1–5 EXPLICIT FORMULAS FOR NUMBERS OF ...

ˇ FACTA UNIVERSITATIS (NIS) Ser. Math. Inform. 9 (1994), 1–5

EXPLICIT FORMULAS FOR NUMBERS OF CARLITZ AND TOSCANO Gradimir V. Milovanovi´ c and Nenad P. Caki´ c

Abstract. In this paper we give some explicit exspressions for numbers treated by Carlitz [1]. Also, we consider the similar problems for the generalized Stirling numbers of the second kind, introduced by Toscano [2].

1. Introduction In [1] L. Carlitz introduced arrays of numbers an,t and bn,t in the following way µ(n−1)+1

(1.1)

(xλ+µ Dµ )n =

X

an,t (xt+µ−1+nλ Dt+µ−1 )

t=1

and µ(n−1)+1

(1.2)

µ

(x D

λ+µ n

) =

X

bn,t (xt+µ−1 Dt+µ−1+nλ ).

t=1

As a generalization of the Stirling numbers of the second kind, L. Toscano (see [2] and [3]) definied the numbers Kn,k,p with the recurrence relation (p ∈ R+ ): Kn,1,p = Kn,n,p = 1,

Kn,k,p = Kn−1,k−1,p + k p Kn−1,k,p .

In this paper we give explicit expressions for the numbers an,t , bn,t , and Kn,k,p . As a consequence of that, we also give two combinatorial identities. Received November 17, 1993. 1991 Mathematics Subject Classification. Primary 11B73, 11B75. This work was supported in part by the Science Fund of Serbia under grant 0401F. 1

G. V. Milovanovi´c and N. P. Caki´c

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2. Representation of an,t and bn,t For numbers a ∈ R and k, is ∈ N0 (s = 0, 1, . . . , n − 1) we introduce the notations a(k) = a(a − 1) · · · (a − k + 1) (k > 0), and Sn =

n−1 X

a(0) = 1,

a(k) = 0 (k > a)

is .

s=0

Theorem 2.1. Let n, t, µ ∈ N. The explicit representation of the numbers an,t defined in (1.1) is given by (2.1)

n−1 Y

X

an,t =

i1 +···+in−1 =(n−1)µ−t+1 k=1

µ ik

 k(λ + µ) − Sk


(ik )

,

where 0 = i0 ≤ i1 ≤ i2 ≤ · · · ≤ in−1 ≤ µ. Proof. We start with the following expansion (see [4]) ! n−1 X Y s (xr Ds )n = (kr − Sk )(ik ) xnr−Sn Dns−Sn , ik 0≤i1 ≤i2 ≤···≤in−1 ≤s

k=1

where i0 = 0. If we introduce supstitution ns − Sn = t + s − 1 (i0 = 0), then t ∈ {1, 2, . . . , (n − 1)s + 1}, and (n−1)s+1

(xr Ds )n =

X

Qt ,

t=1

where Qt =

X

n−1 Y

i1 +···+in−1 =(n−1)s−t+1

k=1

! ! s (ik ) (kr − Sk ) x(r−s)n+s+t−1 Dt+s−1 ik

and 0 = i0 ≤ i1 ≤ · · · ≤ in−1 ≤ s. For µ = s and λ = r − s, we have µ(n−1)+1

(2.2)

(x

λ+µ

µ n

D ) =

X t=1

Rt ,

Explicit Formulas for Numbers of Carlitz and Toscano

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where n−1 Y

X

Rt =

i1 +···+in−1 =(n−1)µ+1−t k=1

! ! µ (ik ) (k(λ + µ) − Sk ) xt+µ−1+nλ Dt+µ−1 . ik

So, from (1.1) and (2.2) we obtain the exspression (2.1).  Similarly, we can formulate the following result: Theorem 2.2. The explicit representation for the numbers bn,t defined by (1.2) is given by n−1 Y

X

bn,t =

i1 +···+in−1 =(n−1)µ−t+1 k=1

 λ+µ (µk − Sk )(ik ) , ik

where 0 = i0 ≤ i1 ≤ · · · ≤ in−1 ≤ λ + µ. The Carlitz’s explicit exspression for an,t is (2.3) an+1,t =

t X (−1)t−j 1 (j + λ + µ − 1)(µ) · · · (j + nλ + µ − 1)(µ) , (t − 1)! (t − j)!(j − 1)! j=1

and therefore, if we identify the right sides of (2.1) and (2.3) (substitute n with n + 1), we get the following Corollary 2.3. If n, t, λ, µ are natural numbers and 0 = i0 ≤ i1 ≤ · · · ≤ in−1 ≤ µ, then the identity n   Y µ

X

i1 +···+in =nµ−t+1 k=1

=

t X j=1

ik

(k(λ + µ) − Sk )(ik )

(−1)t−1 (j + λ + µ − 1)(µ) · · · (j + nλ + µ − 1)(µ) (t − j)!(j − 1)!

is valid.

3. Representation of Kn,k,p L. Toscano [2–3] introduced the numbers Kn,k,2 in the following way (3.1)

(xD)2n =

n X i=1

Kn,i,2 xi D2i−1 xi−1

G. V. Milovanovi´c and N. P. Caki´c

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The explicit exspression for Kn,k,2 is Kn,k,2

  k 2k 2(−1)k X i (−1) i2n . = (2k)! i=1 k−i

But, Toscano did not give any explicit representation for the generalized numbers Kn,k,p . Theorem 3.1. The explicit representation for the numbers Kn,k,p (with Kn,1,p = Kn,n,p = 1) is given by (3.2)

Kn,k,p =

X n−1 Y s − Ss p s=1

is

,

where the sumation on the right is over all i1 , . . . , in−1 ∈ {0, 1} such that i1 + i2 + · · · + in−1 = n − k and i0 = 0. Proof. For in−1 ∈ {0, 1} we get from (3.2) that n−2 Y

X

Kn,k,p =

i1 +···+in−2 =n−k s=1

+

X

s − Ss is

p

(n − 1 − i1 − · · · − in−2 )

p

n−2 Y s=1

i1 +···+in−2 =n−k−1

s − Ss is

p ,

i.e., Kn,k,p = Kn−1,k−1,p + k p Kn−1,k,p , because n − 1 − i1 − · · · − in−2 = k and n − k = (n − 1) − (k − 1). This completes the proof.  For p = 2 we have (3.3)

Kn,k,2 =

X n−1 Y s − Ss 2 s=1

is

,

where the sumation on the right is over all i1 , . . . , in−1 ∈ {0, 1} such that i1 + i2 + · · · + in−1 = n − k and i0 = 0. Notice that for bigger values of k (k > n/2) this formula is simpler than the Toscano expression (3.1). For example, if we take k = n − 1 then from (3.1) we find Kn,n−1,2

  n−1 2n − 2 2(−1)n−1 X i (−1) i2n , = (2n − 2)! i=1 n−i−1

Explicit Formulas for Numbers of Carlitz and Toscano

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while representation (3.3) gives Kn,n−1,2 =

n−1 Y

X

i1 +···+in−1 =1 s=1

s − Ss is

2

= 12 + 22 + · · · + (n − 1)2 ,

i.e., Kn,n−1,2 = n(n − 1)(2n − 1)/6. From (3.1) and (3.2) we have: Corollary 3.2. The identity X

n−1 Y

i1 +···+in−1 =n−k s=1

s − Ss is

2

  k 2(−1)k X 2k i = (−1) i2n (2k)! i=1 k−i

is valid (with i1 , i2 , . . . , in−1 ∈ {0, 1}). Remark 3.1. Many authors have studied “new” numbers related to numbers an,t , bn,t and Kn,k,p . In a forthcoming paper we will analyse several results of other authors which are consequences of fundamental results of L. Carlitz and L. Toscano.

REFERENCES 1. L. Carlitz: On arrays of numbers. Amer. J. of Math. 54 (1932), 739–752. 2. L. Toscano: Sulla iterazione dell’ operatore xD. Rendiconti di Matematica e delle sue applicazioni (V) 8 (1949), 337–350. 3. L. Toscano: Su gli operatori permutabili di secondo ordine. Giornale di Matematiche di Battaglini 90 (1962), 55–71. ´: On some combinatorial identities. Univ. Beograd. Publ. 4. N. P. Cakic Elektrotehn. Fak. Ser. Mat. Fiz. No 678 – No 715 (1980), 91–94. University of Niˇs Faculty of Electronic Engineering Department of Mathematics P. O. Box 73, 18000 Niˇs Yugoslavia

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G. V. Milovanovi´c and N. P. Caki´c

University of Niˇs Faculty of Technology Department of Mathematics 16000 Leskovac Yugoslavia EXPLICITNE FORMULE ZA CARLITZ-OVE I TOSCANO-OVE BROJEVE

Gradimir V. Milovanovi´ c i Nenad P. Caki´ c U radu se izvode eksplicitni izrazi za brojeve koje je definisao L. Carlitz [1]. Takod¯e, razmatraju se sliˇcni problemi za generalisane Stirlingove brojeve druge vrste koje je uveo L. Toscano [2].