EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 1122-1137 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global The r-Dowling Numbers and Matrices Containing r-Whitney Numbers of the Second Kind and Lah Numbers Roberto B. Corcino1,∗, Charles B. Montero2, Maribeth B. Montero2, Jay M. Ontolan1 1 Research Institute for Computational Mathematics and Physics, Cebu Normal University, 6000 Cebu City, Philippines 2 Department of Mathematics, Mindanao State University, 9700 Marawi City, Philippines Abstract. This paper derives three forms of explicit formula for r-Dowling numbers. One of these is expressed in terms of exponential polynomial. The other two formulas are derived using an inverse relation and Faa di Bruno’s formula together with certain identity of Bell polynomials of the second kind. These two formulas are expressed in terms of the r-Whitney numbers of the second kind, r-Whitney-Lah numbers, and the ordinary Lah numbers. As a consequence, a relation between r-Dowling numbers and the sums of row entries of the product of matrices containing the r-Whitney numbers of the second kind, r-Whitney-Lah numbers, and the ordinary Lah numbers is established. Moreover, a q-analogue of the explicit formula is obtained. 2010 Mathematics Subject Classifications: 05A15, 11B65, 11B73 Key Words and Phrases: r-Dowling numbers, (r, β)-Bell numbers, Bell polynomials, Lah numbers, r-Whitney numbers, Faa di Bruno’s formula, r-Whitney-Lah numbers 1. Introduction The Bell numbers, denoted by Bn, were defined in [5] as the sum of Stirling numbers of the second kind Bn := n∑ k=0 S(n, k). (1) Since the numbers S(n, k) are interpreted as the number of ways to partition an n-set into k nonempty subsets, Bn can then be interpreted as the total number of ways to partition an n-set. Several properties and application were obtained for these numbers including ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3494 Email addresses: rcorcino@yahoo.com (R. B. Corcino), charlesmontero@yahoo.com (C. Montero), bette myb@yahoo.com (M. Montero), ontolanjay@gmail.com (J. Ontolan) http://www.ejpam.com 1122 c© 2019 EJPAM All rights reserved. R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1123 generating functions, recursive formulas, explicit formula, and expression in terms of a moment of the Poisson random variable [19, 20]. By adding one parameter r, A.Z. Broder [3] defined combinatorially a generalization of S(n, k), the r-Stirling numbers of the second kind, denoted by { n k } r , as follows:{ n k } r := the number of partitions of an n-set into k nonempty subsets such that the numbers 1, 2, . . . , r are in distinct subsets. These numbers possessed several properties parallel to those of the classical Stirling num- bers of the second kind, which can be found in [3]. In the same paper [3] , Broder was able to derive a relation expressing { n k } r in terms of the classical Stirling numbers of the second kind: { n k } r = n∑ j=k ( n j ) S(j, k)rn−j . (2) Letting r = 0, equation (2) gives { n k } 0 = S(n, k) with 00 defined to be 1. Parallel to the definition of Bell numbers in (1), Mezo [17] defined the r-Bell numbers as Bn,r = n∑ k=0 { n+ r k + r } r . (3) Mezo [17] obtained several interesting properties for these numbers analogous to those of the classical Bell numbers. It is worth mentioning that r-Bell numbers were first