A NOTE ON q-ANALOGUE OF SANDOR’S FUNCTIONS

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

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

by Taekyun Kim1 , C. Adiga2 and Jung Hun Han2 1

Department of Mathematics Education Kongju National University, Kongju 314-701 South Korea e-mail:tkim@kongju.ac.kr / tkim64@hanmail.net 2

Department 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 functions 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, ∞),

2 and S∗ (x) = max{m ∈ N : m! ≤ x},

x ∈ [1, ∞).

He has studied many important properties of S∗ relating to continuity, differentiability and Riemann integrability and also proved the following theorems: Theorem 1.1 S∗ (x) âˆź

logx loglogx

(x → ∞).

Theorem 1.2 The series ∞ X n=1

1 n(S∗ (n))Îą

is convergent for ι > 1 and divergent for ι ≤ 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-SomashekaraFathima [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 follows:

m(m + 1) Z(x) = min m ∈ N : x ≤ , x ∈ (0, ∞), 2

m(m + 1) ≤ x , x ∈ [1, ∞), Z∗ (x) = max m ∈ N : 2 P (x) = min{m ∈ N : px ≤ m!},

p > 1, x ∈ (0, ∞),

P∗ (x) = max{m ∈ N : m! ≤ px },

p > 1, x ∈ [1, ∞).

and

He has also proved the following theorems:

3 Theorem 1.3 Z∗ (x) ∼

1√ 8x + 1 (x → ∞).

2

Theorem 1.4 The series ∞ X n=1

1 (Z∗ (n))α

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

α ∞ X 1 loglogn n logP∗ (n) n=1 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 Theorems 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 + q 2 ) · · · (1 + q + q 2 + · · · + q n−1 ),

cf [3],

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

4 the gamma function as (q; q)∞ (1 − q)1−x , (q x ; q)∞

Γq (x) =

0 < q < 1,

and x (q â&#x2C6;&#x2019;1 ; q â&#x2C6;&#x2019;1 )â&#x2C6;&#x17E; Î&#x201C;q (x) = â&#x2C6;&#x2019;x â&#x2C6;&#x2019;1 (q â&#x2C6;&#x2019; 1)1â&#x2C6;&#x2019;x q (2) , (q ; q )â&#x2C6;&#x17E;

q > 1,

where (a; q)â&#x2C6;&#x17E; =

â&#x2C6;&#x17E; Y

(1 â&#x2C6;&#x2019; aq n ).

n=0

It is well known that Î&#x201C;q (x) â&#x2020;&#x2019; Î&#x201C;(x) as q â&#x2020;&#x2019; 1, where Î&#x201C;(x) is the ordinary gamma function.

2

Main Theorems

We now define the q-analogues of Z and Zâ&#x2C6;&#x2014; as follows: Zq (x) = min

Î&#x201C;q (m + 2) 1 â&#x2C6;&#x2019; qm :xâ&#x2030;¤ 1â&#x2C6;&#x2019;q 2Î&#x201C;q (m)

,

m â&#x2C6;&#x2C6; N, x â&#x2C6;&#x2C6; (0, â&#x2C6;&#x17E;),

and Zqâ&#x2C6;&#x2014; (x)

= max

1 â&#x2C6;&#x2019; q m Î&#x201C;q (m + 2) : â&#x2030;¤x , 1â&#x2C6;&#x2019;q 2Î&#x201C;q (m)

Î&#x201C;q (3) m â&#x2C6;&#x2C6; N, x â&#x2C6;&#x2C6; ,â&#x2C6;&#x17E; , 2Î&#x201C;q (1)

where 0 < q < 1. Clearly, Zq (x) â&#x2020;&#x2019; Z(x) and Zqâ&#x2C6;&#x2014; (x) â&#x2020;&#x2019; Zâ&#x2C6;&#x2014; (x) as q â&#x2020;&#x2019; 1â&#x2C6;&#x2019; . From the definitions of Zq and Zqâ&#x2C6;&#x2014; , it is clear that

