2 . . .  c1,a1  c2,1 . . .  c2,a2 . . .  cα,1 . . .  cα,aα (3)

(2)

where  ci,j = ci,j − 1. For instance, if σ = (175)(284)(36) ∈ P5,3 then ψ(σ) = 6417325 ∈ F5,1 and inv(σ) = inv(ψ(σ)) = 12. (α) Clearly the map ψ is injective and inv(σ) = inv (ψ(σ)) for all σ ∈ Pn,α . Now it is enough to show that (α−1)

(α) |Pn,α | = |Fn,1

It is known [2] that the α-Stirling numbers particular,

n+α k

α

| for α ≥ 2.

of the first kind are defined by

(8) n+α k

  n+α (n + α − 1)! (α) = |Pn,α |= (α − 1)! α α

α

(α)

= |Pn,k |. In

(9)

(α−1)

(see [2, p. 243]). Since an element in Fn,1 is a permutation of the set [n + α − 1] in which the first α − 1 elements are right-to-left minima, we obtain (α−1)

|Fn,1

|=

(n + α − 1)! . (α − 1)!

By (9) and (10), we obtain (8) and complete the proof. 2 The following lemma is obtained from [4, (ii) of Lemma 2.1] when s → ∞ and r = α + 1.

(10)

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Lemma 3.3. For α ∈ N0 , we have α 

1 1 − qj z j=0

⎛ ⎜ = ⎝ n≥0

⎞

⎟ zn . q inv(σ) ⎠ [n]q ! (α+1)

σ∈Pn,α+1

We are now ready to give a combinatorial interpretation for the coefficients of the q-Laguerre polynomials. (α)

Theorem 3.4. The q-Laguerre polynomials Ln (x; q) can be expressed as ⎛ L(α) n (x; q) =

n

⎜ (−1)n−k ⎝

k=0

⎞

⎟ q ρ −α+inv(ρ) ⎠ xk

(α)

ρ∈Fn,k+1

where ρ is the length of the first factor in ρ. (i)

Proof. Let α ≥ 1. For an integer i with 1 ≤ i ≤ n, we denote by the set Bn+α the collection of bipartitions {A, B} ∈ Πn+α,2 of the set [n + α] such that 1, . . . , α belong to A and |B| = i. Then every (k + 1)-factorized (α) permutation ρ ∈ Fn,k+1 may be expressed as ρ = ρA /ρB where ρA is a permutation in which the elements (i)

1, . . . , α are right-to-left minima and ρB is a k-factorized permutation of B for some i and {A, B} ∈ Bn+α . (i) Thus, for each k ≤ i ≤ n and {A, B} ∈ Bn+α , if we denote by F(A, B) the collection of (k + 1)-factorized permutations ρ which can be expressed as ρ = ρA /ρB then 

(α)

Fn,k+1 =

F(A, B).

(11)

(i) {A,B}∈Bn+α

k≤i≤n (α)

Let Ln,k = (−1)n−k

 (α)

ρ∈Fn,k+1

q ρ −α+inv(ρ) . We note that inv({A, B}) equals the number of pairs (a, b) with (i)

a ∈ A, b ∈ B, and a > b. Since for {A, B} ∈ Bn+α we have inv(ρ) = inv(ρA /ρB ) = inv({A, B}) + inv(ρA ) + inv(ρB ), it follows from (11) that (α)

Ln,k = (−1)n−k



q ρ −α+inv(ρ)

(α)

ρ∈Fn,k+1

= (−1)n−k

n





q |ρA |−α+inv({A,B})+inv(ρA )+inv(ρB )

i=k {A,B}∈B(i) ρA /ρB ∈F (A,B) n+α

= (−1)n−k

n



i=k {A,B}∈B(i) n+α (i)

q inv({A,B})



q n−i+inv(ρA )+inv(ρB ) .

