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arXiv:math/0511131v1 [math.NT] 5 Nov 2005

A NOTE ON q-ANALOGUE OF S´ANDOR’S FUNCTIONS

by

Taekyun Kim1, C. Adiga2and Jung Hun Han2

1Department of Mathematics Education

Kongju National University, Kongju 314-701

South Korea

e-mail:tkim@kongju.ac.kr / tkim64@hanmail.net

2Department of Studies in Mathematics

University of Mysore, Manasagangotri

Mysore 570006, India

e-mail:c-adiga@hotmail.com

Dedicated to Sun-Yi Park on 90th birthday

ABSTRACT

The additive analogues of Pseudo-Smarandache, Smarandache-simple func-

tions and their duals have been recently studied by J. S´ andor. In this note,

we obtain q-analogues of S´ andor’s theorems [6].

Keywords and Phrases: q-gamma function, Pseudo-Smarandache function,

Smarandache-simple function, Asymtotic formula.

2000 AMS Subject Classification: 33D05,40A05.

1 Introduction

The additive analogues of Smarandache functions S and S∗ have been

introduced by S´ andor [5] as follows:

S(x) = min{m ∈ N : x ≤ m!},x ∈ (1,∞),

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and

S∗(x) = max{m ∈ N : m! ≤ x},x ∈ [1,∞).

He has studied many important properties of S∗relating to continuity, differ-

entiability and Riemann integrability and also proved the following theorems:

Theorem 1.1

S∗(x) ∼

logx

loglogx

(x → ∞).

Theorem 1.2 The series

∞

?

n=1

1

n(S∗(n))α

is convergent for α > 1 and divergent for α ≤ 1.

In [1], Adiga and Kim have obtained generalizations of Theorems 1.1 and

1.2 by the use of Euler’s gamma function. Recently Adiga-Kim-Somashekara-

Fathima [2] have established a q-analogues of these results on employing the

q-analogue of Stirling’s formula. In [6], S´ andor defined the additive analogues

of Pseudo-Smarandache, Smarandache-simple functions and their duals as fol-

lows:

Z(x) = min

?

m ∈ N : x ≤m(m + 1)

2

?

,x ∈ (0,∞),

Z∗(x) = max

?

m ∈ N :m(m + 1)

2

≤ x

?

,x ∈ [1,∞),

P(x) = min{m ∈ N : px≤ m!}, p > 1,x ∈ (0,∞),

and

P∗(x) = max{m ∈ N : m! ≤ px}, p > 1,x ∈ [1,∞).

He has also proved the following theorems:

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Theorem 1.3

Z∗(x) ∼1

2

√8x + 1(x → ∞).

Theorem 1.4 The series

∞

?

n=1

1

(Z∗(n))α

is convergent for α > 2 and divergent for α ≤ 2. The series

∞

?

n=1

1

n(Z∗(n))α

is convergent for all α > 0.

Theorem 1.5

logP∗(x) ∼ logx

(x → ∞).

Theorem 1.6 The series

∞

?

n=1

1

n

?loglogn

logP∗(n)

?α

is convergent for α > 1 and divergent for α ≤ 1.

The main purpose of this note is to obtain q-analogues of S´ andor’s Theo-

rems 1.3 and 1.5. In what follows, we make use of the following notations and

definitions. F. H. Jackson defined a q-analogue of the gamma function which

extends the q-factorial

(n!)q= 1(1 + q)(1 + q + q2)···(1 + q + q2+ ··· + qn−1),

cf [3],

which becomes the ordinary factorial as q → 1. He defined the q-analogue of

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the gamma function as

Γq(x) =

(q;q)∞

(qx;q)∞(1 − q)1−x,

0 < q < 1,

and

Γq(x) =(q−1;q−1)∞

(q−x;q−1)∞(q − 1)1−xq(x

2), q > 1,

where

(a;q)∞=

∞

?

n=0

(1 − aqn).

It is well known that Γq(x) → Γ(x) as q → 1, where Γ(x) is the ordinary

gamma function.

2 Main Theorems

We now define the q-analogues of Z and Z∗as follows:

Zq(x) = min

?1 − qm

1 − q

: x ≤Γq(m + 2)

2Γq(m)

?

,m ∈ N,x ∈ (0,∞),

and

Z∗

q(x) = max

?1 − qm

1 − q

:Γq(m + 2)

2Γq(m)

≤ x

?

,m ∈ N,x ∈

?Γq(3)

2Γq(1),∞

?

,

where 0 < q < 1. Clearly, Zq(x) → Z(x) and Z∗

the definitions of Zqand Z∗

q(x) → Z∗(x) as q → 1−. From

q, it is clear that

Zq(x) =

1,

if

x ∈

x ∈

?

?

0,

Γq(3)

2Γq(1)

?

1−qm

1−q,

if

Γq(m+1)

2Γq(m−1),Γq(m+2)

2Γq(m)

?