introduced by C.B. Corcino in [6]. Furthermore, by adding one more parameter m, Mező [16] defined the r-Whitney numbers of the first and second kind, denoted by wm,r(n, k) and Wm,r(n, k), as coefficients of the following expansions mn(x)n = n∑ k=0 (−1)n−kwm,r(n, k)(mx+ r)k, (4) and (mx+ r)n = n∑ k=0 W (n, k)mk(x)k, (5) where (x)k = x(x− 1) . . . (x− k + 1) if k ≥ 1, with (x)0 = 1. Below are the few values of wm,r(n, k) and Wm,r(n, k) with m = r = 2: n/k 0 1 2 3 4 n/k 0 1 2 3 4 0 1 0 1 1 2 1 1 2 1 2 8 6 1 2 4 6 1 3 48 44 12 1 3 8 28 12 1 4 384 400 140 20 1 4 16 120 100 20 1 R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1124 Table 1: Few values of w2,2(n, k) Table 2: Few values of W2,2(n, k) It would be interesting to note that the numbers Wm,r(n, k) are equivalent to the (r, β)- Stirling numbers [7] and the numbers wm,r(n, k) are equivalent to the numbers that ap- peared in [10]. One can easily verify that these numbers satisfy the following inverse relation fn = n∑ j=0 (−1)n−jwβ,r(n, j)gj ⇐⇒ gn = n∑ j=0 Wβ,r(n, j)fj . (6) Analogous to (2), Cheon and Jung [4] expressed the r-Whitney numbers of the second kind Wm,r(n, k) in terms of the classical Stirling numbers of the second kind S(n, k) as Wm,r(n, k) = n∑ i=k ( n i ) mi−krn−iS(i, k). (7) Replacing m by β, k by j and r by −r in equation (7), yield Wβ,−r(n, j) = n∑ k=j ( n k ) βk−j(−r)n−kS(k, j). (8) Moreover, Cheon and Jung [4] defined the r-Dowling polynomials, denoted by Dm,r(n, x), as follows Dm,r(n, x) = n∑ k=0 Wm,r(n, k)xk. (9) Taking x = 1, equation (9) reduces to Dm,r(n, 1) = n∑ k=0 Wm,r(n, k), the r-Dowling numbers. These numbers are equivalent to the (r, β)-Bell numbers in [8], denoted by Gn,β,r, and have also been considered in the paper [12] using the same notation Gn,β,r. Throughout this paper, we use Gn,β,r to denote the r-Dowling numbers. It is worth mentioning that Gn,β,r satisfy the following generating function∑ n≥0 Gn,β,r tn n! = erte 1 β (eβt−1) . (10) The Lah numbers, denoted by L(n, k), were defined in [5], combinatorially, as the number of ways to partition an n-set into k nonempty linearly ordered subsets. These numbers have been shown to satisfy the following relations L(n, k) = ( n− 1 k − 1 ) n! k! (11) R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1125 L(n, k) = n∑ j=k s(n, j)S(j, k). (12) On the other hand, the r-Whitney-Lah numbers, denoted by Lm,r(n, k), were defined by Cheon and Jung [4] parallel to (12) as follows Lm,r(n, k) = n∑ j=k wm,r(n, j)Wm,r(j, k). (13) Several properties of Lm,r(n, k) have been derived through factorization of the r-Whitney- Lah matrix [Lm,r(n, k)]n,k≥0 (see [4]) including the triangular relation Lm,r(n, k) = Lm,r(n− 1, k − 1) + (2r + (n+ k − 1)m)Lm,r(n− 1, k) Below is a triangular array of values for Lm,r(n, k) with m = r = 2: n/k 0 1 2 3 4 0 1 1 4 1 2 24 12 1 3 192 144 24 1 4 1920 1920 480 40 1 Table 3: Few values of L2,2(n, k). In this paper, two explicit formulas for Gn,β,r are derived using the two methods applied by Feng Qi [21] in expressing the Bell numbers in terms of Stirling numbers of the second kind and Lah numbers. The two methods yield exactly the same explicit formula when they are applied by Feng Qi to Bell numbers. However, when these methods are applied here to Gn,β,r, they give two equivalent formulas of different forms. These formulas imply two matrix relations involving r-Dowling numbers, r-Whitney numbers of the second, r-Whitney-Lah numbers and Lah numbers. 