(12)

ρA /ρB ∈F (A,B)

Now fix {A, B} ∈ Bn+α for each i, 1 ≤ i ≤ n. For each ρ = ρA /ρB ∈ F(A, B), we correspond ρA to the (α) permutation ρ∗A in Fn−i,1 by mapping aj to j for each j = 1, . . . , n + α − i where aj is the jth smallest symbol of ρA among a1 , . . . , an+α−i , and ρB to the permutation ρ∗B in Fi,k by mapping bj to j for each j = 1, . . ., i where bj is the jth smallest symbol of ρB among b1 , . . . , bi . Then the mappings ρA → ρ∗A (α) and ρB → ρ∗B give bijections from {ρA | ρ ∈ F(A, B)} to Fn−i,1 and from {ρB | ρ ∈ F(A, B)} to Fi,k , respectively. Obviously inv(ρA ) = inv(ρ∗A ) and inv(ρB ) = inv(ρ∗B ). Thus

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q inv(ρA )+inv(ρB ) =

ρA /ρB ∈F(A,B)

7

∗

∗

q inv(ρA )+inv(ρB ) .

(13)

(α)

∗ ρ∗ A ∈Fn−i,1 ,ρB ∈Fi,k

Using the formula [9, Lemma 5.1] given by

q

  n = i q

inv({A,B})

(i) {A,B}∈Bn+α

(1 ≤ i ≤ n)

(14)

where inv({A, B}) = 0 when B = ∅, together with (12) and (13), we obtain

(α)

Ln,k

⎛   n ⎜ n−i n (−1) = ⎝ i q

⎞⎛



i=k

∗ ⎟ q n−i+inv(ρA ) ⎠ ⎝(−1)i−k

⎞



∗

q inv(ρB ) ⎠ .

(15)

ρ∗ B ∈Fi,k

(α) ρ∗ A ∈Fn−i,1

By Lemma 3.2,



∗

q inv(ρA ) =

(α)

q inv(σ) .

(α+1)

ρ∗ A ∈Fn−i,1

σ∈Pn−i,α+1

Thus the first factor in the summand in (15) equals n−i

(−1)

  n n−i q i q



q inv(σ) .

(α+1) σ∈Pn−i,α+1

By Lemma 3.3, α 

1 1 + q j+1 z j=0

⎛ ⎜ = ⎝ n≥0

⎞

⎛ ⎟ (−qz) ⎜ n n = q inv(σ) ⎠ ⎝(−1) q [n] ! q (α+1)



n

n≥0

σ∈Pn,α+1

As the (n, i)-entry of the q-Riordan matrix



zn n≥0 gn [n]q ! , z

summand in (15) is the (n, i)-entry of the q-Riordan matrix

 is q α

⎞

⎟ zn . q inv(σ) ⎠ [n]q ! (α+1)

σ∈Pn,α+1

n

i q gn−i

j=0

in general, the first factor in the  1 , z . j+1 1+q z q

It remains to show  that  the second factor in the summand in (15) is equal to the (i, k)-entry of the z q-Riordan matrix 1, 1+z . Let pmtn(Bj ) be the set of permutations on Bj . For a given k-partition q

π = {B1 , B2 , . . . , Bk } ∈ Πi,k , we obtain |B1 |!|B2 |! · · · |Bk |! k-factorized permutations on [i]. For instance, k ρπ := ρ1 ρ2 · · · ρk where ρj ∈ pmtn(Bj ). Since inv(ρπ ) = j=1 inv(ρj ) + inv(π), it follows that ρ∈Fi,k



q inv(ρ) =

π={B1 ,...,Bk }∈Πi,k

⎛ ⎝



⎞

⎛

q inv(ρ1 ) ⎠ · · · ⎝

ρ1 ∈pmtn(B1 )



⎞ q inv(ρk ) ⎠ q inv(π) .