,m ≥ 2,

(2.1)

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and

Z∗

q(x) =1 − qm

1 − q

if

x ∈

?Γq(m + 2)

2Γq(m)

,Γq(m + 3)

2Γq(m + 1)

?

.

(2.2)

Since

1 − qm−1

1 − q

≤1 − qm

1 − q

=1 − qm−1

1 − q

+ qm−1≤1 − qm−1

1 − q

+ 1,

(2.1) and (2.2) imply that for x ≥

Γq(3)

2Γq(1),

Z∗

q(x) ≤ Zq(x) ≤ Z∗

q(x) + 1.

Hence it suffices to study the function Z∗

q. We now prove our main theorems.

Theorem 2.1 If 0 < q < 1, then

√1 + 8xq − (1 + 2q)

2q2

< Z∗

q(x) ≤

√1 + 8xq − 1

2q

,x ≥

Γq(3)

2Γq(1).

Proof. If

Γq(k + 2)

2Γq(k)

≤ x <

Γq(k + 3)

2Γq(k + 1),

(2.3)

then

Z∗

q(x) =1 − qk

1 − q

and

(1−qk)(1−qk+1)−2x(1−q)2≤ 0 < (1−qk+1)(1−qk+2)−2x(1−q)2. (2.4)

Consider the functions f and g defined by

f(y) = (1 − y)(1 − yq) − 2x(1 − q)2

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and

g(y) = (1 − yq)(1 − yq2) − 2x(1 − q)2.

Note that f is monotonically decreasing for y ≤1+q

for y <1+q

2qand g is strictly decreasing

2q2. Also f(y1) = 0 = g(y2) where

y1=(1 + q) − (1 − q)√1 + 8xq

2q

,

y2=(q + q2) − q(1 − q)√1 + 8xq

2q3

.

Since y1<1+q

2q,y2<1+q

2q2 and qk<1+q

2q<1+q

2q2, from (2.4), it follows that

f(qk) ≤ f(y1) = 0 = g(y2) < g(qk).

Thus y1≤ qk< y2and hence

1 − y2

1 − q

<1 − qk

1 − q

≤1 − y1

1 − q.

i.e.

√1 + 8xq − (1 + 2q)

2q2

< Z∗

q(x) ≤

√1 + 8xq − 1

2q

.

This completes the proof.

Remark.

Letting q → 1−in the above theorem, we obtain S´ andor’s

Theorem 1.3.

We define the q-analogues of P and P∗as follows:

Pq(x) = min{m ∈ N : px≤ Γq(m + 1)}, p > 1,x ∈ (0,∞),

and

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P∗

q(x) = max{m ∈ N : Γq(m + 1) ≤ px}, p > 1,x ∈ [1,∞),

where 0 < q < 1. Clearly, Pq(x) → P(x) and P∗

the definitions of Pqand P∗

q(x) → P∗(x) as q → 1−. From

q, we have

P∗

q(x) ≤ Pq(x) ≤ P∗

q(x) + 1.

Hence it is enough to study the function P∗

q.

Theorem 2.2 If 0 < q < 1, then

P∗(x) ∼

xlogp

?

log

1

1−q

?

(x → ∞).

Proof. If Γq(n + 1) ≤ px< Γq(n + 2), then

P∗

q(x) = n

and

logΓq(n + 1) ≤ logpx< logΓq(n + 2).

(2.5)

But by the q-analogue of Stirling’s formula established by Moak [4], we

have

logΓq(n + 1) ∼

?

n +1

2

?

log

?qn+1− 1

q − 1

?

∼ nlog

?

1

1 − q

?

.

(2.6)

Dividing (2.5) throughout by nlog(

1

1−q), we obtain

logΓq(n + 1)

nlog(

1

1−q)

≤

xlogp

P∗

q(x)log(

1

1−q)<logΓq(n + 2)

nlog(

1

1−q)

.

(2.7)

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Using (2.6) in (2.7), we deduce

lim

x→∞

xlogp

P∗

q(x)log(

1

1−q)= 1.

This completes the proof.

References

[1] C. Adiga and T. Kim, On a generalization of S´ andor’s function, Proc.

Jangjeon Math. Soc., 5, (2002), 121–124.

[2] C. Adiga, T. Kim, D. D. Somashekara and N. Fathima, On a q-analogue

of S´ andor’s function, J. Inequal. Pure and Appl. Math., 4,No.4 Art.84

(2003), 1–5.

[3] T. Kim, Non-archimedean q-integrals associated with multiple Changhee

q-Bernoulli polynomials, Russian J. Math. Phys., 10 (2003), 91–98.

[4] D. S. Moak, The q-analogue of Stirling’s formula, Rocky Mountain J.

Math., 14 (1984), 403–413.

[5] J. S´ andor, On an additive analogue of the function S, Notes Numb. Th.

Discr. Math., 7(2001), 91–95.

[6] J. S´ andor, On additive analogues of certain arithmetic functions, Smaran-

dache Notions J.,14(2004),128-133.