2. Expression in Terms of Exponential Polynomials The exponential polynomial [2], denoted by Φn(x), appeared in the resulting expression in applying Mellin derivative ( x d dx )n to the function ex. The notation for Mellin derivative would mean that the differential operator x d dx is applied n times to ex. The first two applications of the operator give x d dx ex = xex( x d dx )2 ex = ( x d dx )( x d dx ex ) R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1126 = x d dx (xex) = ( x2 + x ) ex. Continuing in this manner yields( x d dx )n ex = Φn(x)ex. The exponential polynomial satisfies the following generating function ex(e t−1) = ∞∑ n=0 Φn(x) tn n! , (14) which can be expressed in polynomial form as Φn(x) = n∑ k=0 S(n, k)xk, (15) whose coefficients are the Stirling numbers of the second kind. Note that, when x = 1/β and t = βt, (16) reduces to e 1 β (et−1) = ∞∑ n=0 Φn (1/β) (βt)n n! , β 6= 0. (16) Hence, the exponential generating function in (10) can be written as ∑ n≥0 Gn,β,r tn n! = ∑ n≥0 (rt)n n! ∑ n≥0 Φn (1/β) (βt)n n!  = ∑ n≥0 { n∑ k=0 Φk (1/β) (βt)k k! (rt)n−k (n− k)! } = ∑ n≥0 { n∑ k=0 ( n k ) Φk (1/β)βkrn−k } tn n! . Comparing the coefficients of tn n! yields the following explicit formula. Theorem 2.1. The r-Dowling numbers can be expressed as Gn,β,r = n∑ k=0 ( n k ) Φk (1/β)βkrn−k, (17) which is a kind of binomial combination of Φk(1/β). R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1127 Using (15), the explicit formula in (17) can further be written as Gn,β,r = n∑ k=0  k∑ j=0 S(k, j)(1/β)j  ( n k ) βkrn−k. (18) This gives the following matrix relation. Theorem 2.2. For n ∈ N, the r-Dowling numbers Gi,β,r equal to the sum of the entries of the ith row of the product of two matrices[( i j ) βjri−j ] (n+1)×(n+1) [ S(i, j)(1/β)j ] (n+1)×(n+1) . (19) 3. r-Whitney Numbers of the Second Kind and r-Whitney-Lah Numbers In this section, a new explicit formula for r-Dowling numbers expressed in terms of r-Whitney Lah numbers and r-Whitney numbers of the second kind is established. As a consequence, a relation in terms of matrices involving the r-Dowling numbers, the r- Whitney-Lah numbers and the r-Whitney numbers of the second kind is obtained. Note that equation (13) can be rewritten as follows (−1)nLβ,r(n, k) = n∑ j=0 wβ,r(n, j)Wβ,r(j, k). (20) Using the inverse relation of r-Whitney numbers in (6) with fn = (−1)nLβ,r(n, k) and gj = (−1)jWβ,r(j, k), equation (20) yields (−1)nWβ,r(n, k) = n∑ j=0 Wβ,r(n, j)(−1)jLβ,r(j, k); that is, S(n, k;β, r) = Wβ,r(n, k) = n∑ j=0 (−1)n−jWβ,r(n, j)Lβ,r(j, k). Summing up both sides over k from 0 to n, gives the following theorem. Theorem 3.1. The explicit formula for r-Dowling numbers is given by Gn,β,r = n∑ j=0 (−1)n−j { j∑ k=0 Lβ,r(j, k) } Wβ,r(n, j). (21) R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1128 For instance, when β = r = 2 and n = 4, we get G4,2,2 = 4∑ j=0 (−1)4−j { j∑ k=0 L2,2(j, k) } W2,2(4, j) = (1)(16)− (5)(120) + (37)(100)− (361)(20) + (461)(1) = 257. Now, we can rewrite the sum in (21) as Gn,β,r = S0 + S1 + S2 + . . .