ρk ∈pmtn(Bk )

 Using τ q inv(τ ) = [m]q ! where τ runs through all permutations of an m-element set, the above equation is equivalent to ρ∈Fi,k

q inv(ρ) =

π={B1 ,...,Bk }∈Πi,k

where mj = |Bj |. Note that m1 + · · · + mk = i. Since

[m1 ]q ! · · · [mk ]q !q inv(π)

(16)

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F (z) :=

z zn = , (−1)n−1 [n]q ! 1+z [n]q ! n≥1

  z it follows from Proposition 3.1 and (16) that the (i, k)-entry of the q-Riordan matrix 1, 1+z is q

⎛



⎝

π={B1 ,...,Bk }∈Πi,k

k 

⎞ (−1)mj −1 [mj ]q !⎠ q inv(π) = (−1)i−k

j=1



q invρ ,

ρ∈Fi,k

as required. Since every q-Riordan matrix (g, f )q may be factorized into (g, z)q (1, f )q by means of usual matrix multiplication [4, p. 243], we obtain 

(α)

Ln,k

⎞ ⎛ ⎞

 α  z z 1 1 ⎠ . , z ⎠ 1, , =⎝ =⎝ j+1 z j+1 z 1 + z 1 + q 1 + z 1 + q q j=0 j=0 ⎛

 n,k∈N0

α 

q

(α)



(α)

(α)

By (7), we obtain Ln,k = n,k . Thus Ln,k

q



(α)

n,k∈N0

is the coefficient matrix of Ln (x; q). Hence the theorem

holds for α ≥ 1. (i) Next consider the case α = 0. In this case, as the set Bn+1 , we take the collection of bipartitions {A, B} of the set [n + 1] in which the element 1 belongs to A and |B| = i where the elements of the blocks A and (0) B are in increasing order for each i, 1 ≤ i ≤ n. Then every (k + 1)-factorized permutation ρ ∈ Fn,k+1 may be expressed as ρ = ρA /ρB where ρA is a permutation in which the entry 1 is written first and ρB is a (i) k-factorized permutation of B. It is easy to see that the proof for α ≥ 1 works for the newly taken Bn+1 . 2 (α)

Remark. By Theorem 3.4, Ln =

n

(α) n−k |Fn,k+1 |xk . k=0 (−1)

Then, by (1),

 n! n + α (α) = |Fn,k+1 |. k! k + α (2)

(2)

(2)

Example 1. The sets F2,1 , F2,2 and F2,3 are as follows: (2)

F2,1 = {1234, 1243, 3124, 4123, 3412, 4312, 1324, 1423, 3142, 4132, 1342, 1432}; (2)

F2,2 = {12/34, 12/43, 123/4, 132/4, 312/4, 124/3, 142/3, 412/3}; (2)

F2,3 = {12/3/4}. Therefore,

q ρ −2+inv(ρ) = q 2 (1 + 2q + 3q 2 + 3q 3 + 2q 4 + q 5 ),

(2)

ρ∈F2,1



q ρ −2+inv(ρ) = 1 + 2q + 2q 2 + 2q 3 + q 4 ,

(2)

ρ∈F2,2



q ρ −2+inv(ρ) = 1.

(2)

ρ∈F2,3

Then, by Theorem 3.4, (2)

L2 (x; q) = q 2 (1 + 2q + 3q 2 + 3q 3 + 2q 4 + q 5 ) − (1 + 2q + 2q 2 + 2q 3 + q 4 )x + x2 .

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4. A connection with q-rook numbers In this section, we translate the combinatorial interpretation obtained in the previous section into the language of q-rook numbers. A board is a subset of an n × n grid of squares. We label the square in the ith row from the top and the jth column from the left as (i, j) in the same way as referring to elements of an n × n matrix. In 1986, Garsia and Remmel [8] introduced the q-rook number rk (B, q) for a Ferrers board B. It is defined as rk (B, q) =



q inv(C)

(17)

