DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND ...

DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS ´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN Abstract. In the present article we extend several arithmetical results to a family of generalized Bell numbers called r-Bell numbers. In particular, we generalize some congruences such as Touchard’s congruence, Sun-Zagier congruence for Bell and derangement numbers and polynomials, among others. We also describe the behaviour of the 2-adic valuation of the r-Bell numbers. Published in: Journal of Number Theory, 177(2017), 136–152. DOI: http://dx.doi.org/10.1016/j.jnt.2017.01.022

1. Introduction The Stirling numbers are one of the most important combinatorial sequences, with several applications in Number Theory, Combinatorics, Special Functions, among others. The  Stirling numbers of the second kind nk count the number of set partitions of [n] := {1, . . . , n} into k non-empty blocks (cf. [5]). The total number of set partitions of [n] is given by the Bell number Bn , then n   X n . Bn = k k=0 Moreover, the Bell polynomials, Bn (x), are defined by n   X n k x . Bn (x) = k k=0 Many kinds of generalizations of Stirling and Bell numbers have been studied in the literature (cf. [15]). In particular, in the present article we are interested in the  r-Stirling numbers defined by Broder [4]. The r-Stirling numbers of the second kind, nk r , are defined as the number of set partitions of [n+r] into k +r blocks such that the first r elements are in distinct blocks. It is clear that if r = 0 we obtain the Stirling numbers of the second kind. Date: March 13, 2017. 2010 Mathematics Subject Classification. Primary 11B83; Secondary 11B73, 05A15, 05A19, 05A16. Key words and phrases. r-Bell numbers, r-Stirling numbers, Touchard’s congruence, p-adic valuation. The research of Istv´an Mez˝ o was supported by the Scientific Research Foundation of Nanjing University of Information Science & Technology, The Startup Foundation for Introducing Talent of NUIST. Project no.: S8113062001, and the National Natural Science Foundation for China. Grant no. 11501299. 1

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´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

The r-Stirling numbers of the second kind satisfy the following recurrence       n−1 n−1 n (1) , n≥k + =(k + r) k−1 r k k r r   with nk r = 0 if n < k and nk r = 1 if n = k. The r-Bell numbers were defined in [16] by n   X n . Bn,r := m r m=0 Thus Bn,r counts the number of partitions of an n+r-element set where the first r elements are in different blocks. If r = 0 we obtain the Bell numbers Bn . The r-Bell numbers are related to Bell numbers by the formula [16]   n X n−k n (2) Bk . r Bn,r = k k=0 The r-Bell polynomials are defined by Bn,r (x) :=

n   X n k=0

k

xk . r

It is clear that Bn,r (1) = Bn,r . The r-Bell polynomials are related to Bell polynomials by the formula [16]   n X n−k n (3) Bk (x). Bn,r (x) = r k k=0 Several results on divisibility of Bell numbers have been given in the literature. For example, the famous congruence of Touchard [22] states that Bn+p ≡ Bn+1 + Bn

(mod p).

Here and throughout of the paper p denotes a prime number. Gessel [8] showed that for each positive integer n, there exist integers a0 , a1 , . . . , an−1 such that for all m ≥ 0, Bm+n + an−1 Bm+n−1 + · · · + a0 Bm ≡ 0

(mod n!).

Sun and Zagier [28] showed that for any prime p not dividing m (4)

p−1 X k=1

Bk ≡ (−1)m−1 Dm−1 (−m)k

(mod p),

where Dn is the n-th derangement number, i.e., Dn counts the number of fixed-point free n-permutations. For additional results about divisibility properties of Bell numbers and polynomials see [1, 6, 10, 11, 14, 25, 26, 28, 29].

DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS

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The goal of the present article is to show several arithmetical results on the r-Bel numbers and polynomials. In particular, in Sections 2 and 3 we generalize the Touchard’s congruence to the r-Bell numbers and polynomials by using an umbral symbolic approach and some elementary recurrence relations of the r-Bell numbers. We also give a generalization of the Sun-Zagier congruence for Bell and derangement numbers and polynomials. In Section 4 we show that the r-Bell numbers are a periodic sequence of period Np = (pp − 1)(p − 1). Finally, in Section 5 we give an explicit expression for the 2-adic valuation of the r-Bell numbers. 2. Generalized Touchard’s congruence In this section we generalize the Touchard’s congruence to the r-Bell number case. First we need a divisibility lemma for the r-Stirling numbers. Lemma 1. For any prime p the following congruences for the r-Stirling numbers hold true: npo ≡ r p ≡ r (mod p), 0 n p or n p o ≡ 1 (mod p), ≡ 1 n p1 or ≡ 0 (mod p) (1 < k < p). k r  Note that the first congruence is actually an identity: p0 r = r p . The proof of this lemma straightforwardly follows from the identity [16, eq. (3)] n    nno X i n r n−i . = k k r i i=0 and the little Fermat’s congruence ap ≡ a (mod p) with a ∈ Z. Theorem 2. For any n and r non-negative integers Bn+p,r ≡ rBn,r + Bn,r+1 + Bn,p+r

(mod p).

Proof. In order to prove our theorem, we invoke the formula [17]     n X m X n m n−k Bk . Bn+m,r = (j + r) k j r k=0 j=0 By the previous lemma this reduces to three terms in j (j ∈ {0, 1, p}) modulo p:       n n n X X X n−k n n−k n n−k n Bk . Bk + (r + p) Bk + (r + 1) Bn+p,r ≡ r r k k k k=0 k=0 k=0 From Equation (2) we get the result.



Setting r = 0 we get that Bn,p ≡ Bn+p − Bn+1

(mod p),

´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

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from where, by Touchard’s congruence, we get that Bn,p ≡ Bn

(mod p).

Theorem 3. For any n and r non-negative integers Bn+p,r ≡ (r + 1)Bn,r + Bn,r+1

(mod p).

Proof. The proof runs like in Theorem 2.



In particular, if r = 0 we recover the Touchard’s congruence. 3. Generalizations of Touchard’s congruence by Using an Umbral Symbolic Approach The goal of this section is to study the congruence of Touchard and its generalization for the r-Bell polynomials by using an umbral symbolic approach. For this purpose, we first introduce some basic ideas of the classical umbral calculus, which was introduced by Blissard in the 1860’s [3], and formalized by Rota and Taylor [21]. For more information on the umbral calculus see for example [7]. This method has been used successfully to study several combinatorial sequences and polynomials related to Stirling and Bell numbers, see for example [24, 25, 26, 27]. A sequence (an ) is umbrally represented if each term an is formally replaced by the powers αn , where α is a new variable called umbra of the sequence (an ). The original sequence is recovered by the evaluation map eval(αn ) = an . We are going to write αn = an instead of eval(αn ) = an . Let us introduce the Bell umbra B defined by Bn = Bn and the generalized Bell umbra Bx given by Bnx = Bn (x) (cf. [25, 26]). From identities n   n   X X n n (−1)n−k Bk+1 (x) Bk (x) and xBn = Bn+1 (x) = x k k k=0 k=0 we obtain the following equalities in terms of the generalized Bell umbra: (5)

Bn+1 = x(Bx + 1)n , x

(6)

xBnx = Bx (Bx − 1)n .

Now we introduce the r-Bell umbra Br and the generalized r-Bell umbra Bx,r by Bnr = Bn,r and Bnx,r = Bn,r (x), respectively. Then Equation (3) can be rewritten as Bnx,r = (Bx + r)n .

(7) Moreover, from (6) we have xBnx,r

n

= x(Bx + r) = x

n   X n i=0

i

Bix r n−i

n   X n Bx (Bx − 1)i r n−i = Bx (Bx + r − 1)n . = i i=0

DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS

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The above identity can be rewritten as Bn,r (x) =

i    n X X i n i=0 j=0

i

j

(−1)i−j r n−i Bj+1 (x).

