A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS

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1 J. Korean Math. Soc. 47 (2010), No. 3, pp DOI /JKMS A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS Seok-Zun Song, Gi-Sang Cheon, Young-Bae Jun, and LeRoy B. Beasley Abstract. In this paper, more generalized q-factorial coefficients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-stirling numbers of both kinds and the extended q-lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences. 1. Introduction During the last several decades, the Stirling numbers of the first and second kinds have been studied from many diverse viewpoints (see for example [2]- [17]). A viewpoint of Carlitz [2], motivated by the enumeration problem for Abelian groups, is to study the Stirling numbers as specializations of the q- Stirling numbers. Originally, he defined the q-stirling numbers of the second kind as the numbers S q (n, k) in our notation such that (1) [x] n = q (k 2) Sq (n, k)[x] k, where [x] and [x] k denote the q-number and the (falling) q-factorial of order k, respectively defined as and [x] = (1 q x )/(1 q) = 1 + q + + q x 1, (2) [x] k = [x][x 1] [x k + 1] (k 1), [x] 0 = 1 for real numbers x and q. Recently there has been interested in generalizing the q-factorial as well as q-stirling numbers, see for example [3, 4, 14, 17]. This interest is largely motivated by Carlitz [2]. He found for some purposes it is convenient to generalize Received November 19, Mathematics Subect Classification. Primary 05A30; Secondary 05A15. Key words and phrases. q-factorial, q-stirling numbers, q-lah numbers. 645 c 2010 The Korean Mathematical Society

2 646 S.-Z. SONG, G.-S. CHEON, Y.-B. JUN, AND L. B. BEASLEY the q-stirling numbers, and studied the coefficients a n,k (r) such that (3) [x + r] n = q (k+r 2 ) an,k (r)[x] k for any real number r. The expression (3) may be written as (4) [x] n = q (k2)+kr S q (n, k; r)[x r] k, which reduces to (1) when r = 0, where S q (n, k; r) := q (r 2) an,k (r). In 2004, Charalambides [4] (also see [12]) called [x r] k and S q (n, k; r) the non-central q-factorial of order k with non-centrality parameter r and the non-central q-stirling numbers of the second kind, respectively. Moreover, he (see [3, 4]) intensively investigated the generalized q-factorial coefficients with increment h, R q (n, k; h), and the non-central generalized q-factorial coefficients, C q (n, k; s, r), where (5) [x kh] = q h(n 2) n q (k 2) Rq (n, k; h)[x] k and (6) [sx + r k] = q rn (n 2) n q s(k 2) Cq (n, k; s, r)[x] k,q s. In the present paper, more generalized q-factorial coefficients which involve R q (n, k; h) and C q (n, k; s, r) as well as central or non-central q-stirling numbers of both kinds and q-lah numbers are examined by a natural extension of the q-factorial. This immediately leads to the notion of the extended q-stirling numbers of both kinds and the extended q-lah numbers. Specifically, in Section 2 we first define the extended q-factorial for a sequence α = (α n ) n 0 of any numbers. Then we develop the extension of the generalized q-factorial coefficients and obtain the explicit formula by aid of Newton s general interpolation formula based on the divided differences. In Section 3, we give the definition of the extended q-stirling numbers of both kinds and the extended q-lah numbers, and we examine the connections with the generalized q-factorial coefficients. In Section 4, the non-central q-stirling numbers of both kinds with an increment which generalize the non-central q-stirling numbers examined in [4] are developed. In Section 5 we obtain some interesting matrix factorizations arising from the extended q-factorial coefficients, q-stirling numbers of both kinds and q-lah numbers. In particular, the LDU-factorization of the Vandermonde matrix is given. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences for α. Finally we should mention that similar generalizations of the factorial coefficients can be found in the literature, see for example [6, 8, 14, 17]. However it should also be emphasized that we have focused our attention on q-analogies of