C∈Ck (B)

where Ck (B) is the collection of all placements of k non-attacking rooks on B and the statistic inv(C) is defined as follows: first cross out all squares which either contain a rook, or are below or to the left of any rook. Then the statistic inv(C) is the number of squares of B not crossed out. In this paper, we are interested in the q-rook numbers of the Ferrers board related to q-Laguerre polyno(α) mials. In particular, we show that the q-Laguerre polynomial Ln (x; q) can be expressed in terms of q-rook numbers rk (B, q) defined in (17). A board B is a Ferrers board if there exists a nondecreasing finite sequence (λ1 , λ2 , . . . , λ ) of positive integers λi such that B = {(i, j) | 1 ≤ i ≤ m, 1 ≤ j ≤ λi } where we are using ordinary cartesian coordinates so the (1, 1) square is at the bottom left. We denote such Ferrers board by B = (λ1 , λ2 , . . . , λ ). If  non-attacking rooks can be placed on the board B then B is called complete. Two rooks placed in squares (i1 , j1 ) and (i2 , j2 ) are noncrossing if i1 < i2 and j1 > j2 or i1 > i2 and j1 < j2 , that is, the upper rook is to the right of the lower rook. (α) Let Bn,h be the set of complete Ferrers boards of length n + α with h different heights of the form B = (λ1 . . . , λn+α−m , n + α, . . . , n + α)    m times (α)

where α ≤ m ≤ n + α − h + 1, λn+α−m ≤ n, and let Cn,h (B) denote the collection of all placements of n + α (α)

non-attacking rooks on a board B ∈ Bn,h satisfying the following conditions: (i) there are h rooks which have no squares above them; (ii) rooks in the first α rows are pairwise noncrossing. Example 2. Let α = 3, n = 2 and h = 2. Then (3)

B2,2 = {B1 = (2, 2, 5, 5, 5), B2 = (2, 5, 5, 5, 5), B3 = (1, 5, 5, 5, 5)}. Thus there are 10 placements: (3)

(i) C2,2 (B1 )

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(3)

(ii) C2,2 (B2 )

(3)

(iii) C2,2 (B3 )

(α)

(α)

Now we define the q-rook number rn,h (B, q) on a Ferrers board B ∈ Bn,h as (α)

rn,h (B, q) =



q inv(C) .

(α)

C∈Cn,h (B)

n (α) Theorem 4.1. Let Ln (x; q) = h=0 (−1)n−h an,h xh be the q-Laguerre polynomial of degree n for α ≥ 1. Then the coefficients an,h can be expressed in terms of the q-rook numbers as follows:

an,h =

(α)

q #(B)−α rn,h+1 (B, q)

(α)

B∈Bn,h+1

where #(B) is the number of squares on the top row of the board B. (α)

Proof. Let ρ = ρ1 /ρ2 / · · · /ρh ∈ Fn,h where ρi = ai,1 ai,2 · · · ai,ji . We define the map (α)

φ : Fn,h →



(α)

Cn,h (B)

(α)

B∈Bn,h (α)

so that φ maps ρ to the placement of n + α non-attacking rooks on a board B ∈ Bn,h such that (i) the board B is given by B = (λ1 , . . . , λ1 , λ2 , . . . , λ2 , . . . , λh , . . . , λh ) where ci is the number of the          c1 times

c2 times

ch times

symbols of ρh+1−i and λi = n + α + 1 − minρh+1−i ; (ii) the rook corresponding to the jth element a in ρ is placed in the square labeled (a, n + α + 1 − j) of B, or more precisely, the rook corresponding to each symbol ai,j of ρi is placed on B in the square with i−1 label (ai,j , n + α + 1 − j) if i = 1 and (ai,j , n + α + 1 − j − m=1 ci ) if i ≥ 2. (2)

For instance, consider 142/63/5 ∈ F4,3 . Then φ maps this 3-factorized permutation to the placement of 6 rooks on the Ferrers board B = (2, 4, 4, 6, 6, 6) given in Fig. 1.

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(2)

Fig. 1. The placement on B = (2, 4, 4, 6, 6, 6) corresponding to 142/63/5 ∈ F4,3 .

