A34 INTEGERS 14 (2014) the (R1,...,Rp)-BELL POLYNOMIALS

#A34 INTEGERS 14 (2014)

THE (r1, . . . , rp)-BELL POLYNOMIALS

Mohammed Said Maamra Faculty of Mathematics, RECITS’s Laboratory, USTHB, Algiers, Algeria [email protected] or [email protected]

Miloud Mihoubi Faculty of Mathematics, RECITS’s laboratory, USTHB, Algiers, Algeria. [email protected] or [email protected]

Received: 12/10/12, Revised: 12/21/13, Accepted: 4/19/14, Published: 7/10/14

Abstract

In a previous paper, Mihoubi et al. introduced the (r1, . . . , rp)-Stirling numbers and the (r1, . . . , rp)-Bell polynomials and gave some of their combinatorial and algebraic properties. These numbers and polynomials generalize, respectively, the r- Stirling numbers of the second kind introduced by Broder and the r-Bell polynomials introduced by Mez˝o. In this paper, we prove that the (r1, . . . , rp)-Stirling numbers of the second kind are log-concave. We also give generating functions and generalized recurrences related to the (r1, . . . , rp)-Bell polynomials.

1. Introduction

In 1984, Broder [2] introduced and studied the r-Stirling number of the second kind n , which counts the number of partitions of the set [n] = 1, 2, . . . , n k r { } into k non-empty subsets such that the r ﬁrst elements are in distinct subsets. In 2011, Mez˝o [8] introduced and studied the r-Bell polynomials. In 2012, Mihoubi et al. [12] introduced and studied the (r1, . . . , rp)-Stirling number of the second kind n , which counts the number of partitions of the set [n] into k non- k r1,...,rp empty subsets such that the elements of each of the p sets R := 1, . . . , r , 1 { 1} R2 := r1 + 1, . . . , r1 + r2 , . . . , Rp := r1 + rp 1 + 1, . . . , r1 + + rp are in { } { · · · · · · } distinct subsets.

This work is motivated by the study of the r-Bell polynomials [8] and the (r1, . . . , rp)- Stirling numbers of the second kind [12], in which we may establish INTEGERS: 14 (2014) 2

the log-concavity of the (r , . . . , r )-Stirling numbers of the second kind, • 1 p generalized recurrences for the (r , . . . , r )-Bell polynomials, and • 1 p the ordinary generating functions of these numbers and polynomials. •

To begin, by the symmetry of the (r1, . . . , rp)-Stirling numbers with respect to r , . . . , r , let us suppose that r r r , and throughout this paper we use 1 p 1  2  · · ·  p the following notation and deﬁnitions

r := (r , . . . , r ) , r := r + + r , p 1 p | p| 1 · · · p t r1 rp 1 Pt (z; rp) := (z + rp) (z + rp) (z + rp) , t R, · · · 2 n+ rp 1 | | n + r B (z; r ) := | p| zk, n 0 n p k + r p rp kX=0 ⇢ p and ei denotes the i-th vector of the canonical basis of R . In [12], the following were proved:

zk B (z; r ) = exp ( z) P (k; r ) , (1) n p n p k! k 0 X n+ rp 1 | | n + r P (z; r ) = | p| zk. (2) n p k + r p rp kX=0 ⇢ For later use we deﬁne the following numbers

rp 1 k r1 rp 1 ak (rp 1) = ( 1)| | , jp 1 = j1 + + jp 1, j1 · · · jp 1 | | · · · jp 1 =k  | X | n where k are the absolute Stirling numbers of the ﬁrst kind. Upon using the known identity ⇥ ⇤ r r r j r j (u) = ( 1) u j j=0 X  we may state that we have

rp 1 | | k r1 rp 1 ak (rp 1) u = (u) (u) . (3) · · · kX=0

In our contribution, we give more properties for the rp-Stirling numbers and the rp-Bell polynomials. The paper is organized as follows. In the next section we prove INTEGERS: 14 (2014) 3

n+ rp that the sequence k+|r | ; 0 k n + rp 1 is strongly log-concave and we p rp   | | ✓ ◆ n+ rp give an approximation of k+|r | when n for a ﬁxed k. In the third section p rp ! 1 we write Bn (z; rp) in the basis Bn+k (z; rp) : 0 k rp 1 and Bn+m (z; rp) {   | |} in the family of bases zjB (z; r + je ) : 0 j n . As consequences, we also m p p   give some identities for the r -Stirling numbers. In the fourth section we give the p ordinary generating functions of the rp-Stirling numbers of the second kind and the rp-Bell polynomials.

