ˇ FACTA UNIVERSITATIS (NIS) Ser. Math. Inform. Vol. 22, No. 2 (2007), pp. 105–108
ON THE NUMBERS RELATED TO THE STIRLING NUMBERS OF THE SECOND KIND Nenad P. Caki´ c
Abstract. In this paper we give the new explicit expression for the numbers cnk treated by Mijajlovi´c and Markovi´c [Facta Univ. Ser. Math. Inform. 13 (1998), 7–17]. The new combinatorial identities are also given.
1.
Introduction
In this paper, we consider the numbers cnk , which may be defined by the operational relation n X (xα+1 D)n = cnk xnα+k Dk , k=1
and by the recurrence relation (triangular or Pascal-type relation) cn+1k = cnk−1 + (nα + k)cnk , 1 ≤ k ≤ n, c11 = 1, cnk = 0 for k > n, where α ∈ R (see [5]). It is clear that for α = 0, cnk are the usual Stirling numbers of the second kind. The aim of this paper is to prove the new explicit expression for the numbers cnk . Also, we give some additional remarks and comments. 2.
Main Result
In [5] Mijajlovi´c and Markovi´c studied the numbers cnk and proved that (x| − α)n =
n X
cnk (x|1)k ,
k=0
where n ∈ N, α ∈ R and (x|α)n denotes the generalized factorial of the form (x|α)n =
n−1 Y
(x − jα), n ≥ 1, (x|α)0 = 1.
j=0
Received June 20, 2007. 2000 Mathematics Subject Classification. Primary 11B73, 11B75; Secondary 05A19.
105
106
Nenad P. Caki´c
and in particular (x|1)n = (x)(n) = x(x − 1) · · · (x − n + 1), (x)(0) = 1. Theorem. The explicit expression for the numbers cnk defined in [5] is given by:
! ! 1+α (n − 1)(1 + α) − i1 − · · · − in−2 ··· . i1 in−1
X
(2.1) cnk =
i1 + · · · + in−1 = n − k ij ∈ {0, 1}
Proof. Starting from (2.1), for in ∈ {0, 1} we get cn+1k = ! ! 1+α n(1 + α) − i1 − · · · − in−1 ··· i1 in
X i1 + · · · + in = n + 1 − k ij ∈ {0, 1}
! ! 1+α (n − 1)(1 + α) − i1 − · · · − in−2 ··· + i1 in−1
X
=
i1 + · · · + in = n + 1 − k ij ∈ {0, 1}
X
(nα + k)
i1 + · · · + in−1 = n − k ij ∈ {0, 1}
! ! 1+α (n − 1)(1 + α) − i1 − · · · − in−2 ··· i1 in−1
i.e., cn+1k = cnk−1 + (nα + k)cnk , because n(1 + α) − i1 − · · · − i(n − 1) = n(1 + α) − n + k = nα + k. This completes the proof.
2
Corollary. Unsigned Lah numbers, L(n, k) (see, for example [2]and [3] ) defined by triangular recurrence relation L(n, k) = L(n − 1, k − 1) + (n + k − 1)L(n − 1, k), L(n, n) = L(n, 1) = 1, L(n, k) = 0 for n < k, have the explicit representation
L(n, k) = X i1 + i2 + · · · + in−1 = n − k ij ∈ {0, 1}
2 i1
!
4 − i1 i2
!
6 − i1 − i2 i3
This expression follows from (2.1) for α = 1.
!
! 2(n − 1) − i1 − · · · − in−2 ··· . in−1
107
On the Numbers Related to the Stirling Numbers of the Second Kind Naturally, it is well-known that (see [3]) n! L(n, k) = k!
„
n−1 k−1
« .
So, the combinatorial identity
n! n − 1 k! k − 1 X
! = 2 i1
i1 + · · · + in−1 = n − k ij ∈ {0, 1}
!
4 − i1 i2
!
6 − i1 − i2 i3
!
2(n − 1) − i1 − · · · − in−2 ··· in−1
!
is valid. Remark 2.1. Explicit expression for the numbers cnk from [5] is
cnk =
(2.2)
! n 1 X k (−1)k−i i(i + α) · · · (i + (n − 1)α), k! i k=1
and therefore, if we identify the right sides of (2.1) and (2.2), we get the following combinatorial identity
X i1 + · · · + in−1 = n − k ij ∈ {0, 1}
1+α i1
!
2(1 + α) − i1 i2
!
(n − 1)(1 + α) − i1 − · · · − in−2 ··· in−1
!
! n 1 X k−i k = (−1) i(i + α) · · · (i + (n − 1)α). k! i k=1
Remark 2.2. For α = 0 we get the explicit formula for the Stirling numbers of the second kind (see [1] and comment by L. Carlitz therein)
S(n, k) =
X i1 + · · · + in−1 = n − k ij ∈ {0, 1}
1 i1
!
2 − i1 i2
!
! n − 1 − i1 − · · · − in−2 ··· . in−1
Remark 2.3. The numbers cnk are the special case of a generalized Stirling numbers S(n, k; α, β, γ), introduced by Hsu and Shiue ([4]). Namely, cn,k = S(n, k; −α, 1, γ), where the following relations holds (x|α)n =
n X
S(n, k; α, β, γ)(x − γ|β)k ,
k=0
S(n + 1, k; α, β, γ) = S(n, k − 1) + (kβ − nα + γ)S(n, k; α, β, γ).
108
Nenad P. Caki´c
Evidently, the classical Stirling numbers s(n, k) and S(n, k) are in fact S(n, k; 1, 0, 0) and S(n, k; 0, 1, 0) respectively. Also, the binomial coefficients are given by S(n, k; 0, 0, 1). In a forthcoming paper we give an explicit formula for generalized Stirling numbers S(n, k; α, β, γ).
REFERENCES ´: On some combinatorial identities. Univ. Beograd. Publ. Elek1. N. P. Cakic trotehn. Fak. Ser Mat. Fiz. No. 678–715 (1980), 91-94. 2. L. Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions. D.Reidel Publ. Co., Holland 1974. 3. R.L. Graham, D.E. Knuth and O. Patashnik: Concrete Mathematics. AddisonWesley, Reading, Mass. 1989. 4. L.C. Hsu and P.J.-S. Shiue: A unified approach to generalized Stirling numbers. Advanced in Applied Math. 20 (1998), 366–384. ˇ Mijajlovic ´ and Z. Markovic ´: Some recurrence formulas related to the dif5. Z. ferential operator θD. Facta Univ. Ser. Math. Inform. 13 (1998), 7–17.
Faculty of Electrical Engineering Department of Applied Mathematics P. O. Box 35–54 11120 Belgrade, Serbia
[email protected]