Since 1, 2, . . . , α belong to ρ1 and are right-to-left minima, the rooks in the first α rows are noncrossing mutually. Moreover, by the way in which φ is defined, there is no square above the rook corresponding to (α) min ρi for each i = 1, . . . , k. Thus the placement φ(ρ) belongs to Cn,h (B) and so φ is well-defined.  (α) (α) (α) Take C ∈ B∈B(α) Cn,h (B). Then C ∈ Cn,h (B) for some B ∈ Bn,h . Let B = (λ1 , . . . , λ1 , λ2 , . . . , λ2 , . . . ,       n,h c1 times

c2 times

λ , . . . , λ ). If there is a rook in the square labeled (i, j), then we place i in the (n + α + 1 − j)th position.  h  h ch times

Since the rooks in C are nonattacking, we obtain a permutation of [n +α]. Now we factorize this arrangement by putting the first c1 elements into the first factor, the next c2 elements into the second factor, and so on to have the arrangement, say ρ, factorized into h factors. Since the rooks in the first α rows of B are noncrossing mutually, 1, 2, . . . , α are right-to-left minima. Each of h rooks above which no squares exist is the minimum of the factor of ρ which it belongs to. Since λ1 ≤ λ2 ≤ · · · ≤ λh , the minimum in each factor (α) of ρ is located in increasing order. Thus ρ belongs to Fn,h . It is easy to see that φ(ρ) = C. Hence φ is a bijection. (α) It is known [8] that the statistic inv(C) for C ∈ Cn,h (B) may be viewed as the number of inversions of a permutation of [n]. Indeed, if B is an n × n square board, ρ = a1 a2 · · · an is a permutation of [n] and C is the configuration obtained by placing the ith rook in the square labeled (ai, n + 1 − α) then a simple argument gives inv(C) = inv(ρ). (α)

Furthermore, by the way in which φ is defined, the length of the first factor of the permutation ρ in Fn,h+1 equals the number of squares in the top row of Bρ for which φ(ρ) ∈ Cn,h+1 (Bρ ). Then, by Theorem 3.4, an,h =





q ρ −α+inv(ρ) =

(α)

q #(Bρ )−α+inv(φ(ρ)) .

(α)

ρ∈Fn,h+1

ρ∈Fn,h+1

(α)

As φ(ρ) ∈ Cn,h+1 (Bρ ) and Bρ ∈ Bn,h+1 , the right hand side of the second equality equals ⎛

⎜ q #(Bρ )−α ⎝

(α)

B∈Bn,h+1

⎞ (α)

C∈Cn,h+1 (B)

⎟ q inv(C) ⎠ =



(α)

q #(B)−α rn,h+1 (B, q).

(α)

B∈Bn,h+1

Hence the proof follows. 2 (α)

When q = 1 we obtain a new combinatorial interpretation for the Laguerre polynomials Ln (x) as follows.

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(α)

Corollary 4.2. For α ≥ 1, let Ln (x) = an,h =

n

n−h an,h xh . h=0 (−1)



(α)

|Cn,h+1 (B)| =

(α)

Then

 n! n + α . h! h + α

B∈Bn,h+1

Example 3. To obtain q-rook numbers given in Example 2, we cross out all squares which either contain a rook, or are below or to the left of any rook. Then we have following: (3)

(i) r2,2 (B1 , q) = 1 + q

(3)

(ii) r2,2 (B2 , q) = q + q 2 + q 3 + q 4

(3)

(iii) r2,2 (B3 , q) = 1 + q + q 2 + q 3

(3)

It follows from Theorem 4.1 that the coefficient of x in L2 (x; q) is

(3)

q #(B)−α r2,2 (B) = (1 + q) + q(q + q 2 + q 3 + q 4 ) + q(1 + q + q 2 + q 3 )

(3)

B∈B2,2

= 1 + 2q + 2q 2 + 2q 3 + 2q 4 + q 5 . 5. Algebraic properties of the q-Laguerre polynomials In this section, we investigate some algebraic properties of the q-Laguerre polynomials which give the well-known results when q → 1. (α) We first show that the q-Laguerre polynomials Ln (x; q) are orthogonal by using the following lemma.