2. Log-Concavity of the rp-Stirling Numbers

In this section we discuss the real roots of the polynomial Bn (z; rp) , the log-

n+ rp concavity of the sequence k+|r | , 0 k n + rp 1 , the greatest maximiz- p rp   | | ✓ ◆ n n+ rp ing index of k and we give an approximation of m+| r | when n tends to rp p rp inﬁnity. The case p = 1 was studied by Mez˝o [9] and another study is done by Zhao [15] for a large class of the Stirling numbers.

In what follows, for illustration or if the order of r1, . . . , rp is unknown, we write the polynomial Bn (z; rp) as Bn (z; r1, . . . , rp) for which r1, . . . , rp are taken in any order.

Theorem 1. The roots of the polynomial Bn (z; rp) are real and non-positive.

To prove this theorem, we use the following lemma.

Lemma 2. Let j, p be nonnegative integers and set

dj B(j) (z; r ) := exp ( z) (zrp exp (z) B (z; r )) , n p dzj n p n n k where r0 := 0 and Bn (z; r0) := Bn (z) = k z . Then, we have k=0 P (j) rp j Bn (z; rp) = z Bn (z; r1, . . . , rp, j) if j < rp, B(j) (z; r ) = B (z; r , . . . , r , j) if j r , n p n 1 p p

(j) (rp+1) with deg Bn = n + r . In particular, we have Bn (z; r ) = B (z; r ) . | p| p n p+1 INTEGERS: 14 (2014) 4

(j) Proof. The deﬁnition of Bn (z; rp) and the identity (1) show that we have

(j) exp (z) Bn (z; rp)

dj zk+rp = P (k; r ) dzj 0 n p k! 1 k 0 X @ A k+rp j n r1 rp 1 j z = (k + r ) (k + r ) (k + r ) (k + r ) . p p · · · p p k! k max(0,j r ) X p Then, for 0 j < r we obtain  p

k+rp j (j) n r1 rp 1 j z exp (z) B (z; r ) = (k + r ) (k + r ) (k + r ) (k + r ) n p p p · · · p p k! k 0 X rp j = z exp (z) Bn (z; r1, . . . , rp, j) and for j r we obtain p

k+rp j (j) n r1 rp 1 j z exp (z) B (z; r ) = (k + r ) (k + r ) (k + r ) (k + r ) n p p p · · · p p k! k j r X p k n r1 rp 1 rp z = (k + j) (k + j) (k + j) (k + j) · · · k! k 0 X = exp (z) Bn (z; r1, . . . , rp, j) .

(j) It is obvious that we have deg Bn = n + rp and for j = rp+1 rp we obtain (rp+1) | | Bn (z; rp) = Bn (z; r1, . . . , rp, rp+1) = Bn (z; rp+1) .

Proof of Theorem 1. We will show by induction on p that the roots of the polynomials Bn (z; rp) are real and non-positive. Indeed, for p = 0 the classical Bell polynomial Bn (z; r0) = Bn (z) has only real non-positive roots and for p = 1 the polynomial Bn (z; r1) is the r1-Bell polynomial introduced in [8] and has only real non-positive roots. Assume, for 1 r r r , that the roots of the poly-  1  2  · · ·  p nomial Bn (z; rp) are real and negative, denoted by z1, . . . , zn+ rp 1 with 0 > z1 (j) | | zn+ rp 1 . We will prove that the polynomial Bn (z; rp) has only real non- · · · | | (rp+1) positive roots and we conclude that the polynomial Bn (z; rp+1) = Bn (z; rp) (see Lemma 2) has only real non-positive roots. (j) Firstly, we examine the polynomials Bn (z; rp) for j < rp. Indeed, the above statements show that the function