+ Sn where Sj = n∑ k=0 (−1)n−kWβ,r(n, k)Lβ,r(k, j). As a consequence, we have the following theorem. Theorem 3.2. For n ∈ N, the r-Dowling numbers Gi,β,r equal to the sum of the entries of the ith row of the product of two matrices[ (−1)i−jWβ,r(i, j) ] (n+1)×(n+1) [Lβ,r(i, j)](n+1)×(n+1) , (22) whose entries are respectively the r-Whitney numbers of the second kind and the r-Whitney Lah numbers. For instance, when β = r = 2 and n = 3, we get[ (−1)i−jW2,2(i, j) ] 4×4 [Lβ,r(i, j)]4×4 =  1 0 0 0 −2 1 0 0 4 −6 1 0 −8 28 −12 1   1 0 0 0 4 1 0 0 24 12 1 0 192 144 24 1  =  1 0 0 0 2 1 0 0 4 6 1 0 8 28 12 1  Summing up the entries of each row of the above matrix product, we obtain the column vector whose entries are the r-Dowling numbers 1 2 + 1 4 + 6 + 1 8 + 28 + 12 + 1  =  1 3 11 49  =  G0,2,2 G1,2,2 G2,2,2 G3,2,2  . R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1129 Corollary 3.3. For 0 ≤ i, l ≤ n, the r-Whitney numbers of the second kind satisfy the following explicit formula Wβ,r(i, l) = i∑ j=0 (−1)i−jWβ,r(i, j)Lβ,r(j, l); that is, [Wβ,r(i, j)]n+1×n+1 = [ (−1)i−jWβ,r(i, j) ] n+1×n+1 [Lβ,r(i, j)]n+1×n+1 . It can easily be shown that n∑ j=i (−1)n−jwβ,r(n, j)Wβ,r(j, i) = n∑ j=i Wβ,r(n, j)(−1)j−iwβ,r(j, i) = δni, (23) where δni is the Kronecker delta. This relation implies that [Wβ,r(i, j)] −1 n+1×n+1 = [ (−1)i−jwβ,r(i, j) ] n+1×n+1 . (24) Thus, we have[ (−1)i−jwβ,r(i, j) ] n+1×n+1 [ (−1)i−jWβ,r(i, j) ] n+1×n+1 [Lβ,r(i, j)]n+1×n+1 = In+1. 4. r-Whitney Numbers of the Second Kind and Lah Numbers In this section, we will find a new explicit formula for computing r-Dowling numbers Gn,β,r in terms of r-Whitney numbers of the second kind and the ordinary Lah numbers using the Faa di Bruno’s formula and certain identity of Bell polynomials of the second kind. The following theorem contains the desired formula. Theorem 4.1. For n ∈ N, the r-Dowling numbers Gn,r,β equal Gn,r,β = n∑ j=0 (−1)n−jWβ,−r(n, j) j∑ i=0 βj−iL(j, i). (25) Proof. Let us recall the following identity from [1, 13] on the nth derivative of the exponential function e± 1 t expressed in terms of the Lah numbers( e± 1 t )(n) = (−1)ne± 1 t n∑ k=1 (±1)kL(n, k) 1 tn+k , (26) the identity from [5] on Bell polynomials of the second kind Bn,k(abx1, ab 2x2, . . . , ab n−k+1xn−k+1) = akbnBn,k(x1, x2, . . . , xn−k+1), (27) R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1130 and the famous identity from [5] on Faá di Bruno formula described in terms of the Bell polynomials of the second kind dn dtn f ◦ h(t) = n∑ k=0 f (k)(h(t))Bn,k(h ′(t), h′′(t), . . . , h(n−k+1)(t)). (28) Replacing t by −t in the generating function for the r-Dowling numbers Gn,β,r in equation (10), yields ∑ n≥0 Gn,β,r (−t)n n! = e−rt · e 1 βeβt e 1 β ; equivalently, e 1 β ∑ n≥0 (−1)nGn,β,r tn n! = e 1 βeβt · e−rt. (29) Then taking kth derivative both sides of (29) with respect to t yields e 1 β ∞∑ n=k (−1)kGn,β,r tn−k (n− k)! = dk dtk ( e 1 βeβt · e−rt ) . (30) Taking f(u) = e 1 u and h(t) = βeβt in (28) and making use of (26) give dk ( e 1 βeβt ) dtk = dk ( f ◦ h(t) ) dtk = k∑ j=1 dj(e1/u) duj Bk,j(β(βeβt), β2(βeβt), . . . , βk−j+1(βeβt)) = k∑ j=1 (−1)je1/u j∑ i=1 L(j, i) · 1 uj+i Bk,j(β(βeβt), β2(βeβt), . . . , βk−j+1(βeβt)) = e 1 βeβt k∑ j=1 (−1)j j∑ i=1 L(j, i) · 1 (βeβt)j+i Bk,j(β(βeβt), β2(βeβt), . . . , βk−j+1(βeβt)), where u(t) = βeβt. Further by virtue of Bk,j(abx1, ab 2x2, . . . , ab k−j+1xk−j+1) = ajbkBk,j(x1, x2, . . . , xk−j+1) and Bk,j( k−j+1︷ ︸︸ ︷ 1, 1, . . . , 1) = S(k, j) R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1131 listed in [5], [p.135], where a and b are complex numbers, we