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Lemma 5.1. [5] Let {sn (x; q)}n∈N0 be a q-Sheffer for a pair (g(z), f (z)). If g(z) ∈ Eq (0) and f (z) ∈ Eq (1) satisfy a0 + a1 z Dq g(z) =− g(z) 1 + b1 z + b2 z 2

and

Dq f (z) =

g(z) (1 + b1 z + b2 z 2 )g(qz)

(18)

for any functions a0 , a1 , b1 and b2 of q such that a1 = −[n − 1]q b2 for all n ≥ 1, then sn (x; q) are orthogonal and satisfy the three-term recurrence relation: sn+1 (x; q) = (x − tn )sn (x; q) − λn sn−1 (x; q)

(n ≥ 1)

(19)

with initial values s0 (x; q) = 1 and s1 (x; q) = x −a0 where tn = a0 +[n]q b1 and λn = [n]q (a1 +[n −1]q b2 ) = 0. (α)

Theorem 5.2. The polynomials Ln (x; q) are orthogonal and satisfy the three-term recurrence relation: (α)

(α)

Ln+1 (x; q) = (x − q[n + α + 1]q − [n]q )L(α) n (x; q) − q[n]q [n + α]q Ln−1 (x; q), (α)

(20)

(α)

with initial values L0 (x; q) = 1 and L1 (x; q) = x − q[α + 1]q .   α (α) Proof. Recall that Ln (x; q) is the q-Sheffer for (g, f ) where g = j=0 n∈N0 α+1 Dq g = −q[α + 1]q j=0 1+q1j+1 z it follows that

1 1+q j+1 z

and f =

q[α + 1]q q[α + 1]q (1 + z) q[α + 1]q + q[α + 1]q z Dq g =− =− =− . α+2 α+2 g 1+q z (1 + z)(1 + q z) 1 + (1 + q α+2 )z + q α+2 z 2

z 1+z .

Since

(21)

Since (1 + q α+2 z)g(qz) = (1 + qz)g, we similarly obtain Dq f = Now

Dq g in g α+2

g 1 = . α+2 (1 + z)(1 + qz) (1 + (1 + q )z + q α+2 z 2 )g(qz)

(21) and Dq f are in the form given in (18) where a0 = a1 = q[α + 1]q , b1 = 1 + q α+2 and (α)

b2 = q . Since α is a nonnegative integer, a1 = −[n − 1]q b2 . Thus Ln (x; q) are orthogonal by Lemma (α) 2.1. Furthermore, substituting Li (x; q) for si (x; q) into (19) for i = n − 1, n, n + 1 gives rise to (20) with tn = q[α + 1]q + [n]q (1 + q α+2 ) = q[n + α + 1]q + [n]q and λn = [n]q (q[α + 1]q + [n − 1]q q α+2 ) = q[n]q [n + α]q . Hence the theorem follows. 2 (α)

With the recurrence relation (20), we obtain the first few polynomials Ln (x; q): (α)

L0 (x; q) = 1, (α)

L1 (x; q) = x − q[α + 1]q , (α)

L2 (x; q) = x2 − [2]q [α + 2]q x + q 2 [α + 1]q [α + 2]q (α)

L3 (x; q) = x3 − (q + [2]q [α + 2]q + [α + 4]q )x2 + [3]q [α + 2]q [α + 3]q x − q 3 [α + 1]q [α + 2]q [α + 3]q .