(0) rp fn (z; rp) := exp (z) Bn (z; rp) = z exp (z) Bn (z; rp) INTEGERS: 14 (2014) 5

vanishes at z0, z1, . . . , zn+ rp 1 with z0 = 0 > z1 zn+ rp 1 and z0 = 0 is of | | · · · | | multiplicity rp. Lemma 2 gives

d (1) rp 1 (fn (z; rp)) = exp (z) Bn (z; rp) = z exp (z) Bn (z; r1, . . . , rp 1, rp, 1) dz and by applying Rolle’s theorem to the function fn (z; rp) we conclude that its d derivative dz (fn (z; rp)) vanishes at some points x1, . . . , xn+ rp 1 with 0 > x1 | | (1) z1 x2 xn+ rp 1 zn+ rp 1 . Consequently, the polynomial Bn (z; rp) · · · | | | | vanishes at x1, . . . , xn+ rp 1 and at x0 = 0 (with multiplicity rp 1). The number | | (1) of these roots is (n + rp 1 ) + (rp 1) = n + rp 1. Because Bn (z; rp) is of | | | | degree n + r (see Lemma 2), it must have exactly n + r ﬁnite roots; the missing | p| | p| one, denoted by xn+ rp 1 +1, cannot be complex. By the fact that the coecients of k | | z in Bn (z; r1, . . . , rp 1, rp, 1) are positive, the root xn+ rp 1 +1 must be negative (1) | | too. So, the polynomial Bn (z; rp) has n + rp 1 + 1 real negative roots and | | z = 0 is a root with multiplicity rp 1. Similarly, we apply Rolle’s theorem to the d (2) function (fn (z; rp)) to conclude that the polynomial Bn (z; rp) has n+ rp 1 +2 dz | | real negative roots and z = 0 is a root with multiplicity rp 2, and so on. So, the (0) (1) (rp 1) polynomials Bn (z; rp) , Bn (z; rp) , . . . , Bn (z; rp) have only real non-positive roots. (j) Secondly, we examine the polynomials Bn (z; rp) for rp j rp+1. Indeed, we (rp)   have Bn (0; r ) = 0 and consider the function p 6 rp 1 d (rp 1) r 1 fn (z; rp) = exp (z) Bn (z; rp) = z exp (z) Bn (z; r1, . . . , rp 1, rp, rp 1) . dz p As it is shown above, this function has n + r 1 real negative roots and the root | p| z = 0, then Rolle’s theorem shows that its derivative

rp d (rp) fn (z; rp) = exp (z) Bn (z; rp) = exp (z) Bn (z; r1, . . . , rp 1, rp, rp) dzrp has at least n + r 1 real negative roots. This means that the polynomial | p| (rp) Bn (z; rp) = Bn (z; r1, . . . , rp 1, rp, rp) has at least n + r 1 real negative roots and because it is of degree n + r , | p| | p| the missing one cannot be complex. By the fact that the coecients of zk in Bn (z; r1, . . . , rp 1, rp, rp) are positive, this root must be negative too. So, the poly- (rp) nomial Bn (z; rp) has n+ rp real negative roots. Similarly, apply Rolle’s theorem r d p | | (rp+1) to dzrp fn (z; rp) and conclude that the polynomial Bn (z; rp) has n + rp real (rp) (rp+1) | | negative roots and so on. So, the polynomials Bn (z; rp) , . . . , Bn (z; rp) van- (rp+1) ish only at negative numbers. Then, the polynomial Bn (z; rp+1) = Bn (z; rp) (see Lemma 2) has only real negative roots. INTEGERS: 14 (2014) 6

Upon using Newton’s inequality [6, p. 52], which is given by

Theorem 3. (Newton’s inequality) Let a0, a1, . . . , an be real numbers. If all the n i zeros of the polynomial P (x) = aix are real, then the coecients of P satisfy k=0 P 2 1 1 ai 1 + 1 + ai+1ai 1, 1 i n 1, i n i   ✓ ◆ ✓ ◆ we may state that:

n+ rp Corollary 4. The sequence k+|r | , 0 k n + rp 1 is strongly log-concave p rp   | | ⇢ (and thus unimodal).

This property shows that the sequence ( n , 0 k n) admits an index k rp   K 0, 1, . . . , n for which n is the maximum of n . An application of 2 { } K rp k rp Darroch’s inequality [3] will help us to localize this index.

Theorem 5. (Darroch’s inequality) Let a0, a1, . . . , an be real numbers. If all the n i zeros of the polynomial P (x) = aix are real and negative and P (1) > 0, then k=0 the value of k for which ak is maximizeP d is within one of P 0(1)/P (1).