Let f (x) be a polynomial. Then by linearity and from (5) we have Bx f (Bx ) = xf (Bx + 1), therefore by induction on m we obtain that Bx (Bx − 1) · · · (Bx − m + 1)f (Bx ) = xm f (Bx + m). From (7) we have f (Bx,r ) = f (Bx + r). Hence xr f (Bx,r ) = Bx (Bx − 1) · · · (Bx − r + 1)f (Bx ).

(8)

For two polynomials p(x) and q(x) in Zp [x], the congruence p(x) ≡ q(x) (mod pZp [x]) means that the corresponding coefficients of p(x) and q(x) are congruent modulo p. From Lagrange’s congruence (cf. [5, pp. 218]) x(x − 1) · · · (x − p + 1) ≡ xp − x (mod pZp [x]) by setting x = Bx we get xr (Bpx − Bx )f (Bx,r ) ≡ Bx (Bx − 1) · · · (Bx − p + 1)xr f (Bx,r ) (mod pZp [x]) = Bx (Bx − 1) · · · (Bx − p + 1)Bx (Bx − 1) · · · (Bx − r + 1)f (Bx ) = Bx (Bx − 1) · · · (Bx − r + 1)xp f (Bx + p) ≡ Bx (Bx − 1) · · · (Bx − r + 1)xp f (Bx ) (mod pZp [x]) = xp+r f (Bx,r ). Therefore (Bpx − Bx )f (Bx,r ) ≡ xp f (Bx,r )

(mod pZp [x])

for any prime p. From above congruence we obtain the following congruences for any positive integer l (9)

(Bpx − Bx )l f (Bx,r ) ≡ xpl f (Bx,r ) (mod pZp [x]),

(10)

(Bpx − xp )l f (Bx,r ) ≡ Blx f (Bx,r )

(mod pZp [x]).

By the congruence (x + y)p ≡ xp + y p (mod p) and by the little Fermat’s theorem ap ≡ a (mod p) with a ∈ Z, we obtain (Bx − 1 + r)p f (Bx,r ) ≡ (Bx − 1)p f (Bx,r ) + r p f (Bx,r ) ≡ Bpx f (Bx,r ) − f (Bx,r ) + rf (Bx,r ) ≡ (Bx + xp − 1 + r)f (Bx,r )

(mod pZp [x]).

In particular, (11)

(Bx − 1 + r)p Bx ≡ (Bx + xp − 1 + r)Bx

(mod pZp [x]).

´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

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Theorem 4. For any n and r non-negative integers we have Bn+p,r (x) ≡ Bn+1,r (x) + xp Bn,r (x)

(mod pZp [x]).

Proof. From (11) we have n+p xBn+p,r (x) = xBx,r = Bx (Bx − 1 + r)n+p = (Bx − 1 + r)p Bx (Bx − 1 + r)n

≡ (Bx + xp − 1 + r)Bx (Bx − 1 + r)n = Bx (Bx − 1 + r)n+1 + xp Bx (Bx − 1 + r)n p+1 n = xBn+1 Bx,r = xBn+1,r (x) + xp+1 Bn,r (x) x,r + x

(mod pZp [x]). 

If r = 0 we obtain the congruence Bn+p (x) ≡ Bn+1 (x) + xp Bn (x)

(mod pZp [x]).

and if x = 1 we obtain the congruence Bn+p,r ≡ Bn+1,r + Bn,r

(mod p).

A stronger form of Touchard’s congrunce for polynomials is even known [6], which says that (12)

m

Bn+pm (x) ≡ Bn+1 (x) + (xp + · · · + xp )Bn (x)

(mod pZp [x])

for all positive integer m. In particular if x = 0 we obtain Bn+pm ≡ Bn+1 + mBn

(13)

(mod p).

We can generalize the above congruences in terms of r-Bell polynomials and therefore numbers. First we need the following congruence (14)

m

m

(Bx − 1 + r)p ≡ Bx + xp + · · · + xp − 1 + r

(mod pZp [x]).

It is clear by induction on m. Theorem 5. For any n, r and m non-negative integers we have m

Bn+pm ,r (x) ≡ Bn+1,r (x) + (xp + · · · + xp )Bn,r

(mod pZp [x]).