3 A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS 647 the generalized factorial coefficients and related numbers for any sequences. In particular, our extended q-stirling numbers allow a straightforward extension of the notions of the central and non-central q-stirling numbers presented in [4, 12]. 2. Extended q-factorial coefficients We begin this section defining the extended q-factorial of order n associated with a sequence α = (α n ) n 0 of any numbers, denoted by f n (x; α), as { [x α0 ][x α (7) f n (x; α) = 1 ] [x α ] if n 1, 1 if n = 0. For the sake of notational convenience we shall mean α α 0 = (α n α 0 ) n 0, α 1 = (α n 1) n 0 and α = ( α n ) n 0. We note that f n (x; α) may be considered as the central or non-central extended q-factorial along with α 0 = 0 or α 0 0, respectively. Also one may consider f n (x; α α 0 ) as the central extended q-factorial by means of f n (x; α α 0 ) = [x][x (α 1 α 0 )] [x (α α 0 )], n 1. We shall take f n (x; α) as the notation for the extended rising q-factorial of order n associated with α = (α n ) n 0 by means of f n (x; α) = [x + α 0 ][x + α 1 ] [x + α ], f 0 (x; α) = 1. By the q-newton s formula (see [4]), the expansion of the extended q-factorial f n (x; α) into a polynomial of q-factorial [x] k is given by 1 [ (8) f n (x; α) = k [k]! q f n (x; α) ] [x] x=0 k, where k q is the q-difference operator of order k defined by k q = (E 1)(E q) (E q k 1 ) together with the usual shift operator E. We suppose that α = (α n ) n 0 and β = (β n ) n 0 are two distinct sequences. Applying Newton s general interpolation formula [11] based on the divided differences, we easily obtain the expansion of f n (x; α) into a polynomial of f n (x; β) given by (9) f n (x; α) = q C k(β) ( Dq k f n (x; α) ) f x=β k (x; β), 0 where C k (β) = β 1 + β β k 1 and D k q is the q-divided difference operator of order k defined by (10) Dq k f n (x; α) = Dk 1 q f n (x; α) Dq k 1 f n (β k ; α). [x] [β k ] The following theorem provides very useful another generalization for q- Stirling numbers of both kinds and q-lah numbers in the next section.

4 648 S.-Z. SONG, G.-S. CHEON, Y.-B. JUN, AND L. B. BEASLEY Theorem 2.1. Given the sequences α = (α n ) n 0 and β = (β n ) n 0 of any numbers, the followings are equivalent: (i) f n (x; α) = q Cn(α) q Ck(β) Ω q (n, k; α, β)f k (x; β); (ii) Ω q (n, k; α, β) = Ω q (n 1, k 1; α, β) + ([β k ] [α ])Ω q (n 1, k; α, β), Ω q (0, 0; α, β) = 1; (iii) Ω q (n, k; α, β) = k i=0 ([β ] [α i ]) if β k i=0, i ([β ] [β i ]) i β for each i, = 0,..., k, =0 Ω q (n, 0; α, β) = i=0 ([β 0] [α i ]), (n 1), where C n (α) := α 0 + α α and C 0 (α) = 0 are assumed. Proof. Comparing (i) with (9), we get Ω q (n, k; α, β) = q C n(α) [ D k q f n (x; α) ] x=β 0. Using the identity f n (x; α) = q α ([x] [α ])f (x; α), we get Ω q (n, k; α, β) = q C n(α) D k q f n (x; α) x=β0 = q C (α) D k q ([x] [β k ])f (x; α) x=β0 + ([β k ] [α ])q C(α) D k q f (x; α) x=β0 = q C(α) Dq k 1 {D q ([x] [β k ])f (x; α)} x=β0 + ([β k ] [α ])Ω q (n 1, k; α, β) = Ω q (n 1, k 1; α, β) + ([β k ] [α ])Ω q (n 1, k; α, β), which proves (i) (ii). The relation (i) (iii) is easily seen by Dq k f n (β 0 ; α) = f n(β 0 ; α) ω k (β 0, β) + f n(β 1 ; α) ω k (β 1, β) + + f n(β k ; α), (k = 0, 1,..., n), ω k (β k, β) where ω k (β, β) = qc k+1 (β)f k+1 (x;β) [x] [β ] x=β with ω 0 (β, β) = 1. Hence the proof is complete. Theorem 2.1 suggests that many well-known results can be derived when we set q = 1 or use special sequences α and β. We call the coefficients Ω q (n, k; α, β) the extended q-factorial coefficients associated with the sequences α and β. By aid of the formula (11) k q f n (x; α) = k [ ] ( 1) k q (k 2 ) k f n (x + ; α), =0 we can easily derive the following corollary from Theorem 2.1. Corollary 2.2. Given the sequence α = (α n ) n 0 of any numbers, the followings are equivalent:

