Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 5, Article 84, 2003
ON A q-ANALOGUE OF SÁNDOR’S FUNCTION
C. ADIGA, T. KIM, D. D. SOMASHEKARA, AND SYEDA NOOR FATHIMA
D EPARTMENT OF S TUDIES IN M ATHEMATICS
M ANASA G ANGOTHRI , U NIVERSITY OF M YSORE
M YSORE -570 006, INDIA.
I NSTITUTE OF S CIENCE E DUCATION
KONGJU NATIONAL U NIVERSITY, KONGJU 314-701
S. KOREA.
tkim@kongju.ac.kr
Received 19 September, 2003; accepted 29 September, 2003
Communicated by J. Sándor
Dedicated to Professor Katsumi Shiratani on the occasion of his 71st birthday
A BSTRACT. In this paper we obtain a q-analogue of J. Sándor’s theorems [6], on employing the
q-analogue of Stirling’s formula established by D. S. Moak [5].
Key words and phrases: q-gamma function, q-Stirling’s formula, Asymptotic formula.
2000 Mathematics Subject Classification. 33D05, 40A05.
1. I NTRODUCTION
F. H. Jackson defined a q-analogue of the gamma function which extends the q-factorial
(n!)q = 1(1 + q)(1 + q + q 2 ) · · · (1 + q + ... + q n−1 ), cf. [3, 4],
which becomes the ordinary factorial as q → 1. He defined the q-analogue of the gamma
function as
Γq (x) =
(q; q)∞
(1 − q)1−x ,
(q x ; q)∞
0 < q < 1,
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
This paper was supported by Korea Research Foundation Grant (KRF-2002-050-C00001).
132-03
2
C. A DIGA , T. K IM , D. D. S OMASHEKARA , AND S YEDA N OOR FATHIMA
and
Γq (x) =
x
(q −1 ; q −1 )∞
(q − 1)1−x q (2) ,
−x
−1
(q ; q )∞
where
(a; q)∞ =
∞
Y
q > 1,
(1 − aq n ).
n=0
It is well-known that Γq (x) → Γ(x) as q → 1, where Γ(x) is the ordinary gamma function. In
[2], R. Askey obtained a q-analogue of many of the classical facts about the gamma function.
In his interesting paper [6], J. Sándor defined the functions S and S∗ by
S(x) = min{m ∈ N : x ≤ m!},
x ∈ (1, ∞),
S∗ (x) = max{m ∈ N : m! ≤ x},
x ∈ [1, ∞).
and
He has studied many important properties of S∗ and proved the following theorems:
Theorem 1.1.
S∗ (x) ∼
Theorem 1.2. The series
log x
log log x
∞
X
n=1
(x → ∞).
1
n(S∗ (n))α
is convergent for α > 1 and divergent for α ≤ 1.
In [1], C. Adiga and T. Kim have obtained a generalization of Theorems 1.1 and 1.2.
We now define the q-analogues of S and S∗ as follows:
Sq (x) = min{m ∈ N : x ≤ Γq (m + 1)},
x ∈ (1, ∞),
Sq∗ (x) = max{m ∈ N : Γq (m + 1) ≤ x},
x ∈ [1, ∞),
and
where 0 < q < 1.
Clearly Sq (x) → S(x) and Sq∗ (x) → S∗ (x) as q → 1− .
In Section 2 of this paper we study some properties of Sq and Sq∗ , which are similar to those of
S and S∗ studied by Sándor [6]. In Section 3 we prove two theorems which are the q-analogues
of Theorems 1.1 and 1.2 of Sándor [6].
To prove our main theorems we make use of the following q-analogue of Stirling’s formula
established by D.S. Moak [5]:
z
Z −z logn q
1
q −1
1
udu
(1.1) log Γq (z) ∼ z −
log
+
u
2
q−1
log q − log q
e −1
∞
X B2k log q 2k−1
+ Cq +
q z P2k−1 (q z ),
z −1
(2k)!
q
k=1
J. Inequal. Pure and Appl. Math., 4(5) Art. 84, 2003
http://jipam.vu.edu.au/
O N A q-A NALOGUE OF S ÁNDOR ’ S F UNCTION
3
where Cq is a constant depending upon q, and Pn (z) is a polynomial of degree n satisfying,
0
Pn (z) = (z − z 2 )Pn−1
(z) + (nz + 1)Pn−1 (z),
2. S OME P ROPERTIES OF Sq
AND
P0 = 1,
n ≥ 1.
Sq∗
From the definitions of Sq and Sq∗ , it is clear that
(2.1)
if x ∈ (Γq (m), Γq (m + 1)],
Sq (x) = m
for m ≥ 2,
and
(2.2)
Sq∗ (x) = m
if x ∈ [Γq (m + 1), Γq (m + 2)),
for m ≥ 1.
(2.1) and (2.2) imply
Sq (x) =
 ∗
 Sq (x) + 1, if x ∈ (Γq (k + 1), Γq (k + 2)),
 S ∗ (x),
q
if x = Γq (k + 2).
Thus
Sq∗ (x) ≤ Sq (x) ≤ Sq∗ (x) + 1.
Hence it suffices to study the function Sq∗ . The following are the simple properties of Sq∗ .
(1) Sq∗ is surjective and monotonically increasing.