Proof. From (14) we have m

n+p xBn+pm ,r (x) = xBx,r = Bx (Bx − 1 + r)n+p

m

m

≡ (Bx + xp + · · · + xp − 1 + r)Bx (Bx − 1 + r)n = Bx (Bx − 1 + r)

n+1

(mod pZp [x])

pm

p

+ (x + · · · + x )Bx (Bx − 1 + r)n m

= xBn+1 + (xp + · · · + xp )xBnx,r r m

= xBn+1,r (x) + (xp + · · · + xp )xBn,r (x).  In particular if r = 0 we obtain the congruence (12) and if x = 1 we obtain the congruence Bn+pm ,r ≡ Bn+1,r + mBn,r

(mod p).

DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS

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3.1. Congruences for the r-Bell numbers and derangement numbers. Sun and Zagier [28] found the following interesting congruence p−1 m−1 X X (m − 1)! Bk (x) p (−x) ≡ (−x) (−x)k k (−m) k! k=1 k=0 m

(mod pZp [x])

for any prime p not dividing m. The case x = 1 generates the congruence (4). In [25], Sun et al. generalized the above result. In particular, they found the following congruences p−1 n   X X Bn+k (x) n m p (−1)m+k−1 Dm+k−1 (1 − x) (mod pZp [x]), x ≡x k k (−m) k=1 k=0 or equivalently (15)

m

x

p−1 n  X X Bj+k (x) n j=0

j

k=1

(−m)k

≡ (−1)m+n−1 xp Dm+n−1 (1 − x)

(mod pZp [x])

where Dn (x) is the n-th derangement polynomial defined by n   X n j!(x − 1)n−j , Dn (x) = j j=0  n   and k are the Stirling numbers of the first kind. The Stirling numbers nk count the number of permutations on n elements with k cycles. These numbers can also defined by the following equality n   X n k (16) x . x(x − 1) · · · (x − n + 1) = k k=0 In the following theorem we generalize the congruence given by Sun and Zagier for the r-Bell numbers. Theorem 6. For any integers r ≥ 0 and m ≥ 1 and any prime p not dividing m, we have (−x)

m

p−1 X Bk,r (x) k=1

(−m)k

≡ (−x)p−r Dm+r−1 (1 − x)

(mod pZp [x]).

Proof. From (16) by setting x = Bx we get n   X n k Bx . Bx (Bx − 1) · · · (Bx − n + 1) = k k=0

From (8) by setting f (x) = xn we obtain xr Bn,r (x) = xr Bnx,r = Bx (Bx − 1) · · · (Bx − r + 1)Bnx r   r   X r k+n X r Bk+n (x). Bx = = k k k=0 k=0

´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

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Finally, from (15) we have (−x)

p−r

m−r

p−1 r  X X Bj+k (x) r

(mod pZp [x]) k (−m) j=0 k=1 p−1 r   X 1 X r m−r = (−x) Bj+k (x) k (−m) j=0 j k=1

Dm+r−1 (1 − x) ≡ (−x)

= (−x)

m

j

p−1 X Bk,r (x)

(−m)k

k=1

. 

In particular, if x = 1 we obtain the following corollary Corollary 7. For any integers r ≥ 0 and m ≥ 1 and any prime p not dividing m, we have p−1 X Bk,r ≡ (−1)m−r−1 Dm+r−1 k (−m) k=1

(mod p).

4. The Periodicity of the r-Bell Numbers Modulo a Prime Number An integer sequence A = (an )n≥0 is a periodic sequence modulo m, with period t if there exists s ≥ 0 such that an+t ≡ an (mod m), for n ≥ s. The smallest t is called the minimum period of A. The periodicity of the Bell numbers has been studied by several authors. The first result about this was given by Hall [9]. He showed that the sequence of Bell numbers p −1 has period Np = pp−1 . This result was rediscovered by Williams [31], who also show that the minimum period is exactly Np for p = 2, 3 and 5. For example, in Figure 1 (left) we show the first few values of the sequence (Bn mod 3)n≥0 . In this case the period is N3 = (33 − 1)/(3 − 1) = 13. 2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0