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The rescaled Al-Salam–Chihara polynomials Pn (x) := Pn (x; a, b; q) are defined in [15] by

 (q − 1)ax + 1/a + a ; a, b; q Pn (x) = Qn 2

(22)

where Qn (x) are the Al-Salam–Chihara polynomials satisfying the recurrence relation (4). Let (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 ). Then Pn (x) can be expressed as Pn (x) =

n (q −n ; q)k (abq k ; q)n−k

(q; q)k

k=0

(1 − q)k q

k+1 2

a2k−n Φk (x; a)

(23)

k−1  where Φk (x; a) = j=0 x − [j]q (1 − q −j /a2 ) for k ≥ 1 and Φ0 (x; a) = 1. From (4), (20) and (22), it follows that

√

q q−1

L(α) n (x; q) =

n

 1 √ Pn x; √ , qq α+1 ; q . q

√ √ A simple computation after substituting a = 1/ q and b = qq α+1 into (23) yields L(α) n (x; q) =

n

(−1)n−k q (1−k)(n−k)

k=0

  [n]q ! n + α √ Φk (x; 1/ q). [k]q ! k + α q

(24)

√ Since Φk (x; 1/ q) = xk as q → 1, it follows that the explicit formula (24) is reduced to (1) when q → 1. More generally, using the Heine’s Binomial Formula in [14] given by n + α 1 = zn jz 1 − q n q j=0 α 

(25)

n≥0

we obtain the following theorem. Theorem 5.3. For α, β ∈ N0 with α > β, we have L(α) n (x; q) =

  n [n]q ! n − j + α − β − 1 (β) (−1)n−j q (n−j)(β+2) Lj (x; q). [j] ! α − β − 1 q q j=0

Proof. It follows from (6) that

⎛ L(α) n (x; q)

n≥0

n

z =⎝ [n]q !

α 

j=β+1

⎛

α−β−1 

=⎝

j=0

⎞⎛

⎞

  1 1 ⎠⎝ ⎠ eq xz 1 + q j+1 z 1 + q j+1 z 1+z j=0 β 

⎞ zn 1 (β) ⎠ . L (x; q) n 1 − q j (−q β+2 z) [n]q ! n≥0

By Heine’s Binomial Formula in (25), the right hand side of the second equality above equals ⎛ ⎞   (β) n + α − β − 1 zj ⎝ (−1)n q n(β+2) zn⎠ Lj (x; q) n [j]q ! q n≥0

j≥0

⎞ ⎛   n (n−j)(β+2) zn [n]q ! n − j + α − β − 1 (β) ⎝ (−1)n−j q . Lj (x; q)⎠ = [j]q ! [n]q ! α−β−1 q j=0 n≥0

(26)

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Making the coefficients of z n on both sides equal, we obtain (26) as required. 2 Corollary 5.4. Let Sα = {(j1 , j2 , . . . , jα ) ∈ Nα 0 | n ≥ j1 ≥ j2 · · · ≥ jα ≥ 0}. For an integer α ≥ 1, the polyno(α) mials Ln (x; q) can be expressed in terms of q-Laguerre polynomials as follows:

L(α) n (x; q) =

(−1)n−jα q n−jα +N (α)

(j1 ,j2 ,...,jα )∈Sα

where N (α) =

α

i=1 (n

[n]q ! (0) L (x; q) [jα ]q ! jα

− ji ).

Proof. By substituting β = α − 1 into Theorem 5.3, we obtain n

L(α) n (x; q) =

(−1)n−j1 q (n−j1 )(α+1)

j1 =0

[n]q ! (α−1) L (x; q). [j1 ]q ! j1

Then inductively we obtain L(α) n (x; q) =

j1 n

(−1)n−j1 +j1 −j2 q (n−j1 )(α+1)+(j1 −j2 )α

j1 =0 j2 =0

.. .



=

α−1

(−1)n−j1 +

i=1

(ji −ji+1 )

[n]q ! [j1 ]q ! (α−2) L (x; q) [j1 ]q ! [j2 ]q ! j2

q (n−j1 )(α+1)+

α−1 i=1

(ji −ji+1 )(α+1−i)

(j1 ,j2 ,...,jα )∈Sα

× =

[jα−1 ]q ! (0) [n]q ! [j1 ]q ! ··· L (x; q) [j1 ]q ! [j2 ]q ! [jα ]q ! jα [n]q ! (0) L (x; q) (−1)n−jα q n(α+1)−j1 −j2 −···−jα−1 −2jα [jα ]q ! jα