The following corollary gives a small interval for this index.

n Corollary 6. Let Kn,r be the greatest maximizing index of . We have p k rp Bn+1 (1; rp) Kn+ r ,r (rp + 1) < 1. | p| p B (1; r ) ✓ n p ◆ n+ rp Proof. Since the sequence k+|r | is strongly log-concave, there exists an index p rp n+ rp n+ rp n+ rp Kn+ rp ,rp for which r| | < < K | | > > n+|r | . Then, on | | p rp n+ rp ,rp p rp · · · | | rp · · · applying Theorem 1 and Darro ch’s theorem, we obtain

d dz Bn (z; rp) z=1 Kn+ rp ,rp < 1. | | Bn (1; rp) d It remains to use the ﬁrst identity given in [12, Corollary 12] byz dz (Bn (z; rp)) = B (z; r ) (z + r ) B (z; r ) . n+1 p p n p INTEGERS: 14 (2014) 7

3. Generalized Recurrences and Consequences

In this section, di↵erent representations of the polynomial Bn (z; rp) in di↵erent bases or families of basis are given by Theorems 7 and 10. Indeed, a representation in the basis Bn+k (z; rp) : 0 k n + rp 1 is given by the following theorem. {   | |} Theorem 7. We have

rp 1 | | Bn (z; rp) = ak (rp 1) Bn+k (z; rp) , kX=0 rp 1 | | Bn (z; rp+q) = ak (rp 1) Bn+k (z; rp, . . . , rp+q) . kX=0

rm rm rm j r j m Proof. Upon using the fact that (k + rp) = ( 1) j (k + rp) , we get j=0 P ⇥ ⇤ zk B (z; r ) = exp ( z) P (k; r ) n p n p k! k 0 X rm k rm j rm P0 (k; rp) n+j z = exp ( z) ( 1) rm (k + rp) j (k + rp) k! k 0j=0  X X rm r j rm = ( 1) m B (z; r r e ) , m = 1, 2, . . . , p 1, j n+j p m m j=0 X  and with the same process, we obtain

r1 rp 1 rp 1 jp 1 r1 rp 1 Bn (z; rp) = ( 1)| || | Bn+ jp 1 (z; rp) · · · j1 · · · jp 1 | | j1=0 jp 1=0   X X rp 1 | | rp 1 k r1 rp 1 = ( 1)| | Bn+k (z; rp) j1 · · · jp 1 k=0 jp 1 =k  X | X | rp 1 | | = ak (rp 1) Bn+k (z; rp) . kX=0 This implies the ﬁrst identity of the theorem. Now, from Lemma 2 we can write

rp+1 d rp exp ( z) (z exp (z) Bn (z; rp)) dzrp+1 rp 1 | | rp+1 d rp = ak (rp 1) exp ( z) (z exp (z) Bn+k (z; rp)) , dzrp+1 kX=0 INTEGERS: 14 (2014) 8

rp 1 | | which gives by utilizing Lemma 2: Bn (z; rp+1) = ak (rp 1) Bn+k (z; rp, rp+1) . k=0 We can repeat this process q times to obtain the secondP identity of the theorem.

So, the rp-Stirling numbers admit an expression in terms of the usual r-Stirling numbers given by the following corollary.

Corollary 8. We have

rp 1 n + rp | | n + j + rp | | = aj (rp 1) . k + rp k + rp rp j=0 rp ⇢ X ⇢

Proof. Using Theorem 7, the polynomial Bn (z; rp) can be written as follows:

rp 1 rp 1 n+j | | | | n + j + rp k aj (rp 1) Bn+j (z; rp) = aj (rp 1) z k + rp j=0 j=0 rp X X kX=0⇢ n+ rp 1 rp 1 | | | | k n + j + rp = z aj (rp 1) k + rp j=0 rp kX=0 X ⇢

n+ rp 1 | | n+ rp k and since Bn (z; rp) = | | z , the identity follows by identiﬁcation. k+rp k=0 rp P In [12], we proved the following:

tn B (z; r ) = B (z exp (t) ; r ) exp (z (exp (t) 1) + r t) . n p n! 0 p p n 0 X The following theorem gives more details on the exponential generating function of the rp-Bell polynomials and will be used later.

Theorem 9. We have tn B (z; r ) = B (z exp (t) ; r ) exp (z (exp (t) 1) + r t) n+m p n! m p p n 0 X rp 1 | | dm+k = ak (rp 1) (exp (z (exp (t) 1) + rpt)) . dtm+k kX=0 INTEGERS: 14 (2014) 9

