Combinatorial applications of the special numbers and polynomials

0.11 MB·

by  Yilmaz Simsek

7 Combinatorial applications of the special numbers and polynomials
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Yilmaz Simsek
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Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey,

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E-mail: [email protected]
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Abstract

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Inthispaper,byusingsomefamiliesofspecialnumbersandpolynomialswiththeirgenerat-
]
O ing functions,we givevariouspropertiesofthese numbersandpolynomials. These numbers
are related to the well-known numbers and polynomials, which are the Euler numbers, the
C
Stirling numbers of the second kind, the central factorialnumbers and the array polynomi-
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h als. Wealsodiscusssomecombinatorialinterpretationsofthesenumbersrelatedtotherook
at polynomials and numbers. Furthermore, we give computation formulas for these numbers
m and polynomials.
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2010 Mathematics Subject Classification. 11B68; 05A15; 05A19; 12D10; 26C05; 30C15.

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Key Words. Euler numbers; Central factorial numbers; Array polynomials; Stirling numbers; Generating
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functions; Binomial coefficients; Combinatorial sum.
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1 1. Introduction
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ThespecialnumbersandtheirgeneratingfunctionshavemanyapplicationinCombinatorialNumberSys-
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. tem andin ProbabilityTheory. There aremany advantageofthe generatingfunctions. By using generating
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functionsforspecialnumbersandpolynomials,onecangetnotonlyvariouspropertiesofthesenumbersand
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polynomials, but also enumerating arguments such as counting the number of subsets and the number of
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1 total ordering. In this paper, by using generating functions and their functional equations, we derive some
: identities and relations for the special combinatorial numbers such as the Stirling numbers of the first kind,
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the central factorial numbers, the Euler numbers, the array polynomials and the other special numbers. In
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order to give our results, we need some special numbers and polynomials with their generating functions.
r The first kind Apostol-Euler polynomials of order k are defined by means of the following generating
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function:
2 k ∞ tn
F (t,x;k,λ)= etx = E(k)(x;λ) , (1.1)
P1 λet+1 n n!
(cid:18) (cid:19) n=0
X
(|t| < π when λ = 1 and |t| < |ln(−λ)| when λ 6= 1), λ ∈ C, the set of complex numbers, k ∈ N, the set of
natural numbers. By (1.1), we easily see that
E(k)(λ)=E(k)(0;λ),
n n
which denotes the first kind Apostol-Euler numbers of order k. By substituting k = λ = 1 into (1.1), we
have
E =E(1)(1)
n n
which denotes the first kind Euler numbers (cf. [4]-[23], and the references cited therein).
The second kind Euler numbers E∗ of negative order are defined by means of the following generating
n
function:
2 −k ∞ tn
F (t,k)= = E∗(−k) , (1.2)
E2 et+e−t n n!
(cid:18) (cid:19) n=0
X
2 Yilmaz Simsek
where |t|< π (cf. [5], [19], and the references cited therein).
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Theλ-StirlingnumbersofthesecondkindS (n,v;λ)definedbymeansofthefollowinggeneratingfunction:
2
(λet−1)v ∞ tn
F (t,v;λ)= = S (n,v;λ) , (1.3)
S v! 2 n!
n=0
X
where v ∈N and λ∈C (cf. [12], [16], [22], and the references cited therein).
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By using (1.3), we have
v
1 v
S (n,v)= (−1)v−jλjjn
2 v! j
j=0(cid:18) (cid:19)
X
(cf. [12], [16], [22]).
Substituting λ=1 into (1.3), wehave the Stirling numbers ofthe secondkindS (n,v)which denotes the
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number of ways to partition a set of n objects into v groups:
S (n,v)=S (n,v;1).
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(cf. [1]-[23]; see also the references cited in each of these earlier works).
In [16], we defined the λ-array polynomials Sn(x;λ) by means of the following generating function:
v
(λet−1)v ∞ tn
F (t,x,v;λ)= etx = Sn(x;λ) , (1.4)
A v! v n!
n=0
X
where v ∈N and λ∈C (cf. [6], [4], [7], [16], [17], and the references cited therein).
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ThecentralfactorialnumbersT(n,k)(ofthesecondkind)aredefinedbymeansofthefollowinggenerating
function:
∞
F (t,k)= 1 et+e−t−2 k = T(n,k) t2n (1.5)
T
(2k)! (2n)!
n=0
(cid:0) (cid:1) X
(cf. [2], [9], [10], [17], [23], and the references cited therein).
Remark 1. The central factorial numbers are used in combinatorial problems. That is the number of ways
to place k rooks on a size m triangle board in three dimensions is equal to
T(m+1,m+1−k),
where 0≤k ≤m (cf. [1]).
In [19], we defined the numbers y (n,k;λ) by means of the following generating functions:
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∞
F (t,k;λ)= 1 λet+1 k = y (n,k;λ)tn, (1.6)
y1 k! 1 n!
n=0
(cid:0) (cid:1) X
where k ∈N and λ∈C. If we substitute λ=−1 into (1.6), then we get the Stirling numbers of the second
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kind, S (n,k):
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S (n,k)=(−1)ky (n,k;−1)
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(cf. [19], [18]). The numbers y (n,k;λ) is related to following novel combinatorial sum:
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B(n,k)=k!y (n,k;1)= k k jn = dn et+1 k| , (1.7)
1 j dtn t=0
j=0(cid:18) (cid:19)
X (cid:0) (cid:1)
where n=1,2,...(cf. [11], [19]). In the work of Spivey [20, Identity 8-Identity 10], we see that
B(0,k)=2k,B(1,k)=k2k−1,B(2,k)=k(k+1)2k−2,
and also
n
B(m,n)= n j!2n−jS (m,j) (1.8)
j 2
j=0(cid:18) (cid:19)
X
Combinatorialapplicationsofthe specialnumbers and polynomials 3
(cf. [3, p.4,Eq-(7)],[19]; seealso the referencescited in eachofthese earlierworks). In [19], we aconjecture
and two open questions associated with the numbers B(n,k).
In [19], we defined the numbers y (n,k;λ) by means of the following generating functions:
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∞
F (t,k;λ)= 1 λet+λ−1e−t+2 k = y (n,k;λ)tn. (1.9)
y2 (2k)! 2 n!
n=0
(cid:0) (cid:1) X
In [19], we gavesome combinatorialinterpretations for the numbers y (n,k), y (n,k) and B(n,k) as well
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as the generalization of the central factorial numbers. We see that these numbers were related to the rook
numbers and polynomials.
2. Functional equations and related identities
By usinggeneratingfunctions forthe Stirling numbers,the Eulernumbers, the centralfactorialnumbers,
the array polynomials, the numbers y (n,k;λ) and the numbers y (n,k;λ) with their functional equations,
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we derive some identities and relations involving binomial coefficients and these numbers and polynomials.
We also give computation formulas for the first kind and the second kind Euler numbers and polynomials.
By using (1.6) and (1.3), we obtain the following functional equation:
F (2t,k;−λ2)=(−1)kk!F (t,k;λ)F (t,k;λ).
y1 y1 S
By using the above equation, we get
∞ ∞ ∞
tn tn tn
2ny n,k;−λ2 =(−1)kk! y (n,k;λ) S (n,k;λ) .
1 n! 1 n! 2 n!
n=0 n=0 n=0
X (cid:0) (cid:1) X X
By using the Cauchy product in the above equation, we obtain
∞ tn ∞ n n tn
2ny n,k;−λ2 =(−1)kk! S (l,k;λ)y (n−l,k;λ) .
1 n! l 2 1 n!
n=0 n=0l=0(cid:18) (cid:19)
X (cid:0) (cid:1) XX
Comparing the coefficients of tn on both sides of the above equation, we arrive the following theorem:
n!
Theorem 1.
n
y (n,k;−λ2)=(−1)kk!2−n n S (l,k;λ)y (n−l,k;λ).
1 l 2 1
l=0(cid:18) (cid:19)
X
By combining (1.4) with (1.5) and (1.9), we obtain the following functional equation:
F (2t,−k,2k;1)=(2k)!F (t,k)F (t,k;1).
A T y2
Using the above equation, we get
∞ ∞ ∞
tn t2n t2n
2nSn (−k) =(2k)! T(n,k) y (n,k;1) .
2k n! (2n)! 2 (2n)!
n=0 n=0 n=0
X X X
Therefore
∞ tn ∞ n n t2n
2nSn (−k) =(2k)! T(j,k)y (n−l,k;1) .
2k n! l 2 (2n)!
n=0 n=0l=0(cid:18) (cid:19)
X XX
By using the above equation, we arrive at the following theorem:
Theorem 2.
n
n
S2n(−k)=(2k)!2−2n T(l,k)y (n−l,k;1).
2k l 2
l=0(cid:18) (cid:19)
X
4 Yilmaz Simsek
Lemma 1. ([14, Lemma 11, Eq-(7)])
∞ ∞ ∞ [n2]
A(n,k)= A(n,n−2k),
n=0k=0 n=0k=0
XX XX
where [x] denotes the greatest integer function.
By combining (1.4) and (1.5) with (1.6), we get the following functional equation:
k
k! (2l)! t t l
F (t,k)= F ,l F − , ,k−l;1 .
T (2k)! l! T 2 A 2 2
l=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
By using the above equation, we obtain
∞
t2n
T(n,k)
(2n)!
n=0
X
k! k (2l)! ∞ t2n ∞ l tn
= 2−2nT(n,l) Sn ,1 .
(2k)! l! (2n)! k−l 2 n!
l=0 n=0 n=0 (cid:18) (cid:19)
X X X
By using Lemma 1, we get
∞
t2n
T(n,k)
(2n)!
n=0
X
= k! k (2l)! ∞ [n2]T (j,l)Sn−2j l,1 2−2j tn .
(2k)! l! k−l 2 (2j)!(n−2j)!
l=0 n=0j=0 (cid:18) (cid:19)
X XX
Comparing the coefficients on both sides of the above equation, we arrive the following theorem:
Theorem 3. If n is an even integer, we have
[n]
T(n,k)= (2n)!k! k 2 n (2l)!T(j,l)Sn−2j l,1
(2k)!n! 2j 22jl! k−l 2
l=0j=0(cid:18) (cid:19) (cid:18) (cid:19)
XX
and if n is an odd integer, we have
[n]
k 2 n (2l)!T (j,l)Sn−2j l,1 =0.
2j 22jl! k−l 2
l=0j=0(cid:18) (cid:19) (cid:18) (cid:19)
XX
By combining (1.4) with (1.2), we obtain
22k k k
F (2t,k)= (−1)k−jF (t,−2j).
T (2k)! j E2
j=0(cid:18) (cid:19)
X
By using the above functional equation, we get
∞ 2nT(n,k) t2n = 22k ∞ k k (−1)k−jE∗(−2j) t2n .
(2n)! (2k)! j n (2n)!
n=0 n=0j=0(cid:18) (cid:19)
X XX
Comparingthecoefficientsof t2n onbothsidesoftheaboveequation,wearriveatthecomputationformula
(2n)!
for the second kind Euler numbers of negative order which is given by the following theorem:
Theorem 4.
T(n,k)= 22k−n k k (−1)k−jE∗(−2j).
(2k)! j n
j=0(cid:18) (cid:19)
X
Combinatorialapplicationsofthe specialnumbers and polynomials 5
By using (1.9) and (1.3), we get
k
F (t,k;−λ)= k! (−1)kF (t,j;λ)F (−t,k−j;λ−1).
y2 (2k)! S S
j=0
X
By using the above functional equation, we obtain
∞ y (n,k;λ)tn = k! k (−1)k ∞ S (n,j;λ)tn ∞ S n,k−j;λ−1 (−t)n .
2 n! (2k)! 2 n! 2 n!
!
n=0 j=0 n=0 n=0
X X X X (cid:0) (cid:1)
By using the Cauchy product in the right-hand side of the above equation, we obtain
∞
tn
y (n,k;λ)
2 n!
n=0
X
= ∞ k! k n (−1)k+n−d n S (d,j;λ)S n−d,k−j;λ−1 tn.
(2k)! d 2 2 n!
n=0 j=0d=0 (cid:18) (cid:19)
X XX (cid:0) (cid:1)
Comparing the coefficients of tn on both sides of the above equation, we arrive at the following theorem:
n!
Theorem 5.
k n
y (n,k;λ)= k! (−1)k+n−d n S (d,j;λ)S n−d,k−j;λ−1 .
2 (2k)! d 2 2
j=0d=0 (cid:18) (cid:19)
XX (cid:0) (cid:1)
Computation formula for the first kind Euler polynomials of order −k is given by the following theorem:
Theorem 6.
2nλ−k k k k
y (n,k;λ)= E(−k) ;λ2 .
2 (2k)! j n 2
j=0(cid:18) (cid:19) (cid:18) (cid:19)
X
Proof. By using (1.9) and (1.1), we obtain the following functional equation:
λ−k k k k
F (t,k;λ)= F 2t, ;j,λ2 .
y2 (2k)! j P1 2
j=0(cid:18) (cid:19) (cid:18) (cid:19)
X
By using the above equation, we get
∞ y (n,k;λ)tn = ∞ 2nλ−k k k E(−k) k;λ2 tn.
2 n! (2k)! j n 2 n!
n=0 n=0 j=0(cid:18) (cid:19) (cid:18) (cid:19)
X X X
Comparing the coefficients of tn on both sides of the above equation, we arrive at the desired result. (cid:3)
n!
Acknowledgement 1. The paper was supported by the Scientific Research Project Administration of Akd-
eniz University.
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