5 A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS 649 (i) f n (x; α) = q Cn(α) q (k 2) Ωq (n, k; α)[x] k ; (ii) Ω q (n, k; α) = qcn(α) ( k 2) [k]! k =0 ( 1) k q (k 2 ) [ k ] fn (; α); (iii) Ω q (n, k; α) = Ω q (n 1, k 1; α) + ([k] [α ])Ω q (n 1, k; α), Ω q (0, 0; α) = 1. We observe that the formula (i) above corollary may be regarded as an extension of both (5) and (6). If each α n is replaced by nh, then we obtain immediately R q (n, k; h) = Ω q (n, k; α). If each α n is replaced by n r s, then we obtain [ [sx + r k] = [s] n x k r ] s q s = q rn (n 2) n q s(k 2) { } [s] n q (s 1)(nr (n 2)) Ω q s(n, k; α) [x] k,q s. It implies that C q (n, k; s, r) = [s] n q (s 1)(nr (n 2)) Ω q s(n, k; α). 3. Extended q-stirling and q-lah numbers Since the extended q-factorial f n (x; α) is a polynomial of the q-number [x] of degree n 1, by following Carlitz [2] it would be natural to define more generalized q-stirling numbers of the first and second kind related to f n (x; α) denoted by s q (n, k; α) and S q (n, k; α), respectively as follows: n (12) f n (x; α) = q Cn(α) s q (n, k; α)[x] k, and (13) [x] n = q Ck(α) S q (n, k; α)f k (x; α). In a similar way, we may express the extended rising q-factorial f n (x; α) associated with the sequence α = (α n ) n 0 by (14) f n (x; α) = q C n(α) q Ck(α) L q (n, k; α)f k (x; α). We should also mention that (12) and (13) yield the following orthogonality relation: (15) s q (i, k; α)s q (k, ; α) = S q (i, k; α)s q (k, ; α) = δ i, where δ i is the Kronecker s delta.