10

20

30

40

50

0

10

20

30

40

50

Figure 1. The sequences (Bn mod 3)n≥0 and (Bn,7 mod 3)n≥0 . Levine and Dalton [13] showed that the period is exactly Np for p = 7, 11, 13 and 17. Afterwards, Radoux [20] conjectured that for any prime p, the number Np is the minimum

DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS

9

period of the Bell sequence. Since then several authors have showed this conjecture for some particular primes. For example, Wagstaff [30] proved that the period is exactly Np for all primes p < 102 and several larger p. Montgomery et al. [19] improve the previous one for most primes p < 180. In this section, we are going to extend the result of Hall for the r-Bell numbers. In Figure 1 (right) we show the first few values of the sequence (Bn,7 mod 3)n≥0 . In this case the period is N3 = 13. Theorem 8. The r-Bell numbers are a periodic sequence modulo p, with period Np =

pp −1 . p−1

Proof. From (10) we have the following congruence: Bp − B ≡ 1 (mod p). By using the congruence (14) and the Lagrange’s congruence we have (B − 1 + r)

Np

=

p Y

(B − 1 + r)

pp−j

≡

p Y

(B − j − 1 + r)

(mod p)

j=1

j=1

p−1

=

Y j=0 p

(B − j + r) ≡ (B + r)p − (B + r)

≡B −B

(mod p)

(mod p)

≡ 1 (mod p). Therefore Bn+Np ,r = Brn+Np = B(B − 1 + r)n+Np = (B − 1 + r)n B(B − 1 + r)Np ≡ (B − 1 + r)n B = Bnr = Bn,r .

(mod p) 

5. 2-adic Valuation of the r-Bell Numbers For any positive integer Qk nαi> 1, the Fundamental Theorem of Arithmetic implies the prime factorization n = i=1 pi for some positive integers αi ≥ 1. Let p be a prime number, the p-adic valuation of n ∈ N, denoted by νp (n), is the largest nonnegative integer m such that pm divides n, i.e., νpi (n) is the exponent αi in the above prime factorization. The valuation of n = 0 is set to be +∞. Given a positive integer sequence, it is a natural question to explore the behaviour of the factorization of its elements. This can be measured in terms of the p-adic valuation of its elements. For example, Lengyel [12] studied the p-adic valuation of the Fibonacci and Lucas sequence. Straub et al. [23] showed the p-adic valuation of the central binomial coefficients. Zhao et al. [32] studied the 2-adic valuation of the Stirling numbers, see also [2]. In this section motivated by the work of Ambdeberhan, de Angelis and Moll [1] on

´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

10

the valuation of the classical Bell numbers we analyze the 2-adic valuation of the r-Bell numbers. In [1], authors found that   if j ≡ 0, 1, 3, 4, 6, 7, 9, 10 (mod 12); 0, ν2 (B12n+j ) = 1, if j ≡ 2, 11 (mod 12);  2, if j ≡ 5, 8 (mod 12). The 2-adic valuation of the r-Bell numbers follows a similar pattern. In Figure 2 we show the first few values of the 2-adic valuation of the Bell numbers and 5-Bell numbers. 3.0

2.0

2.5 1.5 2.0

1.5

1.0

1.0 0.5 0.5

20

40

60

80

100

20

40

60

80

100

Figure 2. The 2-adic valuation of Bn and Bn,5 . We first get the following useful congruence. We use a similar technique as in [1]. Theorem 9. For all positive integers n and r Bn+24,r ≡ Bn,r

(mod 8).

Before giving the proof, we need to prove the following lemmas. (r)

Lemma 10. Let µj (m) be the polynomials defined recursively by (r)

(r)

(r)

µj+1(m) = (m + r)µj (m) + µj (m + 1), (r)

with µ0 (m) = 1. Then (17)

Bn+j,r =

 n+j  X n+j

m=0

m

= r

n X

m=0

(r) µj (m)

  n . m r

Proof. We proceed by induction on j. From recurrence (1) the result clearly holds for j = 1. Now suppose that the result is true for all positive integers less than or equal to j.

DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS

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We prove it for j + 1. In fact, by using recurrence (1) we obtain that (n+1)+j

  n+1 X X (n + 1) + j  n+1 (r) = µj (m) m r m r m=0 m=0      n+1 X n n (r) + = µj (m) (m + r) m−1 r m r m=0 n  n X (r) (r) = (m + r)µj (m) + µj (m + 1) m r m=0   n X n (r) . = µj+1 (m) m r m=0 (r)



We will express the polynomials µj (m) in terms of the rising factorial. The rising factorial xn , is defined by xn := x(x + 1) · · · (x + n − 1), if n ≥ 1, and x0 = 1. The rising factorials (xi )0≤i≤n are a basis for the vector space of polynomials of degree at most n. Let us denote (r) (r) by aj (n) the coefficients of µj (m) in terms of the basis (xi )0≤i≤n , i.e.,

(r) µj (m)

(18)

=

j X

(r)

aj (s)ms .

s=0

Lemma 11. For all positive integers s, j and r (r)

(r)

(19) aj+1 (s) − (s + 1)aj+1 (s + 1) (r)

(r)

(r)

= aj (s − 1) + (r − 2s)aj (s) + (s + 1 − r)(s + 1)aj (s + 1), (r)

with aj (s) = 0 if s < 0 or s > j. Proof. From identities xn = (x + 1)n − n(x + 1)n−1 and x · xn = xn+1 − nxn we obtain

(r) µj+1(m)

=

j+1 X

(r) aj+1 (s)ms

s=0

=

j+1 X s=0

  (r) aj+1 (s) (m + 1)s − s(m + 1)s−1

j+1

=

X s=0

 (r) (r) aj+1 (s) − (s + 1)aj+1 (s + 1) (m + 1)s ,

´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

12

and (r)

(r)

(m + r)µj (m) + µj (m + 1) =

j X

(r)

aj (s) (m + 1)ms + (m + 1)s

s=0




j

 X (r)  s+1 s s−1 aj (s) (m + 1) + (r − 2s)(m + 1) − s(r − s)(m + 1) = s=0

j+1

=

X s=0

(r) µj+1(m)

Since

 (r) (r) (r) aj (s − 1) + (r − 2s)aj (s) + (s + 1)(s + 1 − r)aj (s + 1) (m + 1)s . (r)

(r)

= (m + r)µj (m) + µj (m + 1), we obtain the desired result.

From Equations (17) and (18) we obtain that the r-Bell numbers are given by (r)

(r)

Bj,r = µj (0) = aj (0). Let us introduce the matrices M := (mi,j )i,j≥0 and N(r) := (ni,j (r))i,j≥0,   1,     if i = j; 1, r − 2i, mi,j = −(i + 1), if i = j − 1; ni,j (r) = (20)   (i + 1 − r)(i + 1), 0,   otherwise.  0, Then from Lemma 11 (r)

where aj

(r)

where if i = j + 1; if i = j; if i = j − 1; otherwise.

(r)

Maj+1 = N(r)aj , iT h (r) (r) (r) := aj (0), aj (1), aj (2), . . . . Taking in count that xs ≡ 0 (mod 8),

for s ≥ 4, then we only have to analyze the following system     (r)    (r)  aj (0) aj+1 (0) r 1−r 0 0 1 −1 0 0   (r) (r)   0 1 −2 0  aj+1 (1) 1 r − 2 2(2 − r) 0  aj (1)    (r)   =  .  0 0 1 r − 4 3(3 − r) a(r) 1 −3 aj+1 (2) 0  (2) j (r) (r) 0 0 1 r − 6 0 0 0 1 a (3) a (3) j

j+1

(r,4)

Let aj

h iT (r) (r) (r) (r) := aj (0), aj (1), aj (2), aj (3) , then (r,4)

(r) (r,4)

aj+1 ≡ X4 aj where (r)

X4



r+1  1 =  0 0

(mod 8),

 1 2 6 r 2 6  . 1 r+7 7  0 1 r+2



DIVISIBILITY PROPERTIES OF THE r-BELL NUMBERS AND POLYNOMIALS

It is clear that

l  (r,4) (r) (r,4) aj+l ≡ X4 aj

13

(mod 8)

for any integer l ≥ 0. By a direct computation we obtain that  24 (r) X4 ≡ I (mod 8). (r,4)

(r,4)

(r,4)

Then aj+24 ≡ aj (mod 8). Since r-Bell numbers are the first entry in the vector aj then congruence in Theorem 9 is obtained.