Proof. Use (1) to get

tn zk tn B (z; r ) = exp ( z) P (k; r ) (k + r )n+m n+m p n! 0 0 p p k! 1 n! n 0 n 0 k 0 X X X @ zk exp ((k +Ar ) t) = exp ( z) P (k; r ) (k + r )m p 0 p p k! k 0 X = B (z exp (t) ; r ) exp (z (exp (t) 1) + r t) . m p p For the second part of the theorem, use Theorem 7 to obtain

rp 1 tn | | tn Bn+m (z; rp) = ak (rp 1) Bn+m+k (z; rp) n! n! n 0 k=0 n 0 X X X rp 1 | | dm+k tn = ak (rp 1) Bn (z; rp) dtm+k 0 n! 1 k=0 n 0 X X rp 1 @ A | | dm+k = ak (rp 1) (exp (z (exp (t) 1) + rpt)) . dtm+k kX=0

Using combinatorial arguments, Spivey [13] established the following identity:

n m m n n k B = j B , n+m j k k j=0 kX=0 X ⇢ ✓ ◆ where Bn is the n-th Bell number, i.e., the number of ways to partition a set of n elements into non-empty subsets. After that, Belbachir et al. [1] and Gould et al. [7] showed, using di↵erent methods, that the polynomial Bn+m (z) = Bn+m (z; 0) i admits a recurrence relation related to the family z Bj (z) as follows:

n m m n n k j B (z) = j z B (z) . (4) n+m j k k j=0 kX=0 X ⇢ ✓ ◆

Recently, Xu [14] gave a recurrence relation on a large family of Stirling numbers and Mihoubi et al. [11] extended the relation (4) to r-Bell polynomials as follows:

n m m + r n n k j B (x) = j x B (x) . (5) n+m,r j + r k k,r j=0 r kX=0 X ⇢ ✓ ◆ INTEGERS: 14 (2014) 10

Other recurrence relations are given by Mez˝o [10]. The following theorem generalizes the Identities (4) and (5), and the Carlitz’s identities [4, 5] given by

m m + r B (1; r) = B (1; k + r) , n+m k + r n k=0 ⇢ r Xs s + r s k Bn (1; r + s) = ( 1) Bn+k (1; r) , k + r r kX=0  and shows that Bn+m (z; rp) admits r-Stirling recurrence coecients in the families of basis

zjB (z; r + je ) : 0 j n , m p p   j z Bm+i (z; r + j) : 0 i rp 1 , 0 j n ,   | |   where Bn (1; r) is the number of ways to partition a set of n elements into non-empty subsets such that the r ﬁrst elements are in di↵erent subsets.

Theorem 10. We have n n + rp j Bn+m (z; rp) = z Bm (z; rp + jep) , j + rp j=0 rp X⇢ rp 1 n | | n + rp j Bn+m (z; rp) = ai (rp 1) z Bm+i (z; rp + j) , j + rp i=0 j=0 rp X X⇢ n n n + rp n j z Bm (z; rp + nep) = ( 1) Bm+j (z; rp) . j + rp j=0 rp X 

n n Proof. Let T (z; r ) := n+rp zjB (z; r + je ) t . The second iden- m p j+rp m p p n! n 0 j=0 rp ! tity given in [12, CorollaryP12]Pby d exp (z) B (z; r + e ) = (exp (z) B (z; r )) m p p dz m p can be used to get

dj exp (z) B (z; r + je ) = (exp (z) B (z; r )) . (6) m p p dzj m p Identity (6) and the exponential generating function of the r-Stirling numbers (see [2]) n n + rp t 1 j = (exp (t) 1) exp (rpt) j + rp n! j! n j⇢ rp X INTEGERS: 14 (2014) 11 prove that 1 T (z; r ) = B (z; r + je ) zj (exp (t) 1)j exp (r t) m p m p p j! p j 0 X dj (z (exp (t) 1))j = exp (r t z) (exp (z) B (z; r )) . p dzj m p j! j 0 X Now, by the Taylor-Maclaurin expansion we have dj (u z)j (exp (z) B (z; r )) = exp (u) B (u; r ) . dzj m p j! m p j 0 X So, this identity and Theorem 9 show that we have tn T (z; r ) = exp (r t z) exp (z exp (t)) B (z exp (t) ; r ) = B (z; r ) . m p p m p n+m p n! n 0 X n By comparing the coecients of t in the two expressions of Tm (z; rp), the ﬁrst identity of this theorem follows. The second identity follows by replacing Bm (z; rp + jep), as given by its expression in Theorem 7 by

rp 1 | | ai (rp 1) Bm+i (z; j + rp) . i=0 X n n+r n j p For the third identity, let A := j=0 j+r ( 1) Bm+j (z; rp) . We use Identity p rp n (1) and the known identity (k +Pr )n⇥= ⇤ n+rp kj (see [2]) to obtain p j+rp j=0 rp ⇥ ⇤ k Pn z n + rp n j j A = exp ( z) Pm (k; rp) ( 1) (k + rp) k! j + rp k 0 j=0  rp X X k n n z n + rp j = ( 1) exp ( z) Pm (k; rp) ( k rp) k! j + rp k 0 j=0  rp X X zk = ( 1)n exp ( z) P (k; r ) ( k r + r )n m p k! p p k 0 X zk = exp ( z) P (k; r ) kn m p k! k n X zk = zn exp ( z) P (k + n; r ) m p k! k 0 X n = z Bm (z; rp + nep) .