obtain dk ( e 1 βeβt ) dtk = e 1 βeβt k∑ j=1 (−1)j j∑ i=1 L(j, i) · 1 (βeβt)j+i · (βeβt)jβkBk,j( k−j+1︷ ︸︸ ︷ 1, 1, . . . , 1) = e 1 βeβt k∑ j=1 (−1)j j∑ i=1 L(j, i) · β k−i (eβt)i S(k, j). Hence, using Leibniz formula, dn dzn ( e 1 βeβt · e−rt ) = n∑ k=0 ( n k ) dk dtk e 1 βeβt dn−k dtn−k e−rt = n∑ k=0 ( n k )e 1 βeβt k∑ j=1 (−1)j j∑ i=1 L(j, i) · β k−i (eβt)i S(k, j)  · (−r)n−ke−rt. Thus, replacing k by n and evaluating at t = 0 in equation (30) give e 1 β (−1)nGn,β,r = n∑ k=0 ( n k ) k∑ j=1 (−1)je 1 β j∑ i=1 L(j, i) · βk−iS(k, j) · (−r)n−k; Rearranging the above sum and using the fact that L(0, i) = 0 for all positive integers i, we get Gn,β,r = n∑ i=0 (−1)n−j i∑ j=0 { n∑ k=j ( n k ) βk−j(−r)n−kS(k, j) } βj−iL(j, i). Applying the property of r-Whitney numbers of the second kind in equation (8) yields Gn,β,r = n∑ i=0 (−1)n−j i∑ j=0 Wβ,−r(n, j)β j−iL(j, i). This is exactly the formula in (25). The following corollary is a direct consequence of Theorem 4.1. Corollary 4.2. For n ∈ N, the r-Dowling numbers Gi,β,r equal to the sum of the entries of the ith row of the product of two matrices[ (−1)i−jWβ,−r(i, j) ] n×n [ βj−iL(i, j) ] n×n , (31) whose entries are respectively r-Whitney numbers of the second kind and the Lah numbers. Proof. We can rewrite the formula in Theorem 4.1 as Gi,β,r = i∑ l=0 Til, i = 0, 1, 2, . . . , n, R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1132 where Til = i∑ j=0 (−1)i−jWβ,−r(i, j)β j−lL(j, l), l = 0, 1, 2, . . . i. Clearly, Til is the (i, l)-entry of the following product of two matrices[ (−1)i−jWβ,−r(i, j) ] n×n [ βj−iL(i, j) ] n×n , (32) containing the r-Whitney numbers of the second kind and Lah numbers, respectively. To illustrate this corollary, let us consider the case where β = 1, r = 2, n = 6. That is,[ (−1)i−jW1,−2(i, j) ] 6×6 [ βi−jL(i, j) ] 6×6 =  1 0 0 0 0 0 2 1 0 0 0 0 4 3 1 0 0 0 8 7 3 1 0 0 16 15 7 2 1 0 32 31 15 5 0 1   1 0 0 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 6 6 1 0 0 0 24 36 12 1 0 0 120 240 120 20 1  =  1 0 0 0 0 0 2 1 0 0 0 0 4 5 1 0 0 0 8 19 9 1 0 0 16 65 55 14 1 0 32 211 285 125 20 1  . (33) Hence, summing up the entries of each row of the matrix in (33) gives the following column vector whose entries are the r-Dowling numbers with β = 1 and r = 2 1 2 + 1 4 + 5 + 1 8 + 19 + 9 + 1 16 + 65 + 55 + 14 + 1 32 + 211 + 285 + 125 + 20 + 1  =  1 3 10 37 151 674  =  G0,1,2 G1,1,2 G2,1,2 G3,1,2 G4,1,2 G5,1,2  . Clearly, the r-Whitney numbers of the second kind Wβ,r(i, l) can be expressed as Wβ,r(i, l) = i∑ j=0 (−1)i−jWβ,−r(i, j)β j−lL(j, l). That is, [Wβ,r(i, j)]n+1×n+1 = [ (−1)i−jWβ,−r(i, j) ] n+1×n+1 [ βi−jL(i, j) ] n+1×n+1 . Using (24), we obtain the following matrix identity.[ (−1)i−jwβ,r(i, j) ] n+1×n+1 [ (−1)i−jWβ,−r(i, j) ] n+1×n+1 [ βi−jL(i, j) ] n+1×n+1 = In+1. R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1133 5. A q-Analogue A q-analogue is a generalization of a known expression parameterized by a quantity q that reduces to the known expression in the limit, as q → 1. For example, the q-analogue of n, n!, (n)k and ( n k ) are respectively given by [n]q = 1 + q + q2 + · · ·+ qn−1 = 1− qn 1− q ; [n]q! = [n]q[n− 1]q · · · [2]q[1]q; [n]k,q = [n]q[n− 1]q · · · [n− k + 1]q;[ n k ] q = [n]q! [k]q![n− k]q! = [n]k,q k! . Recently, R. Corcino et. al [9] defined a q-analogue of r-Whitney numbers of the second kind by means of the