6 650 S.-Z. SONG, G.-S. CHEON, Y.-B. JUN, AND L. B. BEASLEY Further, given sequences α = (α n ) n 0 and β = (β n ) n 0 of any numbers the extended q-factorial f n (x; α) may be expressed by n k f n (x; α) = q Cn(α) s q (n, k; α) q C(β) S q (k, ; β)f (x; β) = q Cn(α) n q C k(β) =0 s q (n, ; α)s q (, k; β) f k (x; β). Comparing above expression with (i) of Theorem 2.1, we obtain the following theorem. Theorem 3.1. Let α = (α n ) n 0 and β = (β n ) n 0 be sequences of any numbers. Then we have (16) Ω q (n, k; α, β) = s q (n, ; α)s q (, k; β). If α = (n) n 0, then f k (x; α) reduces to q-factorial [x] k. It means s q (n, k; α), S q (n, k; α) and L q (n, k; α) reduce to the ordinary q-stirling numbers of the first and second kind and q-lah numbers, respectively. Further, by setting q = 1 these three numbers reduce to Comtet s generalized Stirling numbers of both kinds [6] and the generalized Lah numbers which are related to the generalized factorial (x; α) n := (x α 0 )(x α 1 ) (x α ) for any sequence α = (α n ) n 0, respectively. For the reason, we will call the coefficients s q (n, k; α), S q (n, k; α) and L q (n, k; α) appearing in (12), (13) and (14), respectively, the extended q-stirling numbers of first and second kind and the extended q-lah numbers associated with the sequence α = (α n ) n 0. We note that these special numbers may be obtained in some connections with the extended q-factorial coefficients Ω q (n, k; α, β): (i) s q (n, k; α) = Ω q (n, k; α, 0); (ii) S q (n, k; α) = Ω q (n, k; 0, α); (iii) L q (n, k; α) = Ω q (n, k; α, α). The following result can be easily derived from Theorem 2.1. Theorem 3.2. Given a sequence α = (α n ) n 0 of any numbers, we have: (i) s q (n, k; α) = s q (n 1, k 1; α) [α ]s q (n 1, k; α), s q (0, 0; α) = 1; (ii) S q (n, k; α) = S q (n 1, k 1; α) + [α k ]S q (n 1, k; α), S q (0, 0; α) = 1; (iii) L q (n, k; α)=l q (n 1, k 1; α)+([α k ] [ α ])L q (n 1, k; α), L q (0, 0; α) = 1; (iv) s q (n, k; α) = ( 1) n k [α 0 ] c0 [α 1 ] c1 [α ] c ; (v) S q (n, k; α) = c 0 +c 1 + +c k =n k c i {0,1,...,n} c 0 +c 1 + +c =n k c i {0,1} [α 0 ] c 0 [α 1 ] c1 [α k ] c k ; (vi) L q (n, k; α) = n s q(n, ; α)s q (, k; α);

7 (vii) (viii) A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS 651 S q (n, k; α)x n = n=0 x k (1 [α 0]x)(1 [α 1]x) (1 [α k ]x) ; s q (n, k; α)x n k = (1 [α 0 ]x)(1 [α 1 ]x) (1 [α ]x). We point out that the above theorem may be also derived from some known results for Comtet s generalized Stirling numbers [6]. Corollary 3.3. Let α = (α n ) n 0 be a sequence of any numbers. Then we have: (i) S q (n, k; α) = n q ( k)α 0 [α 0 ] n ( n ) Sq (, k; α α 0 ); (ii) s q (n, k; α) = n q (n k)α0 [ α 0 ] k( k) sq (n, ; α α 0 ). Proof. We only give the proof of (i) since (ii) may be proved analogously. First, let α 0 0 and let k = 0. Since S q (0, 0; α α 0 ) = 1 and S q (n, 0; α α 0 ) = 0 for n 1, (i) gives S q (n, 0; α) = [α 0 ] n. Hence (i) holds for k = 0. Now the proof is by induction on n. Clearly (i) holds when n = 1. Assume that n 2, k 1. From (ii) of Theorem 3.2 we have S q (n, k; α) = 1 q ( k+1)α 0 [α 0 ] ( n 1 ) S q (, k 1; α α 0 ) ( ) + [α k ] q ( k)α 0 n 1 [α 0 ] S q (, k; α α 0 ). Apply [α k ] = [α 0 ] + q α 0 [α k α 0 ] to the last equation and combine the first and second summation using the Pascal identity and S q (, k 1; α α 0 ) + [α k α 0 ]S q (, k; α α 0 ) = S q ( + 1, k; α a 0 ). Then it is easily seen that ( ) n S q (n, k; α) = q ( k)α0 [α 0 ] n S q (, k; α α 0 ) which completes the proof. + q (n k)α 0 S q (n, k; α α 0 ) ( ) n = q ( k)α0 [α 0 ] n S q (, k; α α 0 ), We conclude this section describing the familiar formulae for the q-stirling numbers s q (n, k) and S q (n, k), and q-lah numbers L q (n, k). By a q-binomial coefficient we shall mean [ ] n = r q [n]! [r]![n r]!, where n and r are nonnegative integers and [n]! = [n][n 1] [1]. The following may be easily deduced from Theorem 3.2 and Corollary 3.3:

8 652 S.-Z. SONG, G.-S. CHEON, Y.-B. JUN, AND L. B. BEASLEY k (i) S q (n, k) = 1 [k]! ( 1) k q (k 2 ) ( k 2) [ ] k =0 q []n ; (ii) s q (n, k) = n ( 1) k q n ( 1 k 1) sq (n 1, 1); (iii) L q (n, k) = q 2) ( (k n 2) [ ] [n]! k 1 q [k]!. We also note that the expression (i), (iii) were obtained in [17] and (ii) was obtained in [16]. 4. Examples Let us define the non-central generalized q-factorial with both increment h and non-centrality parameter r of order n denoted by [x; r, h] n as (17) [x; r, h] n := [x r][x r h] [x r (n 1)h]. By setting h = 1 or r = 0, [x; r, h] n is reduced to the noncentral q-factorial with non-centrality parameter r or the central generalized q-factorial with increment h given in (4) and (5), respectively. In this section, we investigate the q-stirling numbers of both kinds and the q-lah numbers corresponding to the generalized q-factorial [x; r, h] n as the special examples of our extended q-factorial coefficients. First note that [x; r, h] n may be considered as f n (x; α) with α = (r + nh) n 0 by means of f n (x; r + nh) := [x r][x (r + h)] [x (r + (n 1)h)]. Thus (12), (13) and (14) respectively can be expressed by (i) [x; r, h] n = q nr (n 2)h (ii) [x] n = n (iii) [x; r, h] k = q nr+(n 2)h n s q (n, k; r, h)[x] k ; q kr+(k 2)h S q (n, k; r, h)[x; r, h] k ; n q kr+(k 2)h L q (n, k; r, h)[x; r, h] k. We call s q (n, k; r, h), S q (n, k; r, h) and L q (n, k; r, h) the non-central q-stirling numbers of the first and second kind with both increment h and non-centrality parameter r and non-central q-lah numbers with both increment h and noncentrality parameter r, respectively. Theorem 4.1. The S q (n, k; r, h) satisfy the following explicit formula: 2)h kr k [ ] (18) S q (n, k; r, h) = q (k [h] k ( 1) k q (k 2 )h k [r + h] n. [k] q h! q h =0 Proof. From the expression (iii) for Ω q (n, k; α, β) with α 0 in Theorem 2.1, we have k [β ] n (19) S q (n, k; r, h) = k i=0, i ([β ] [β i ]). =0

9 A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS 653 By aid of [x] [y] = q y [x y], [xh] = [x] q h[h] and [ x] = q x [x] with the q h -number [x] q h and β = (r + nh) n 0, we get k i=0, i ([β ] [β i ]) = q C k+1(β) β [h][( 1)h] [h][ h][ 2h] [ (k )h] Hence (18) follows from (19). = ( 1) k q kr ( (k+1 2 )+( k +1 2 ))h [h] k [] q h![k ] q h! = ( 1) k q kr ((k 2 ) ( k 2))h [h] k [k] q h! The following is an immediate consequence of Corollary 3.3. ( []q h![k ] q h! [k] q h! Corollary 4.2. The s q (n, k; r, h) and S q (n, k; r, h) satisfy the following relations: (i) s q (n, k; r, h) = n q (n k)r [ r] k( k) sq (n, ; h); (ii) S q (n, k; r, h) = n q ( k)r [r] n ( n ) Sq (, k; h), where s q (n, k; h) and S q (n, k; h) are the central generalized q-stirling numbers of the first and second kind with an increment h, i.e., r = 0, investigated in [3]. 5. Matrix factorizations In this section, we develop special matrices arising from the extended q- factorial coefficients Ω q (n, k; α, β). Define the n n matrix Ω q (n; α, β) by { Ωq (i, ; α, β) if i 0, [Ω q (n; α, β)] i, = 0 otherwise, where the rows and columns are indexed by 0, 1,...,. Similarly the matrices s q (n; α), S q (n; α) and L q (n; α) corresponding to s q (n, k; α), S q (n, k; α) and L q (n, k; α) respectively can be defined as Ω q (n; α, β). We will see how these matrices are connected with each other and are related to the q-vandermonde matrix V ([α 0 ], [α 1 ],..., [α ]). We first define the n n (n 2) matrix F q (l) (α), l = 0,..., n 2, for any numbers α 0, α 1,... by F (l) q (α) = I n l 2 1 [α 0 ] [α l ] 1 where I n l 2 is the identity matrix of order n l 2, and denotes the direct sum of matrices and unspecified entries are all zeros., ).