(2) Sq∗ is continuous for all x ∈ [1, ∞)\A, where A = {Γq (k + 1) : k ≥ 2}. Since
lim
x→Γq (k+1)+
Sq∗ (x) = k
and
lim
x→Γq (k+1)−
Sq∗ (x) = (k − 1), (k ≥ 2),
Sq∗ is continuous from the right at x = Γq (k + 1), k ≥ 2, but it is not continuous from
the left.
(3) Sq∗ is differentiable on (1, ∞)\A and since
Sq∗ (x) − Sq∗ (Γq (k + 1))
=0
x − Γq (k + 1)
lim
x→Γq (k+1)+
for k ≥ 1, it has a right derivative in A ∪ {1}.
(4) Sq∗ is Riemann integrable on [a, b], where Γq (k + 1) ≤ a < b, k ≥ 1.
(i) If [a, b] ⊂ [Γq (k + 1), Γq (k + 2)], k ≥ 1, then
Z b
Z b
∗
Sq (x)dx =
kdx = k(b − a).
a
a
(ii) For n > k, we have
Z Γq (n+1)
(n−k) Z
X
∗
Sq (x)dx =
Γq (k+1)
m=1
Γq (k+m+1)
Sq∗ (x)dx
Γq (k+m)
(n−k)
=
X
(k + m − 1)[Γq (k + m + 1) − Γq (k + m)]
m=1
(n−k)
=
X
(k + m − 1)Γq (k + m)[q + q 2 + · · · + q k+m−1 ].
m=1
J. Inequal. Pure and Appl. Math., 4(5) Art. 84, 2003
http://jipam.vu.edu.au/
4
C. A DIGA , T. K IM , D. D. S OMASHEKARA , AND S YEDA N OOR FATHIMA
(iii) If a ∈ [Γq (k + 1), Γq (k + 2)) and b ∈ [Γq (n), Γq (n + 1)) then
Z Γq (n)
Z b
Z b
Z Γq (k+2)
∗
∗
∗
Sq (x)dx =
Sq (x)dx +
Sq (x)dx +
Sq∗ (x)dx
a
Γq (k+2)
a
= k[Γq (k + 2) − a] +
× (q + q 2 + ... + q
n−k−2
X
Γq (n)
(k + m)Γq (k + m + 1)
m=1
k+m
) + (n − 1)[b − Γq (n)],
by (ii).
3. M AIN T HEOREMS
We now prove our main theorems.
Theorem 3.1. If 0 < q < 1, then
Sq∗ (x) ∼
log x
.
1
log 1−q
Proof. If Γq (n + 1) ≤ x < Γq (n + 2), then
log Γq (n + 1) ≤ log x < log Γq (n + 2).
(3.1)
By (1.1) we have
1
(3.2)
log Γq (n + 1) ∼ n +
2
n+1
1
q
−1
log
∼ n log
.
q−1
1−q
1
Dividing (3.1) throughout by n log 1−q
, we obtain
(3.3)
log Γq (n + 1)
log x
log Γq (n + 2)
≤
<
.
1
1
1
∗
n log 1−q
Sq (x) log 1−q
n log 1−q
Using (3.2) in (3.3) we deduce
log x
= 1.
n→∞ ∗
1
Sq (x) log 1−q
lim
This completes the proof.
Theorem 3.2. The series
(3.4)
∞
X
1
n(Sq∗ (n))α
n=1
is convergent for α > 1 and divergent for α ≤ 1.
J. Inequal. Pure and Appl. Math., 4(5) Art. 84, 2003
http://jipam.vu.edu.au/
O N A q-A NALOGUE OF S ÁNDOR ’ S F UNCTION
5
Proof. Since
Sq∗ (x) ∼
we have
A
log x
,
1
log 1−q
log n
log n
,
< Sq∗ (n) < B
1
1
log 1−q
log 1−q
for all n ≥ N > 1, A, B > 0. Therefore to examine the convergence or divergence of the series
(3.4) it suffices to study the series
log
P
1
1−q
X
∞
n=1
1
.
n(log n)α
1
n(log n)α
By the integral test,
converges for α > 1 and diverges for 0 ≤ α ≤ 1. If α < 0, then
P
1
1
1
> n for n ≥ 3. Hence
diverges by the comparison test.
n(log n)α
(n log n)α
R EFERENCES
[1] C. ADIGA AND T. KIM, On a generalization of Sándor’s function, Proc. Jangjeon Math. Soc., 5
(2002), 121–124.
[2] R. ASKEY, The q-gamma and q-beta functions, Applicable Analysis, 8 (1978), 125–141.
[3] T. KIM, Non-archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials,
Russian J. Math. Phys., 10 (2003), 91–98.
[4] T. KIM, An another p-adic q-L-functions and sums of powers, Proc. Jangjeon Math. Soc., 2 (2001),
35–43.
[5] D.S. MOAK, The q-analogue of Stirlings formula, Rocky Mountain J. Math., 14 (1984), 403–413.
[6] J. SÁNDOR, On an additive analogue of the function S, Notes Numb. Th. Discr. Math., 7 (2001),
91–95.
J. Inequal. Pure and Appl. Math., 4(5) Art. 84, 2003
http://jipam.vu.edu.au/