(j1 ,j2 ,...,jα )∈Sα

=



(−1)n−jα q n−jα +N (α)

(j1 ,j2 ,...,jα )∈Sα

[n]q ! (0) L (x; q), [jα ]q ! jα

which completes the proof. 2 (α)

We now turn to the zeros of the q-Laguerre polynomials Ln (x; q). A sequence {ak }nk=1 is a chain sequence if and only if there exists a parameter sequence {gk }nk=0 such that the ak ’s admit a factorization ak = gk (1 − gk−1 ),

0 ≤ g0 < 1, 0 < gk < 1, 0 < k ≤ n.

(27)

Let sn (x; q) be the orthogonal polynomials satisfying the recurrence relation (19). It is known [11,12] −1 that the polynomials sN (x; q), N ≥ 2, have only positive real zeros if and only if {λn /(tn−1 tn )}N n=1 is a chain sequence with tn > 0 and λn > 0 for N > n ≥ 1. It is also known [11] that the Laguerre polynomials (α) Ln (x) have only positive real zeros. More generally we have the following theorem. (α)

Theorem 5.5. Let q be a positive real number. Then the polynomials Ln (x; q) with degree n ≥ 1 have only positive real zeros. −1 Proof. It suffices to show that {λn /(tn−1 tn )}N n=1 is a chain sequence with tn > 0 and λn > 0 for N > n ≥ 1 where tn and λn are the same as in (19). From (20), we obtain

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λk

q[k]q [k + α]q tk−1 tk (q[k + α]q + [k − 1]q )(q[k + α + 1]q + [k]q ) 

[k − 1]q [k]q . 1− = (q[k + α + 1]q + [k]q ) q[k + α]q + [k − 1]q =

−1 By (27), {λn /(tn−1 tn )}N n=1 is a chain sequence so that Ln (x; q) has only positive real zeros. (α)

2

6. Closing remarks   (α) Consider the sequence Ln (x, y; q)

n≥0

⎛

defined as the q-Sheffer for the pair

⎞ ln 1 (1 + z) − ln (1 + yz) q q ⎝ ⎠ , j yz 1 + q 1 − y j=0 where lnq (1 + z) =



α 

n−1 z n n≥1 (−1) [n]q ,

a q-analogue of the logarithmic function ln(1 + z). By applying an (α)

argument similar to the one used in proving Theorem 5.2, one can show that the polynomials Ln (x, y; q) are also orthogonal and that they satisfy the three-term recurrence relation: (α)

(α)

Ln+1 (x, y; q) = (x − y[n + α + 1]q − [n]q )L(α) n (x, y; q) − y[n]q [n + α]q Ln−1 (x, y; q) (α)

(28)

(α)

with initial values L0 (x, y; q) = 1 and L1 (x, y; q) = x − y[α + 1]q . (α) (α) In fact, the polynomials Ln (x, y; q) generalize both the polynomials Ln (x; q) and Ln (x, y; q) in (3). Comparing the recurrence relations (5), (20) and (28), we see that (α) (α) L(0) n (x, y; q) = Ln (x; q) and Ln (x, q; q) = Ln (x; q). (α)

(α)

In particular, for any α ∈ N0 , the polynomials Ln (x, y; q) are reduced to Ln (x) when q → 1 and y = 1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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D.S. Moak, q-Analog of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981) 20–47. G. Polya, Collected papers, in: R.P. Boas (Ed.), Location of Zeros, vol. II, MIT Press, Cambridge, MA, 1974. R. Simion, D. Stanton, Specializations of generalized Laguerre polynomials, SIAM J. Math. Anal. 25 (1994) 712–719. R. Simion, D. Stanton, Octabasic Laguerre polynomials and permutation statistics, J. Comput. Appl. Math. 68 (1996) 297–329. [21] E.C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed., Oxford University Press, Oxford, 1986 (revised by D.R. Heath-Brown).