Journal of Inequalities in Pure and
Applied Mathematics
ON A Q-ANALOGUE OF SÁNDOR’S FUNCTION
1
C. ADIGA, 2 T. KIM, D. D. SOMASHEKARA AND
SYEDA NOOR FATHIMA
1 Department
of Studies in Mathematics
Manasa Gangothri, University of Mysore
Mysore-570 006, INDIA.
volume 4, issue 5, article 84,
2003.
Received 19 September, 2003;
accepted 29 September, 2003.
Communicated by: J. Sándor
2 Institute
of Science Education
Kongju National University,
Kongju 314-701
S. KOREA.
EMail: tkim@kongju.ac.kr
Abstract
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2000
Victoria University
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Abstract
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].
2000 Mathematics Subject Classification: 33D05, 40A05.
Key words: q-gamma function, q-Stirling’s formula, Asymptotic formula.
On a q-Analogue of Sándor’s
Function
This paper was supported by Korea Research Foundation Grant (KRF-2002-050C00001).
Dedicated to Professor Katsumi Shiratani on the occasion of his 71st birthday
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Some Properties of Sq and Sq∗ . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
9
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1.
Introduction
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) =
and
(q; q)∞
(1 − q)1−x ,
(q x ; q)∞
−1
Γq (x) =
0 < q < 1,
−1
(q ; q )∞
1−x (x2 )
(q
−
1)
q ,
(q −x ; q −1 )∞
where
(a; q)∞ =
∞
Y
q > 1,
On a q-Analogue of Sándor’s
Function
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
Title Page
n
(1 − aq ).
Contents
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, ∞),
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and
S∗ (x) = max{m ∈ N : m! ≤ x},
x ∈ [1, ∞).
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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
On a q-Analogue of Sándor’s
Function
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:
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
Title Page
Sq (x) = min{m ∈ N : x ≤ Γq (m + 1)},
x ∈ (1, ∞),
and
Sq∗ (x)
= max{m ∈ N : Γq (m + 1) ≤ x},
x ∈ [1, ∞),
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].
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To prove our main theorems we make use of the following q-analogue of
Stirling’s formula established by D.S. Moak [5]:
(1.1)
log Γq (z) ∼
1
z−
2
Z −z logn q
qz − 1
udu
1
log
+
u
q−1
log q − log q
e −1
∞
2k−1
X B2k
log q
+ Cq +
q z P2k−1 (q z ),
z −1
(2k)!
q
k=1
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),
P0 = 1,
n ≥ 1.
On a q-Analogue of Sándor’s
Function
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
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2.
Some Properties of Sq and Sq∗
From the definitions of Sq and Sq∗ , it is clear that
(2.1)
Sq (x) = m
if x ∈ (Γq (m), Γq (m + 1)],
for m ≥ 2,
Sq∗ (x) = m
if x ∈ [Γq (m + 1), Γq (m + 2)),
and
(2.2)
for m ≥ 1.
(2.1) and (2.2) imply
 ∗
 Sq (x) + 1, if x ∈ (Γq (k + 1), Γq (k + 2)),
Sq (x) =
 ∗
Sq (x),
if x = Γq (k + 2).
Thus
On a q-Analogue of Sándor’s
Function
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
Sq∗ (x) ≤ Sq (x) ≤ Sq∗ (x) + 1.
Title Page
Hence it suffices to study the function Sq∗ . The following are the simple properties of Sq∗ .
Contents
(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.
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(3) Sq∗ is differentiable on (1, ∞)\A and since
lim
x→Γq (k+1)+
Sq∗ (x) − Sq∗ (Γq (k + 1))
=0
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
Sq∗ (x)dx
Z
a
b
kdx = k(b − a).
=
a
(ii) For n > k, we have
Z Γq (n+1)
Sq∗ (x)dx
Contents
(n−k) Z Γq (k+m+1)
X
m=1
Sq∗ (x)dx
Γq (k+m)
=
(k + m − 1)[Γq (k + m + 1) − Γq (k + m)]
m=1
=
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(n−k)
X
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(n−k)
X
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
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Γq (k+1)
=
On a q-Analogue of Sándor’s
Function
(k + m − 1)Γq (k + m)[q + q 2 + · · · + q k+m−1 ].
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(iii) If a ∈ [Γq (k + 1), Γq (k + 2)) and b ∈ [Γq (n), Γq (n + 1)) then
Z
a
b
Sq∗ (x)dx
Z Γq (k+2)
Z
∗
=
Sq (x)dx +
a
Γq (n)
Sq∗ (x)dx
Γq (k+2)
= k[Γq (k + 2) − a] +
n−k−2
X
Z
b
+
Sq∗ (x)dx
Γq (n)
(k + m)Γq (k + m + 1)
m=1
× (q + q 2 + ... + q k+m ) + (n − 1)[b − Γq (n)],
by (ii).
On a q-Analogue of Sándor’s
Function
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
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3.
Main Theorems
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
log Γq (n + 1) ∼ n +
2
n+1
q
−1
1
log
∼ n log
.
q−1
1−q
1
Dividing (3.1) throughout by n log 1−q
, we obtain
(3.2)
(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.
On a q-Analogue of Sándor’s
Function
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
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Theorem 3.2. The series
∞
X
(3.4)
1
n(Sq∗ (n))α
n=1
is convergent for α > 1 and divergent for α ≤ 1.
Proof. Since
Sq∗ (x) ∼
we have
log x
,
1
log 1−q
log n
log n
< Sq∗ (n) < B
,
A
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
1
1−q
X
∞
n=1
1
.
n(log n)α
P
1
By the integral test,
converges for α > 1 and diverges for 0 ≤ α ≤ 1.
n(log n)α
P
1
1
1
If α < 0, then n(log n)α > n for n ≥ 3. Hence
diverges by the
(n log n)α
comparison test.
On a q-Analogue of Sándor’s
Function
C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
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References
[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.
On a q-Analogue of Sándor’s
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C. Adiga, T. Kim, D. D.
Somashekara and Syeda Noor
Fathima
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