10 654 S.-Z. SONG, G.-S. CHEON, Y.-B. JUN, AND L. B. BEASLEY Lemma 5.1. The n n matrix Ω q (n; α, β), n 2, may be factorized by (20) Ω q (n; α, β) = (F (n 2) q (α)) 1 (I 1 Ω q (n 1; α, β))f q (n 2) (β). Proof. From the recurrence relation for Ω q (n, k; α, β) in Theorem 2.1, we have which implies that F (n 2) q Hence the proof is complete. Ω q (n, k; α, β) + [α ]Ω q (n 1, k; α, β) = Ω q (n 1, k 1; α, β) + [β k ]Ω q (n 1, k; α, β), (α)ω q (n; α, β) = (I 1 Ω q (n 1; α, β))f q (n 2) (β). The following is an immediate consequence of (20) and (16). Corollary 5.2. The n n matrix Ω q (n; α, β), n 2, may be factorized by (i) Ω q (n; α, β) = (F q (0) (α) F (n 2) (ii) Ω q (n; α, β) = s q (n; α)s q (n; β). q (α)) 1 (F q (0) (β) F q (n 2) (β)); For example, if n = 4, then we have: Ω q (n; α, β)= [α 0 ] [α 0 ] [α 1 ] [α 0 ] [α 1 ] [α 2 ] [β 0 ] [β 0 ] [β 1 ] [β 0 ] [β 1 ] [β 2 ] = [α 0 ] [α 0 ][α 1 ] ([α 0 ]+[α 1 ]) 1 0 [α 0 ][α 1 ][α 2 ] [α 0 ][α 1 ]+[α 0 ][α 2 ]+[α 1 ][α 2 ] ([α 0 ]+[α 1 ]+[α 2 ]) [β 0 ] [β 0 ] 2 [β 0 ] + [β 1 ] 1 0 [β 0 ] 3 [β 0 ] 2 + [β 0 ][β 1 ] + [β 1 ] 2 [β 0 ] + [β 1 ] + [β 2 ] 1 =s q (n; α)s q (n; β). The following is an immediate consequence of (15), Theorem 3.1, Theorem 3.2, Corollary 3.3, and Corollary 5.2. Theorem 5.3. The n n matrices s q (n; α), S q (n; α), L q (n; α) have the following matrix factorizations:

11 A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS 655 (i) s q (n; α)s q (n; α) = I n ; (ii) S q (n; α) = { F q (0) (α)f q (1) (α) F q (n 2) (α); (I1 ŝ (iii) s q (n; α) = q (n 1; α α 0 ))P n ( [α 0 ]) if α 0 0, I 1 (ŝ q (n 1; α 1)P 1 ) if α 0 = 0; { Pn ([α (iv) S q (n; α) = 0 ])(I 1 Ŝq(n 1; α α 0 )) if α 0 0, I 1 (P Ŝ q (n 1; α 1)) if α 0 = 0; (v) L q (n; α) = s q (n; α)s q (n; α); (vi) (L q (n; α)) 1 = L q (n; α), where ŝ q (n 1; α α 0 ) and Ŝq(n 1; α α 0 ) are (n 1) (n 1) matrices defined by { q (i )α 0 s q (n 1; i, ; α α 0 ) if n 1 i 1, [ŝ q (n 1; α α 0 )] i, = 0 otherwise, { q (i )α 0 S q (n 1; i, ; α α 0 ) if n 1 i 1, [Ŝq(n 1; α α 0 )] i, = 0 otherwise. Further, the extended q-factorial f n (x; α) can be used to obtain the LDUdecomposition of the n n q-vandermonde matrix V q (n; α) := V ([α 0 ], [a 1 ],..., [α ]) = ([α ] i ) i 0. Let U q (n; α) denote the n n matrix defined by [U q (n; α)] i = f i (α ; α), and let D q (n; α) = diag(q C0(α), q C1(α),..., q C(α) ). Note that U q (n; α) is an upper triangular matrix since f i (α ; α) = 0 if i >. The following theorem may be easily obtained by setting x = α and n = i for each i, = 0, 1,..., n 1 in (13) (also see [16]). Theorem 5.4. Given a sequence α = (α n ) n 0 of any different numbers, the q-vandermonde matrix V q (n; α) may be factorized by V q (n; α) = S q (n; α)d q (n; α)u q (n; α), where S q (n; α) is the generalized q-stirling matrix of the second kind. Thus given a sequence α = (α n ) n 0 of any different numbers, V q (n; α) is nonsingular and by using q αi [α α i ] = [α ] [α i ] we obtain det V q (n; α) = ([α ] [α i ]). 0 i< n Remark. If we take α k x for all k = 0, 1,..., n 1, then S q (n; α) is exactly the same as P n ([x]) = [ ( i ) [x] i ]. Thus our results examined in the present paper may be used to obtain a q-analogue of the Pascal matrix. We omit the details here.