,

Example 12. By a direct computation we obtain that (ν2 (Bn,3 ))0≤n≤23 = {0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0}. Then from Theorem 9 we conclude that   if j ≡ 0, 2, 3, 5, 6, 8, 9, 11 (mod 12); 0, ν2 (B12n+j,3 ) = 1, if j ≡ 7, 10 (mod 12);  2, if j ≡ 1, 4 (mod 12). Example 13. By a direct computation we obtain that

(ν2 (Bn,4 ))0≤n≤23 = {0, 0, 1, 0, 0, 7, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 1}. Then from Theorem 9 we conclude that   if j ≡ 0, 1, 3, 4, 6, 7, 9, 10 (mod 12); 0 ν2 (B12n+j,4 ) = 1, if j ≡ 2, 11 (mod 12);  2, if j ≡ 8 (mod 12).

Since ν2 (B5,4 ) = 7, in this case we can not use the congruence in Theorem 9. In Figure 3 we show the first few values of ν2 (B12n+5,4 ). 8

7

6

5

4

5

10

15

20

25

30

Figure 3. The 2-adic valuation of B12n+5,4 .

´ MEZO ˝ AND JOSE ´ L. RAM´IREZ ISTVAN

14

Theorem 14. For all positive integers n, k and r Bn,r ≡ Bn,8k+r

(mod 8).

Proof. The proof of this congruence follows from the identity [18, Theorem 1] n   X n n−i r Bi,s . Bn,r+s = i i=0 Then Bn,8k+r =

n   X n i=0

i

(8k)n−i Bi,r ≡ Bn,r

(mod 8). 

From congruences in Theorems 9 and 14 we obtain the following theorem. Theorem 15. For any n and k non-negative integers we have ν2 (Bn,r ) = ν2 (Bn,8k+r ) for r = 0, 1, 3, 6. And ν2 (Bn,2 ) = ν2 (Bn,8k+2), n 6≡ 11 (mod 12), ν2 (Bn,4 ) = ν2 (Bn,8k+4), n 6≡ 5 (mod 12), ν2 (Bn,5 ) = ν2 (Bn,8k+5), n 6≡ 7 (mod 12), ν2 (Bn,7 ) = ν2 (Bn,8k+7), n 6≡ 1 (mod 12). 6. Acknowledgments The first author was on a workshop at the Nanjing University of Information Science and Technology where he talked about the divisibility problems studied in the present article. His colleague, Hao Pan (Nanjing University), proved that the strong Touchard congruence (13) holds true without modification for the r-Bell numbers. We thank him for this result. References [1] T. Amdeberhan, V. De Angelis, V. Moll, Complementary Bell numbers: Arithmetical properties and Wilf’s conjecture. Advances in Combinatorics Waterloo Workshop in Computer Algebra, W80 Springer, 2013, 23–56. [2] T. Amdeberhan, D. Manna, V. Moll, The 2-adic valuation of Stirling numbers, Exp. Math. 17 (2009) 69–82. [3] J. Blissard, Theory of generic equations, Quart. J. Pure Appl. Math. 4 (1861) 279–305. [4] A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984) 241–259. [5] L. Comtet, Advanced Combinatorics. D. Reidel Publishing Company, 1977. [6] A. Gertsch, A. M. Robert, Some congruences concerning the Bell numbers, Bull. Belg. Math. Soc. 3 (1996) 467–475. [7] I. M. Gessel, Applications of the classical umbral calculus, Algebra Universalis 49 (2003) 397–434. [8] I. M. Gessel, Congruences for Bell and tangent numbers, Fibonacci Quart. 19 (1981) 137–144. [9] M. Hall, Arithmetic properties of a partition function, Bull. Amer. Math. Society. 40 (1934).

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