Applied Mathematical Sciences, Vol. 9, 2015, no. 114, 5687 - 5696
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.57481
An Explicit Formula for Generalized
Arithmetic-Geometric Sum
Roberto B. Corcino, Cristina B. Corcino
Department of Mathematics
Cebu Normal University
Cebu City, Philippines 6000
Cyril A. Flores
Department of Mathematics
Mindanao State University-Main Campus
Marawi City, Philippines 9700
c 2015 Roberto B. Corcino, Cristina B. Corcino and Cyril A. Flores. This
Copyright article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Abstract
A short survey on different derivations of the explicit formula for
the arithmetic - geometric sums and their generalizations is given in the
introductory part of the paper. Moreover, an explicit formula for certain
generalized arithmetic - geometric sums is established using the concept
of k th differentiation and Leibniz formula. The formula is expressed in
terms of the generalized Eulerian polynomials.
Mathematics Subject Classification: 11B68, 11B83, 11B25
Keywords: Arithmetic-Geometric sum, degenerate Stirling numbers, Eulerian numbers, Eulerian polynomials, Leibniz formula
1
Introduction
The generalized arithmetic-geometric sum studied by L.C. Hsu [10] is
defined in terms of the generalized falling factorial of k of degree p and
5688
R.B. Corcino, C.B. Corcino and C.A. Flores
increment h. That is,
Sa,h,p (n) =
n
X
ak (k|h)p
(1)
k=1
where
(k|h)p =
p−1
Q
(k − jh), p = 1, 2, 3, . . . and (k|h)0 = 1. We see that, if
j=0
h = 0, (2.1) reduces to the form
Sa,p (n) =
n
X
ak k p .
(2)
k=1
The sum Sa,p (n) has been investigated by many mathematicians using different
method to find its explicit formula. R.A. Khan [9] introduced a simple integral
approach that can be presented in all generality with just a basic knowledge
in calculus. D. Sullivan [14] used a simple and elegant recursion formula to
study this problem. A.W.F. Edwards [6] employed a matrix formulation which
is very intimately connected to Pascal’s Triangle and Binomial Theorem. N.
Gauthier [7] presented a calculus-based method, writing the sum as an times
a polynomial of degree p in n plus a term which is n - independent in which
the coefficients are then determined recursively. G.F.C. de Bruyn [5] derived
the explicit expressions for Sa,p (n) in determinant form from the recurrence
formulas in terms of powers of n and n+1. The work of L.C. Hsu [10] expressed
(1) in terms of the degenerate Stirling numbers S(p, j/h) of Carlitz [3] which
is defined by
∞
j X
1
xn
1
h
S(p, j|h)
(1 + hx) − 1 =
j!
n!
n=j
and, consequently, expressed (2) in terms of the classical Stirling numbers of
the second kind [4].
The polynomials commonly called Eulerian today have been introduced
by Euler himself back in 1755. They have been since thoroughly studied, extended, and applied. The historical origin of the classical Eulerian polynomial
Ap (a) is the following summation formula
∞
X
k=0
ak k p =
Ap (a)
,
(1 − a)p+1
|a| < 1,
a 6= 0
(3)
where Ap (a) is the Eulerian polynomial in a of degree p, and p is a positive
integer. Another way of defining the classical Eulerian polynomial Ap (a) is
through the form given by
Ap (a) =
p
X
k=1
A(p, k)ak
(4)
5689
The explicit formula for generalized arithmetic-geometric sum
where A(p, k) are called Eulerian numbers given explicitly by
k
X
p+1
A(p, k) =
(k − j)p ,
(−1)
j
j=0
j
1≤k≤p
(5)
The polynomials defined in (3) and (4) appeared in the explicit formula of the
arithmetic-geometric sum in the paper of L.C. Hsu and E.L. Tan [12]. Recently,
Cirnu [2] obtained this Eulerian numbers representation (3) in a simple manner
using Cauchy-Mertens Theorem.
In 1998, L.C. Hsu and P. J. Shiue [11] introduced two kinds of generalized
Eulerian polynomials which is associated in the constructions of certain explicit
formulas for an extended arithmetic-geometric sum. They formulated closed
summation formula for the expressions
S(n) =
n
X
(k + λ|θ)p ak
(6)
k=0
and
S(∞) =
∞
X
(k + λ|θ)p ak
(7)
k=0
where λ and θ are real or complex numbers and |a| < 1 for S(∞). Evidently,
(6) and (7) will reduce to
n
X
p k
k a
and
k=0
∞
X
k p ak ,
k=0
respectively, when λ = θ = 0. It is worth mentioning that L.C. Hsu [10] already
obtained explicit formula for (6) when λ = 0. The explicit formula for (6) is
given by
p
X
S(n) =
S(p, j, λ|θ)φ(a, n, j),
a 6= 1
j=0
where
1
φ(a, n, j) =
1−a
"
a
1−a
j
r
i X
n+1
a
n+1
−x
j−r
1−a
r=0
#
while the explicit formula for (7) is given by
S(∞) =
p
X
j!S(p, j, λ|θ)aj
j=0
(1 − a)j+1
,
|a| < 1.
(8)
5690
R.B. Corcino, C.B. Corcino and C.A. Flores
The numbers S(p, j, λ|θ) (0 ≤ j ≤ p) are called the Howard’s degenerate
weighted Stirling numbers [8] which are defined by either the generating function
∞
j
X
λ
1
tp
(9)
(1 + θt) θ (1 + θt) θ − 1 = j!
S(p, j, λ|θ)
p!
p=j
or, by the basis - transformation relation
(t + λ|θ)p =
p
X
S(p, j, λ|θ)(t)j .
(10)
j=0
In fact, the equivalence between (9) and (10) may be verified by means of the
recurrence relations
S(p + 1, j, λ|θ) = S(p, j − 1, λ|θ) + (j − pθ + λ)S(p, j, λ|θ)
for p ≥ j ≥ 1 with S(0, 0, λ|θ) = S(p, p, λ|θ) = 1 and S(p, 0, λ|θ) =
(λ|θ)p . Carlitz’s degenerate Stirling numbers S(p, j|θ) and the classical Stirling
numbers S(p, j) of the second kind are special cases of Howard’s degenerate
weighted Stirling numbers with λ = 0, and with λ = 0, θ → 0 respectively.
Parallel to the definition of the classical Eulerian polynomials in (3), the
generalized Eulerian polynomials Ap (a, λ|h) may be defined, based on (8), as
∞
X
ak (k + λ|h)p =
k=0
Ap (a, λ|h)
(1 − a)p+1
(11)
where |a| < 1 and Ap (a, λ|h) is a polynomial in a of degree p units with
Ap (a, 0|h) = Ap (a|h), that is,
∞
X
ak (k|h)p =
k=0
Ap (a|h)
.
(1 − a)p+1
(12)
If h = 0, Ap (a|0) = Ap (a) and (12) reduces to (3).
Using (8) and (11), one can easily see that the polynomials Ap (a, λ|θ) satisfy
Ap (a, λ|θ) =
p
X
j!S(p, j, λ|θ)aj (1 − a)p−j
(13)
j=0
or, equivalently
Ap (a, λ|θ) =
p
X
j=0
A(p, j, λ|θ)aj
(14)
The explicit formula for generalized arithmetic-geometric sum
5691
where the numbers A(p, j, λ|θ) were shown in [11] to be equal to
j
X
r p+1
(j − r + λ|θ)p .
(−1)
A(p, j, λ|θ) =
r
r=0
L.C. Hsu and P.J. Shiue [11] called Ap (a, λ|θ) as the generalized Eulerian polynomial of the first kind. In particular when λ = θ = 0 we obtain the classical
Eulerian polynomial Ap (a) in (3) and (4) and the classical Eulerian numbers
A(p, j, 0|0) = Ap (p, j) in (5).
On the other hand, the second kind of generalized Eulerian polynomials is
given by
p
X
j!S(p, j, α)aj (1 − a)p−j
Ap (a, α) =
j=0
or, equivalently
Ap (a, α) =
p
X
A(p, j, α)xj
j=0
where Ap (a, α) and A(p, j, α) (0 ≤ j ≤ p) may be called Dickson - Eulerian
Polynomial and Dickson - Eulerian numbers, respectively.
Here, we derive an explicit formula for the generalized arithmetic - geometric sum
n
X
ak (k + λ|h)p
k=0
which will be expressed in terms of the first kind generalized Eulerian polynomial. Moreover, some examples will be given to illustrate the usefulness of the
formula. For better representation of the sum, we use the following notation
Sa,h,p (n, λ) =
n
X
ak (k + λ|h)p
k=0
throughout the discussion.
2
Main Results
The explicit formula that we are going to derive for Sa,h,p (n, λ) is parallel to
those formulas obtained in [12]. Now, let us consider first the following lemma
which is useful in obtaining the desired explicit formula.
Lemma 2.1. For |a| < 1, we have
k −1 λ
1
d
Ak (a, λ|h)
h
h
θ 1 − aθ
= k
k
dθ
h (1 − a)k+1
θ=1
5692
R.B. Corcino, C.B. Corcino and C.A. Flores
1
Proof. Choose > 0 very small so that |θ − 1| < with |a| < 1, |aθ h | < 1.
j
∞ P
1
Hence, we can differentiate
aθ h
term by term any number of times
j=0
in a neighboorhood of |θ − 1| < . Since,
∞
X
k
a θ
k+λ
h
k=0
= θ
λ
h
∞
X
k
a θ
k
h
= θ
λ
h
k=0
∞ X
1
aθ h
k
−1
1
λ
= θ h 1 − aθ h
k=0
we have
"
#
∞
−1 1
λ
dk X j j+λ
dk
=
θ h 1 − aθ h
a θ h
dθk
dθk j=0
θ=1
θ=1
( k
)
∞
X
Y
1
= k
aj
(j + λ − qh) ,
h j=0
q=0
k∈N
∞
1 X j
= k
a (j + λ|h)k .
h j=0
By (11) we have
−1 1
λ
1 Ak (a, λ|h)
dk
= k·
.
θ h 1 − aθ h
k
dθ
h (1 − a)k+1
θ=1
Now, we are ready to introduce the main result of this paper which is given in
the following theorem.
Theorem 2.2. For any given integer p ≥ 0, there holds the explicit formula
"
#
p X
p
A
(a,
λ|h)
A
(a,
λ|h)
1
k
p
an+1
(n + 1|h)p−k −
Sa,h,p (n, λ) =
a−1
k (1 − a)k
(1 − a)p
k=0
(15)
where a 6= 0, a 6= 1.
Proof. Consider the sum
S(n, λ, θ) =
n
X
k=0
ak θ
k+λ
h
.
5693
The explicit formula for generalized arithmetic-geometric sum
Taking the pth derivative of S(n, λ, θ) with respect to θ, we have
n
k+λ
dp
1 X k
S(n, λ, θ) = p
a (k + λ|h)p θ h −p .
p
dθ
h k=0
Now, we set
p
h
n
X
k+λ
dp
S(n, λ, θ) =
ak (k + λ|h)p θ h −p .
p
dθ
k=0
Evaluating this at θ = 1, we obtain
p
h
n
X
dp
=
ak (k + λ|h)p = Sa,h,p (n, λ).
S(n, λ, θ)
dθp
θ=1
k=0
Note that S(n, λ, θ) can be expressed further as
S(n, λ, θ) = θ
λ
h
n
X
k
k
h
a θ =θ
λ
h
1 − an+1 θ
k=0
=
n+1
h
!
1
1 − aθ h
ih λ
i
h
n+1
1
θ h (1 − aθ h )−1 .
1 − an+1 θ h
(16)
Hence, using (16) and applying Leibniz’s product formula, we get,
p
Sa,h,p (n, λ) = h
λ
1
dp −1
n+1 n+1
h
h
h
1−a θ
θ (1 − aθ )
dθp
θ=1
X
p−1 p−k dk λ
1
p
d
n+1 n+1
−1
=h
1−a θ h
θ h (1 − aθ h )
p−k
k
k
dθ
dθ
θ=1
θ=1
k=0
p
+ 1−a
n+1
1
dp λ
−1
θ h (1 − aθ h )
.
dθp
θ=1
But
dp−k n+1
n+1
n+1
n+1 n+1
n+1
1 − a θ h = −a
− 1 ···
− (p − k − 1) .
dθp−k
h
h
h
By Lemma 2.1 with |a| < 1, we easily obtain (15).
5694
R.B. Corcino, C.B. Corcino and C.A. Flores
Remark 2.3. When λ = 0, Theorem 2.2 reduces to
"
#
p X
p
A
(a|h)
1
A
(a|h)
k
p
Sa,h,p (n) =
an+1
(n + 1|h)p−k −
k
k
a−1
(1
−
a)
(1
− a)p
k=0
This further gives
"
#
p X
p
A
(a)
1
A
(a)
k
p
Sa,p (n) =
an+1
(n + 1)p−k −
k
k
a−1
(1
−
a)
(1
−
a)p
k=0
when h → 0. The latter is the refined explicit formula in terms of powers of
(n + 1) proposed by E.L. Tan and L.C. Hsu [12].
Example 2.4. Let n = 300, p = 3, h = 1. Then
S2,1,3 (300) =
300
X
2k (k|1)3
k=1
301
=2
3 X
3 Ak (2|1)
k=1
k
(−1)k
(301)3−k −
A3 (2|1)
(−1)3
= 2301 [(301)3 − 6(301)2 + 24(301) − 48] + 48
= 2301 (26465076) + 48
Example 2.5. Consider a pair of trigonometric sums as follows:
R(n, λ) =
n
X
(k + λ|h)p cos kθ
and I(n, λ) =
k=1
n
X
(k + λ|h)p sin kθ
k=1
where λ, h p
are any real or complex numbers and 0 < θ < 2π. Taking a = eiθ
where i = (−1) we have,
n
n
X
X
iθ k
Seiθ ,h,p (n, λ) =
(k + λ|h)p (e ) =
(k + λ|h)p (cos kθ + i sin kθ)
k=1
=
k=1
n
X
(k + λ|h)p cos kθ + i
k=1
= R(n, λ) + iI(n, λ)
Hence by Theorem 2.2, we see that
n
X
k=1
(k + λ|h)p sin kθ
5695
The explicit formula for generalized arithmetic-geometric sum
R(n, λ) = Re
I(n, λ) = Im
"
#!
p iθ
iθ
X
p
A
(e
,
λ|h)
A
(e
,
λ|h)
1
k
p
e(iθ)(n+1)
(n + 1|h)p−k −
k (1 − eiθ )k
eiθ − 1
(1 − eiθ )p
k=0
"
#!
p iθ
iθ
X
p
A
(e
,
λ|h)
1
A
(e
,
λ|h)
k
p
e(iθ)(n+1)
.
(n + 1|h)p−k −
k (1 − eiθ )k
eiθ − 1
(1 − eiθ )p
k=0
Acknowledgements. The authors would like to thank the referee for reading
and evaluating the paper.
References
[1] M. I. Cirnu, Determinantal Formulas for the Sum of Generalized Arithmetic - Geometric Series. Boletin de la Asociacion Matematica Venezolana, 17 (2011), 13 - 25.
[2] M. I. Cirnu, Eulerian Numbers and Generalized Arithmetic-Geometric
Series, U.P.B. Sci. Bull., Series A, 71 (2009), 25 - 30.
[3] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util.
Math., 15 (1979), no. 1, 51 - 88.
[4] L. Comtet, Advanced Combinatorics, Dordrecht: Reidel.
[5] G.F.C. de Bruyn, Formulas for a + a2 2p + a3 3p + . . . + an np , The Fibonacci
Quarterly, 33 (1974), 98 - 103.
[6] A.W.F. Edwards, A Quick Route to the Sums of Powers, Amer. Math.
Monthly, 93 (1986), 451 - 455. http://dx.doi.org/10.2307/2323466
P k r
[7] N. Gauthier, Derivation of a Formula for
r x , The Fibonacci Quarterly, 27 (1987), 402 - 408.
[8] F.T. Howard, Degenerate Weighted Stirling Numbers, Discrete Math, 57
(1985), 45 - 58. http://dx.doi.org/10.1016/0012-365x(85)90155-4
[9] R.A. Khan, A simple derivation of a formula for
n
P
k r ,The Fibonacci
k=1
Quarterly, 19 (1981), 177 - 180.
[10] L.C. Hsu, On A Kind of Generalized Arithmetic - Geometric Progression,
The Fibonacci Quarterly, 35 (1997), 62 - 67.
5696
R.B. Corcino, C.B. Corcino and C.A. Flores
[11] L.C. Hsu, P. J-S. Shiue, On Certain Summation Problems and Generalizations of Eulerian Polynomials and Numbers, Discrete Mathematics,
204 (1999), 237 - 246. http://dx.doi.org/10.1016/s0012-365x(98)00379-3
[12] P
L.C. Hsu, Tan, L. Evelyn, A Refinement of De Bruyn’s Formulas for
ak k p , The Fibonacci Quarterly, 37 (1999), 56 - 60.
[13] Murray H. Protter, B. Morrey Charles, Jr., College Calculus with Analytic Geometry (Third Edition), Addison-Wesley Publishing Company,
Inc. 1977.
[14] D. Sullivan, The Sums of Powers of Integers, Math. Gazette, 71 (1987),
144 - 146. http://dx.doi.org/10.2307/3616508
[15] Chii-Huei Yu, The Derivatives of Two Types of Functions, Chii-Huei,
International Journal of Computer Science and Mobile Applications, 1
(2013), 1 - 8.
Received: July 27, 2015; Published: September 9, 2015