following recurrence relation: Wm,r[n, k]q = qm(k−1)+rWm,r[n− 1, k − 1]q + [mk + r]qWm,r[n− 1, k]q. (34) When q → 1, this will reduce to Wm,r(n, k) = Wm,r(n− 1, k − 1) + (mk + r)Wm,r(n− 1, k). One can easily verify that Wm,r[n, 0] = [r]nq . The q-analogue Wm,r[n, k]q possessed several properties including the following relation n∑ k=0 Wm,r[n, k]q[t− r|m]k,q = [t]nq . (35) For the r-Whitney numbers of the first kind, their q-analogue may be defined by n∑ k=0 (−1)n−kwm,r[n, k]q[t] k q = [t− r|m]n,q. (36) To compute the first values of wm,r[n, k]q, we need to derive the triangular recurrence relation for wm,r[n, k]q. Using (36) and the identity [t− n]q = 1 qn ([t]q − [n]q), we have n+1∑ k=0 (−1)n+1−kwm,r[n+ 1, k]q[t] k q = [t− r|m]n+1,q = [t− (r + nm)]q[t− r|m]n+1,q R. B. Corcino et al. / Eur. J. Pure Appl. Math, 12 (3) (2019), 1122-1137 1134 = ( 1 qr+nm ([t]q − [r + nm]q) ) n∑ k=0 (−1)n−kwm,r[n, k]q[t] k q = n+1∑ k=0 1 qr+nm (−1)n−k+1wm,r[n, k − 1]q[t] k q + n+1∑ k=0 −[r + nm]q qr+nm (−1)n−kwm,r[n, k]q[t] k q = n+1∑ k=0 (−1)n−k+1 qr+nm {wm,r[n, k − 1]q + [r + nm]qwm,r[n, k]q} [t]kq . Thus, comparing the coefficients of [t]kq , we easily obtain the following triangular recurrence relation qr+nmwm,r[n+ 1, k]q = wm,r[n, k − 1]q + [r + nm]qwm,r[n, k]q. (37) Now, to derive the orthogonality relations for wm,r[n, k]q and Wm,r[n, k]q, we first rewrite (36) as k∑ j=0 wm,r[k, j]q[t] j q = [t− r|m]k,q and substituting to (35) yields [t]nq = n∑ k=0 Wm,r[n, k]q[t− r|m]k,q = n∑ k=0 Wm,r[n, k]q k∑ j=0 (−1)k−jwm,r[k, j]q[t] j q = n∑ j=0  n∑ k=j (−1)k−jWm,r[n, k]qwm,r[k, j]q  [t]jq. Hence, we obtain the first form of the desired orthogonality relation n∑ k=j (−1)k−jWm,r[n, k]qwm,r[k, j]q = δn,j , (38) where δn,j is the well-known Kronecker delta. By applying similar argument, that is, by substituting (35) to (36), we obtain the second form of the orthogonality relation n∑ k=j (−1)n−kwm,r[n, k]qWm,r[k, j]q = δn,j , (39) Furthermore, the orthogonality relations in (38) and (39) immediately imply the following inverse relations: fn = n∑ k=0 (−1)n−kwm,r[n, k]qgk ⇐⇒ gn = n∑ k=0 Wm,r[n, k]qfk (40) REFERENCES 1135 fk = ∞∑ n=k (−1)n−kwm,r[n, k]qgn ⇐⇒ gk = ∞∑ n=k Wm,r[n, k]qfn. (41) Parallel to Cheon and Jung [4], a q-analogue of r-Whitney-Lah numbers Lβ,r[n, k]q may be defined by Lβ,r[n, k]q = n∑ j=0 wβ,r[n, j]qWβ,r[j, k]q. (42) This can be written as (−1)nLβ,r[n, k]q = n∑ j=0 (−1)n−jwβ,r[n, j]q(−1)jWβ,r[j, k]q. (43) Using the inverse relation in (40) with fn = (−1)nLβ,r[n, k]q and gj = (−1)jWβ,r[j, k]q, relation (43) implies the following relation (−1)nWβ,r[n, k]q = n∑ j=0 Wβ,r[n, j]q(−1)jLβ,r[j, k]q Wβ,r[n, k]q = n∑ j=0 (−1)n−jWβ,r[n, j]qLβ,r[j, k]q (44) Summing up both sides of (44) over k yields n∑ k=0 Wβ,r[n, k]q = n∑ k=0 Wβ,r[n, k]q n∑ j=0 (−1)n−jWβ,r[n, j]qLβ,r[j, k]q Dβ,r[n]q = n∑ j=0 (−1)n−j { j∑ k=0 Lβ,r[j, k]q } Wβ,r[n, j]q. (45) Remark 5.1. The explicit formula in (45) implies that the (q, r)-Dowling numbers Dβ,r[n]q are equal to eiDLe, where D and L are matrices whose entries areWβ,r[n, j]q and Lβ,r[n, k]q, respectively, ei is the i− th unit vector, and e is the vector with all entries equal to 1. Acknowledgements This research has been funded by Cebu Normal University (CNU) and the Commission on Higher Education - Grants-in-Aid for Research (CHED-GIA). References [1] K. N. 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