12 656 S.-Z. SONG, G.-S. CHEON, Y.-B. JUN, AND L. B. BEASLEY References [1] G. S. Call and D. J. Vellemam, Pascal s matrices, Amer. Math. Monthly 100 (1993), no. 4, [2] L. Carlitz, q-bernoulli numbers and polynomials, Duke Math. J. 15 (1948), [3] Ch. A. Charalambides, On the q-differences of the generalized q-factorials, J. Statist. Plann. Inference 54 (1996), no. 1, [4], Non-central generalized q-factorial coefficients and q-stirling numbers, Discrete Math. 275 (2004), no. 1-3, [5] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl. 329 (2001), no. 1-3, [6] L. Comtet, Nombres de Stirling généraux et fonctions symétriques, C. R. Acad. Sci. Paris Ser. A-B 275 (1972), A747 A750. [7], Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, [8] H. W. Gould, The q-stirling numbers of first and second kinds, Duke Math. J. 28 (1961), [9] L. C. Hsu, A summation rule using Stirling numbers of the second kind, Fibonacci Quart. 31 (1993), no. 3, [10] L. C. Hsu and P. Shiue, A unified approach to generalized Stirling numbers, Adv. in Appl. Math. 20 (1998), no. 3, [11] C. Jordan, Calculus of Finite Differences, Chelsea Pub. Co., New York, [12] M. Koutras, Noncentral Stirling numbers and some applications, Discrete Math. 42 (1982), no. 1, [13] D. E. Loeb, A generalization of the Stirling numbers, Discrete Math. 103 (1992), no. 3, [14] A. De Medicis and P. Leroux, Generalized Stirling numbers, convolution formulae and p, q-analogues, Canad. J. Math. 47 (1995), no. 3, [15] S. C. Milne, A q-analog of restricted growth functions, Dobinski s equality, and Charlier polynomials, Trans. Amer. Math. Soc. 245 (1978), [16] H. Oruc and H. K. Akmaz, Symmetric functions and the Vandermonde matrix, J. Comput. Appl. Math. 172 (2004), no. 1, [17] C. G. Wagner, Generalized Stirling and Lah numbers, Discrete Math. 160 (1996), no. 1-3, Seok-Zun Song Department of Mathematics Jeu National University Cheu , Korea address: szsong@cheu.ac.kr Gi-Sang Cheon Department of Mathematics Sungkyunkwan University Suwon , Korea address: gscheon@skku.edu Young-Bae Jun Department of Mathematics Education Gyeongsang National University Chinu , Korea address: ybun@gsnu.ac.kr

13 A q-analogue OF THE GENERALIZED FACTORIAL NUMBERS 657 LeRoy B. Beasley Department of Mathematics and Statistics Utah State University Logan, UT , U.S.A address: