Hindawi Publishing Corporation
Advances in Numerical Analysis
Volume 2012, Article ID 780646, 13 pages
doi:10.1155/2012/780646
Research Article
Discrete Gamma (Factorial) Function and Its Series
in Terms of a Generalized Difference Operator
G. Britto Antony Xavier, V. Chandrasekar,
S. U. Vasanthakumar, and B. Govindan
Department of Mathematics, Sacred Heart College, Tirupattur 635601, India
Correspondence should be addressed to G. Britto Antony Xavier, shcbritto@yahoo.co.in
Received 17 July 2012; Accepted 5 October 2012
Academic Editor: Rüdiger Weiner
Copyright q 2012 G. Britto Antony Xavier et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The recent theory and applications of difference operator ΔL introduced in M. Maria Susai Manuel
n
et al., 2012 are enriched and extended with a useful tool Δ−m
k e−k ||km−1j for finding the
values of various series of discrete gamma functions in number theory. Illustrative examples show
the effectiveness of the obtained results in finding the values of various gamma series.
1. Introduction
The fractional calculus involving gamma function is a generalization of differential calculus,
allowing to define derivatives of real or complex order 1, 2. It is a mathematical subject that
has proved to be very useful in applied fields such as economics, engineering, and physics
3–7. In 1989, Miller and Ross introduced the discrete analogue of the Riemann-Liouville
fractional derivative and proved some properties of the fractional difference operator 8.
In the general fractional h-difference Riemann-Liouville operator mentioned in 9, 10, the
presence of the h parameter is particularly interesting from the numerical point of view,
because when h tends to zero the solutions of the fractional difference equations can be seen
as approximations to the solutions of corresponding Riemann-Liouville fractional differential
ft Definition 2.8
equation 9, 11. On the other hand, fractional h sum of order m ≥ 1 Δ−m
h
of 9 is very useful to derive many interesting results in a different way in number theory
such as the sum of the mth partial sums on nth powers of arithmetic, arithmetic-geometric
uk
progressions, and products of n consecutive terms of arithmetic progression using Δ−m
12.
We observed that no results in number theory using definition 2.8 of 9 had been
derived. In this paper, we use Definition 2.8 of 9 in a different way and define discrete
gamma factorial function to obtain summation formulas of certain series on gamma function
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and gamma factorial function in number theory by getting closed and summation form of
k
ukm−1j , here we replace ft by uk, h by , and ν by m on the notations used in
Δ−m
9.
2. Preliminaries
Before stating and proving our results, we present some notations, basic definitions, and
preliminary results which will be useful for further subsequent discussions. Let > 0, k ∈
0, ∞, j k − k/ where k/ denotes the integer part of k/, N j {j, j, 2 j, . . .}
and N1 j Nj. Throughout this paper, cj is a constant for all k ∈ N j and for
k
k
any positive integer m, we denote Δ−m
ukm−1j Δ−1
· · · Δ−1
Δ−1
uk|j |
k
k
where Δ−1
uk|j u1 k Δ−1
uk−Δ−1
uj, Δ−1
Δ−1
uk|j |
j, and so on.
k
k
j
· · · |
k
m−1j
,
u2 k Δ−1
u1 k−Δ−1
u1 j
Definition 2.1 see 13. For a real valued function uk, the generalized difference operator
Δ and its inverse are, respectively, defined as
Δ uk uk − uk,
k ∈ 0, ∞, ∈ 0, ∞,
then vk Δ−1
uk cj .
if Δ vk uk,
2.1
2.2
Definition 2.2 see 10. For k, n ∈ 0, ∞, the -factorial function is defined by
n
k n
Γk/ 1
,
Γk/ 1 − n
2.3
n
where Γ is the Euler gamma function and k1 kn .
Remark 2.3. When n ∈ N1, 2.3, and its Δ difference become
n
k n−1
k − t,
t0
n
n−1
Δ k nk
.
2.4
Lemma 2.4 see 13. Let Snr ’s be the Stirling numbers of second kind. Then,
kn n
r1
r
Snr n−r k .
2.5
Theorem 2.5 see 13. Let uk, k ∈ 0, ∞ be real valued function. Then for k ∈ , ∞,
k k/
uk
uk − r.
Δ−1
j
r1
2.6
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3
Lemma 2.6. Let vk and wk be two real valued functions. Then,
−1
−1
−1
Δ−1
vkwk vkΔ wk − Δ Δ wk Δ vk .
2.7
Proof. From 2.1, we find
Δ vkzk zk Δ vk vkΔ zk.
2.8
Applying 2.2 in 2.8, we obtain
−1
Δ−1
vkΔ zk vkzk − Δ zk Δ vk.
2.9
The proof follows by taking wk Δ zk in 2.9.
3. Main Results
In this section, we use the following notations: Lm−1 {1, 2, . . . , m − 1}, 0Lm−1 {φ},
φ is an empty set, 1Lm−1 {{1}, {2}, . . . , {m − 1}}, 2Lm−1 {{1, 2}, {1, 3}, . . . , {1, m −
1}, {2, 3}, . . . , {2, m − 1}, . . . , {m − 2, m − 1}}. In general, tLm−1 set of all subsets of size
t from the set Lm−1 such that if {m1 , m2 , . . . , mt } ∈ tLm−1 , then m1 < m2 < · · · < mt ,
m−1
m − 1Lm−1 {{1, 2, . . . , m − 1}}, ℘Lm−1 m−1
t0 tLm−1 , power set of Lm−1 ,
t1 ft 0
t
for m ≤ 1, and i2 fi 1 for t ≤ 1.
Lemma 3.1 see 13. Let k ∈ , ∞. Then,
k
−k Δ−1
e j
e−j
e−k
− −
.
−1 e −1
e−
3.1
Lemma 3.2 see 13. Let n be any nonnegative integer. Then,
n k
Δ−1
k
j
n1
n1
j
k
− .
n 1 n 1
3.2
In particular, when 1, k r − 1 ∈ N {1, 2, 3, . . .}, j 0, then 3.2 becomes
r−1 r − 1n1
.
Δ−1 kn 0
n 1
3.3
0
Remark 3.3. For any constant c, since 1 k , by 3.2 and linearity of Δ−1
,
⎡
⎤
1
1
k
k
j
k
−1 0 ⎣ − ⎦.
Δ−1
cj cΔ k j c
1! 1!
3.4
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Lemma 3.4. Let n ∈ N0 and r ∈ N3. Then,
r−1
r − 1n1
.
Δ−1 kn r − 2n r − 3n · · · 1n 0
n 1
3.5
Proof. The proof follows by taking 1, uk kn in 2.6 and 3.2.
Theorem 3.5. Let m be a positive integer, ∈ 0, ∞, and k ∈ m, ∞. Then,
k/
r − 1m−1
k
uk − r.
uk
Δ−m
m−1j
rm m − 1!
3.6
Proof. Taking Δ−1
on 2.6, and applying 2.6 for Δ−1
uk − r, we get
Δ−1
k k
uk
Δ−1
j
j
k/
r1
Δ−1
uk − r k/
k−r/
uk − r − s.
r1
3.7
s1
From the notation given this section and ordering the terms uk − r, we find
k
Δ−2
uk
j
k/
r2
r − 11
uk − r.
1!
3.8
on 3.8, by 2.6 for Δ−1
uk − r, we arrive
Again, taking Δ−1
k
Δ−3
uk
2j
k/
r3
r − 21 r − 31
11
··· uk − r,
1!
1!
1!
3.9
which yields by 3.5,
k
Δ−3
uk
2j
k/
r3
r − 12
uk − r.
2!
Now, 3.6 will be obtained by continuing this process and using 3.5.
3.10
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Theorem 3.6. Let m ∈ N2, ∈ 0, ∞, and k ∈ m, ∞. Then,
m−1
k
k
−m
Δ−m
uk m−1j Δ uk m−1j −1t
t1 {m1 ,...,mt }∈tLm−1 m−m t
k
×
− 1 j
m − mt ! m−mt
m −m k
t
mi − 1 j i i−1 ×
.
mi − mi−1 ! mi −mi−1 i2
1
Δ−m
u m1
3.11
m−1j
Proof. Applying the limit j to k on Δ−1
uk, we write
k
−1
−1
Δ−1
uk Δ uk − Δ u j ,
3.12
j
where Δ−1
uj is constant and Δ−1
uk is a function of k. Taking Δ−1
on 3.12, by 3.4 and
applying the limit j to k, we obtain
k
Δ−2
uk
j
k
1 k
k
−1
Δ−2
,
ukj − Δ u j
1! 3.13
j
which can be expressed as
k
Δ−2
uk
j
k
−m Δ−2
uk
−1t Δ 1 u m1 − 1 j
j
k
,
2−m
t
2 − mt !
2−mt k
3.14
j
where t 1 and {m1 mt 1} ∈ 1L1 and is same as
k
Δ−2
uk
j
2−m −1t k t
−2
1
Δ−2
uk
−
Δ
u
j
Δ−m
u m1 − 1 j
2−m
2 − mt ! t
2−m −1t j t −m1 −
Δ u m1 − 1 j ,
2 − mt ! 2−mt
2−mt uk, k
where all the terms except Δ−2
are constants.
3.15
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Advances in Numerical Analysis
on 3.15, by 3.2 and 3.4, we arrive
Again taking Δ−1
k
Δ−3
uk2j
k
Δ−3
uk2j
−
k
1 k j
1! Δ−2
u
2j
−1
1
1
Δ−m
u m1
k
− 1 j
3−m
3 − mt ! t 3−mt k
3.16
2j
2−mt k
1 k
j
1
−12 Δ−m
u m1 − 1 j 2−m
1! 2 − mt ! t ,
2j
which is the same as
k
Δ−3
uk
2j
k
Δ−3
uk
2j
×
3
2
t1 {mt 1},{mt 2}∈1L2 {m1 1,mt 2}∈2L2 1
u m1 − 1 j
−1t Δ−m
m −m k
mi − 1 j i i−1 mi − mi−1 ! mi −mi−1 3−m k t
− mt ! 3−mt
3.17
,
2j
where i 2. In the same way, we find
k
Δ−4
uk
3j
k
Δ−4
uk
3j
×
4
3
t1
{mt }∈1L3 ,
{m1 ,mt }∈2L3 {m1 ,m2 ,mt }∈3L3 4−m 3
k t
− mt ! 4−mt i2
1
u m1 − 1 j
−1t Δ−m
m −m k
mi − 1 j i i−1 mi − mi−1 ! mi −mi−1 3.18
,
3j
which can be expressed as the following:
k
Δ−4
uk
3j
k
Δ−4
uk
3j
3
t1 {m1 ,...,mt }∈tL3 1
u m1 − 1 j
−1t Δ−m
m −m k
3
mi − 1 j i i−1 ×
4 − mt ! 4−mt i2 mi − mi−1 ! mi −mi−1 3.19
4−mt k
3j
The proof completes by continuing this process.
.
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Theorem 3.7 partial summation formula. Let m ∈ N2. Then,
k/
rm
m−1
k
r − 1m−1
uk − r Δ−m
uk
m−1j
m − 1!
t1 {m
−1t
1 ,...,mt }∈tLm−1 m−m t
k
m − mt ! m−mt
m −m k
t
mi − 1 j i i−1 ×
.
mi −mi−1 −
m
m
!
i
i−1
i2
1
× Δ−m
u m1 − 1 j
3.20
m−1j
Proof. The proof follows by equating 3.6 and 3.11.
Corollary 3.8. Let m ∈ N2, ∈ 0, ∞, and k ∈ m, ∞. Then,
k/
rm
r − 1m−1
n
k − r
m − 1!ek−r
⎡
n1−r
n1
m−1
r−1
k
n
m r − 2r−1 r−1
r−1
⎣
×
−1
rm−1
r − 1!
ekr−1 t1 {m1 ,...,mt }∈tLm−1 e− − 1
r1
× −1
n1−r m−mt k
m1 − 1 j r − 2r−1 r−1
rm1 −1 ×
−
m
−1jr−1
m − mt ! m−mt e 1
e −1
t m1
m −m ⎤k
t
mi − 1 j i i−1
⎦
×
mi − mi−1 ! mi −mi−1
i2
.
m−1j
3.21
n
Proof. Taking vk k , wk e−k , uk vkwk in 2.7, by 2.4, 3.1, and 3.2, we
get
Δ−1
uk
1
n−1 −k
n−2 −k
By 3.1 and applying 2.7 for k
Δ−1
uk n1
r1
−k
n e
k −
e −
e , k
−1r−1
−
Δ−1
e−k
n−1
.
nk
e− − 1
3.22
1
e , . . . , k e−k , we arrive
n1−r
k
nr−1
r−1
r − 1r−1 r kr−1 .
−
r − 1!
e −1 e
3.23
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Advances in Numerical Analysis
on and applying 3.23 for m − 1 times, we arrive
Taking Δ−1
Δ−m
uk n1−r
n1
nr−1 m r − 2r−1 r−1 k
.
−1r−1
r − 1! e− − 1rm−1 ekr−1
r1
3.24
n
The proof follows by taking uk k e−k in Theorem 3.7.
The following example illustrates Corollary 3.8.
Example 3.9. Consider the case when m 4 and n 2. In this case, L3 {1, 2, 3}, 1L3 {{1}, {2}, {3}}, 2L3 {{1, 2}, {1, 3}, {2, 3}}, 3L3 {{1, 2, 3}} and 3.21 becomes
k/
r4
2
r − 13 k − r
3!ek−r
⎡
1r
r−1 r−1
3
r−1
2 rr−1 r−1 k
2
t m1 r − 2
× ⎣
−1r−1
−1
r3
rm
−1
1
r − 1!
ekr−1 t1 {m ,...,m }∈tL e− − 1
e− − 1
r1
3
t
1
×
4 − mt
1r
4−m k t
! 4−mt em1 −1jr−1
m1 − 1 j
3
m −m ⎤k
t
mi − 1 j i i−1
⎦
mi − mi−1 ! mi −mi−1
i2
.
3j
3.25
The double summation expression of 3.25 will be obtained by adding the sums corresponds
to 1L3 :
3
2
1r k
r − 1r−1 r−1 1r k
rr−1 r−1
j
j r jr−1
3
r1
−−1
3!
2! 2
e
e
ejr−1
e−−1
r 1
r−1 r−1
r2 2jr−1
e
e−−1
1
1r k
,
2 j 3.26
corresponds to 2L3 :
1
2 r − 1r−1 r−1 1r k j r − 1r−1 r−1 1r
j −−1 r jr−1 j r
2! 2
e
e
e−−1 ejr−1
2
2
1 1 k 2 j 1r k 2 j rr−1 r−1
×
,
r1 jr−1 j 2! 2
2! 2
e
e−−1
3.27
and to 3L3 :
1
r − 1r−1 r−1 1r k
−−1 r jr−1 j e
e
j
1 2 j
1
.
3.28
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9
Corollary 3.10. Let ∈ 0, ∞ and k ∈ , ∞. Then,
k
n1 −nr−1 r−1 k n1−r k k/
k − rn
n
−k
Δ−1
r kr−1 .
k e
k−r
−
j
e
e
r1
r1 1 − e
3.29
j
n
r
r
Proof. Since k e−k 0 as k → ∞ and 0 0 if r /
0 and 0 1 if r 0, the upper limit of
3.23 for k → ∞ will be zero and lower limit for j 0, gives 3.29.
4. Discrete Gamma—Factorial Function
First we derive infinite series formula using Δ−1
, which induces the definition of discrete
gamma factorial function.
uk 0. Then,
Theorem 4.1. Let k ∈ 0, ∞ and limk → ∞ Δ−1
∞
∞ Δ−1
uk
uk r.
k
r0
4.1
Proof. From 2.6, and expressing its terms in reverse order, we find
k
∞
−1
Δ−1
uk
Δ
uk
u j u j · · · uk − uk uk · · · u∞,
j
k
4.2
which is the same as
∞
∞ Δ−1
u j r .
uk j
r0
4.3
u∞ 0 and then replacing j by k.
Now 4.1 follows by given condition Δ−1
Definition 4.2. For n ∈ −∞, ∞ − {0} and k ∈ 0, ∞, the discrete k-gamma factorial function
is defined as
k Γ n
∞
n−1 −k ∞
n−1
k
e
Δ−1
k r e−kr ,
k
r0
4.4
and the discrete k-gamma function is defined as
k Γ n
∞
∞
n−1 −k
k
e
Δ−1
k rn−1 e−kr .
k
r0
4.5
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In particular, when k 0, 4.4 and 4.5 becomes
∞
n−1 −k ∞
n−1
k
Γ n Δ−1
e
r e−r ,
4.6
∞
∞
n−1 −k
k
e
Γ n Δ−1
rn−1 e−r
4.7
k0
k0
r1
r1
which can be called as the discrete gamma factorial function and the discrete gamma function,
respectively.
Theorem 4.3. Let ∈ 0, ∞ and n ∈ N. Then,
⎤
⎡
n nr r k n−r e−kr
∞
⎦ k rn e−kr .
k Γ n 1 ⎣
r1
1 − e−
r0
r0
4.8
Proof. The proof follows by taking j k, k ∞ in 3.29 and multiplying it by .
Theorem 4.4. Let ∈ 0, ∞ and n ∈ N1. Then,
Γ n 1 e−
n1 e−n
nΓ
n!
n
n1 .
1 − e−
1 − e−
4.9
Proof. From 2.7, 3.1, and 4.6, we get
Γ n 1 n
−k
n e
k −
e −
1
−
Δ−1
e−k
n−1
nk
e− − 1
∞
.
4.10
k0
Since k e−k 0 for k 0 and k ∞, 4.10 gives first part of 4.9 by 4.6. Now second part
of 4.9 will be obtained by applying first part of 4.9 again and again and using the identity
Γ 1 /1 − e− .
Advances in Numerical Analysis
11
Theorem 4.5. Let Snr be as given in 2.5 and n ∈ N1. Then,
n
r −k
n −k
n n−r −m
k
k
,
e
S
Δ
e
Δ−m
r
r1
4.11
which gives
Γ n 1 n
n
e− Snr n−r Γ r 1 Sn n−r rΓ r.
1 − e− r1 r
r1
4.12
. Now, 4.12 will be obtained
Proof. The proof of 4.11 follows by 2.5 and linearity of Δ−1
by taking m 1 in 4.11 and using 4.6, 4.7, and 4.9.
Theorem 4.6. Let ∈ 0, ∞ and n ∈ N1. Then,
lim Γ n lim Γ n n − 1! Γn.
→0
→0
4.13
Proof. As → 0, all the terms except the last term of 4.12 will be zero. Also, since
limk → ∞ e− /1 − e− 1, by 4.9, we arrive
lim Γ n lim n − 1Γ n 1 lim Γ n.
→0
→0
→0
4.14
Now, 4.13 follows by replacing n by n − 1 and taking limit → 0 on 4.9.
Theorem 4.7. Let k ∈ , ∞ and j k − k/. Then,
k/
r1
j
k
k
k
−r Γ
−r1 Γ
1 −Γ
1 .
4.15
Proof . From 2.1 and 2.2, we have
k
k
k
Δ Γ
1 Γ
1 ,
4.16
which yields
Δ−1
k
k
k
k
Γ
1
1 .
Γ
j
Now, the proof follows from 2.6 and 4.17.
4.17
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Advances in Numerical Analysis
Theorem 4.8. Let k ∈ , ∞ and j k − k/. Then,
k/
r1
1
1
1
k/ − r
.
−
Γk/ 1
k/ − r 1 Γk/ − r 1 Γ j/ 1
4.18
Proof. From 2.1 and 2.2, we find
Δ
1
k/
−
,
Γk/ 1
k/ 1Γk/ 1
4.19
which yields by 2.2,
−Δ−1
k/
k/ 1Γk/ 1
k
j
1
1
− .
Γk/ 1 Γ j/ 1
4.20
Now, the proof follows from 2.6 and 4.20.
Corollary 4.9. Let k ∈ , ∞ and j k − k/. Then,
∞
1
1
k/ r
.
k/ r 1 Γk/ r 1 Γk/ 1
r0
4.21
Proof. The proof follows from 4.1 and 4.20.
Corollary 4.10. (i) When k 0,
∞
1
r
1.
r 1 Γr 1
r0
4.22
(ii) When k 1 and 0.5,
∞
2r
1
1
1
.
3
r
Γ3
r
Γ3
2!
r0
4.23
(iii) When k n, n ∈ N1,
∞
1
1
1
nr
.
n
1
r
Γn
1
r
Γn
1
n!
r0
4.24
Acknowledgment
This paper is supported by the University Grants Commission, SERO, Hyderabad, India.
Advances in Numerical Analysis
13
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