As consequences of Theorem 10, some identities for the rp-Stirling numbers of the second kind can be deduced as is shown by the following corollary. INTEGERS: 14 (2014) 12

Corollary 11. We have

k m + r n + r n + m + r | p| p = | p| , i + rp k i + rp k + rp i=0⇢ rp ⇢ rp ⇢ rp n X m + j + rp n + rp n j m + rp + n | | ( 1) = | | , k + n + rp j + rp k + rp + n j=0 rp rp rp+nep X ⇢  ⇢ n m + j + rp n + rp n j | | ( 1) = 0, k < n. k + rp j + rp j=0 rp rp X ⇢  Proof. From the ﬁrst identity of Theorem 10 we have n n + rp j Bn+m (z; rp) = z Bm (z; rp + jep) j + rp j=0 rp X⇢ which can be written as

n+m+ rp 1 n m+ rp 1 | | n + m + r n + r | | m + r | p| zk = p zj | p| zi k + rp j + rp i + rp rp j=0 rp i=0 rp kX=0 ⇢ X⇢ X ⇢ n+m+ rp 1 k | | m + r n + r = zk | p| p . i + rp k i + rp i=0 rp rp kX=0 X⇢ ⇢ Then, the desired identity follows by comparing the coecients of zk in the last expansion. Using the deﬁnition of Bn (z; rp) and the third identity of Theorem 10, the second and the third identities of the corollary follow from the deﬁnition

m+ rp 1 | | m + r + n B (z; r + ne ) = | p| zk m p p k + r + n p rp+nep kX=0 ⇢ and the expansion n n n + rp n j Bm (z; rp + nep) = z ( 1) Bm+j (z; rp) j + rp j=0 rp X  n m+j+ rp 1 | | n n + rp n j m + j + rp k = z ( 1) | | z j + rp k + rp j=0 rp rp X  kX=0 ⇢ m+n+ rp 1 n | | k n m + j + rp n + rp n j = z | | ( 1) k + rp j + rp j=0 rp rp kX=0 X ⇢  m+ rp 1 n | | k m + j + rp n + rp n j = z | | ( 1) . k + n + rp j + rp k= n j=0 ⇢ rp  rp X X INTEGERS: 14 (2014) 13

4. Ordinary Generating Functions

The ordinary generating function of the r-Stirling numbers of the second kind [2] is given by

k n + r n k 1 t = t (1 (r + j) t) . (7) k + r n k⇢ r j=0 X Y

An analogous result for the rp-Stirling numbers is given by the following theorem.

Theorem 12. Let n n + r B (z; r ) := | p| zk. n p k + r p rp kX=0⇢ | | Then, we have e

n + r n + r | p| tn = | p| tn k + rp k + rp 1 + rp n k⇢ rp n k⇢ rp X | | X | | rp 1 | | j n + j + rp n+j = aj (rp 1) t t k + rp 1 + rp j=0 n k⇢ rp X X | | rp 1 | | j n + rp n = aj (rp 1) t t , k + rp 1 + rp j=0 n k+j⇢ rp X X | |

n+rp and because = 0 for n = k, . . . , k + rp 1 1, we get k+ rp 1 +rp | | rp | | rp 1 | | n + rp n n + rp n j | | t = t aj (rp 1) t . k + rp r 0 k + rp 1 + rp r 1 0 1 n k⇢ | | p n k+ rp 1 ⇢ | | p j=0 X X| | X @ A @ A The ﬁrst generating function of the theorem follows by using (3) and (7). For the second one, use the deﬁnition of Bn (z; rp) and the last expansion.

e INTEGERS: 14 (2014) 14

Acknowledgments The authors would like to acknowledge the support from the RECITS’s laboratory and the PNR project 8/U160/3172. The authors also wish to thank the referee for reading and evaluating the paper thoroughly.

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