Zq (x) =

  1, 

1â&#x2C6;&#x2019;q m 1â&#x2C6;&#x2019;q

if ,

Î&#x201C;q (3) 2Î&#x201C;q (1)

i



x â&#x2C6;&#x2C6; 0,  i

Î&#x201C;q (m+1) Î&#x201C;q (m+2) , m â&#x2030;¥ 2, if x â&#x2C6;&#x2C6; 2Î&#x201C; , 2Î&#x201C;q (m) q (mâ&#x2C6;&#x2019;1)

(2.1)

5 and

Zq∗ (x)

1 − qm = 1−q

Γq (m + 2) Γq (m + 3) if x ∈ , 2Γq (m) 2Γq (m + 1)

.

(2.2)

Since 1 − qm 1 − q m−1 1 − q m−1 1 − q m−1 m−1 ≤ = +q ≤ + 1, 1−q 1−q 1−q 1−q (2.1) and (2.2) imply that for x ≥

Γq (3) , 2Γq (1)

Zq∗ (x) ≤ Zq (x) ≤ Zq∗ (x) + 1.

Hence it suffices to study the function Zq∗ . We now prove our main theorems.

Theorem 2.1 If 0 < q < 1, then √

1 + 8xq − (1 + 2q) < Zq∗ (x) ≤ 2q 2

√

1 + 8xq − 1 , 2q

x≥

Γq (3) .

2Γq (1)

Proof. If Γq (k + 3) Γq (k + 2) ≤x< , 2Γq (k) 2Γq (k + 1)

(2.3)

then Zq∗ (x)

1 − qk = 1−q

and (1 − q k )(1 − q k+1 ) − 2x(1 − q)2 ≤ 0 < (1 − q k+1 )(1 − q k+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

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

Note that f is monotonically decreasing for y ≤ for y <

1+q .

2q 2

1+q 2q

and g is strictly decreasing

Also f (y1) = 0 = g(y2 ) where √ (1 + q) − (1 − q) 1 + 8xq y1 = , 2q √ (q + q 2 ) − q(1 − q) 1 + 8xq . y2 = 2q 3

Since y1 <

1+q , y2 2q

<

1+q 2q 2

and q k <

1+q 2q

<

1+q , 2q 2

from (2.4), it follows that

f (q k ) ≤ f (y1) = 0 = g(y2) < g(q k ).

Thus y1 ≤ q k < y2 and hence 1 − qk 1 − y1 1 − y2 < ≤ .

1−q 1−q 1−q i.e.

√

1 + 8xq − (1 + 2q) < Zq∗ (x) ≤ 2q 2

√

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)}, and

p > 1, x ∈ (0, ∞),

7

Pq∗ (x) = max{m ∈ N : Γq (m + 1) ≤ px },

p > 1, x ∈ [1, ∞),

where 0 < q < 1. Clearly, Pq (x) → P (x) and Pq∗ (x) → P∗ (x) as q → 1− . From the definitions of Pq and Pq∗, we have Pq∗ (x) ≤ Pq (x) ≤ Pq∗ (x) + 1.

Hence it is enough to study the function Pq∗ .

Theorem 2.2 If 0 < q < 1, then P∗ (x) ∼

xlogp

1 log 1−q

(x → ∞).

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

n+1

1 q −1 1 logΓq (n + 1) ∼ n + log ∼ nlog .

2 q−1 1−q

(2.6)

1 ), we obtain Dividing (2.5) throughout by nlog( 1−q

logΓq (n + 1) xlogp logΓq (n + 2) ≤ ∗ < .

1 1 1 ) nlog( 1−q ) Pq (x)log( 1−q ) nlog( 1−q

(2.7)

8 Using (2.6) in (2.7), we deduce lim

xlogp

x→∞ P ∗ (x)log( 1 ) q 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, Smarandache Notions J.,14(2004),128-133.