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http://jipam.vu.edu.au/

Volume 4, Issue 5, Article 84, 2003

ON Aq-ANALOGUE OF SÁNDOR’S FUNCTION

C. ADIGA, T. KIM, D. D. SOMASHEKARA, AND SYEDA NOOR FATHIMA DEPARTMENT OFSTUDIES INMATHEMATICS

MANASAGANGOTHRI, UNIVERSITY OFMYSORE

MYSORE-570 006, INDIA.

INSTITUTE OFSCIENCEEDUCATION

KONGJUNATIONALUNIVERSITY, KONGJU314-701 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 71stbirthday

ABSTRACT. In this paper we obtain aq-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. INTRODUCTION

F. H. Jackson defined aq-analogue of the gamma function which extends theq-factorial (n!)q = 1(1 +q)(1 +q+q2)· · ·(1 +q+...+qn−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)

(qx;q)

(1−q)1−x, 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

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and

Γq(x) = (q−1;q−1)

(q−x;q−1)(q−1)1−xq(x2), q >1, where

(a;q)=

Y

n=0

(1−aqn).

It is well-known thatΓq(x)→ Γ(x)asq → 1, whereΓ(x)is the ordinary gamma function. In [2], R. Askey obtained aq-analogue of many of the classical facts about the gamma function.

In his interesting paper [6], J. Sándor defined the functionsS andSby S(x) = min{m∈N :x≤m!}, x∈(1,∞), and

S(x) = max{m∈N :m!≤x}, x∈[1,∞).

He has studied many important properties ofSand proved the following theorems:

Theorem 1.1.

S(x)∼ logx

log logx (x→ ∞).

Theorem 1.2. The series

X

n=1

1 n(S(n))α is convergent forα >1and divergent forα≤1.

In [1], C. Adiga and T. Kim have obtained a generalization of Theorems 1.1 and 1.2.

We now define theq-analogues ofSandS as follows:

Sq(x) = min{m ∈N :x≤Γq(m+ 1)}, x∈(1,∞), and

Sq(x) = max{m∈N : Γq(m+ 1)≤x}, x∈[1,∞), where0< q <1.

ClearlySq(x)→S(x)andSq(x)→S(x)asq→1.

In Section 2 of this paper we study some properties ofSqandSq, which are similar to those of SandSstudied by Sándor [6]. In Section 3 we prove two theorems which are theq-analogues of Theorems 1.1 and 1.2 of Sándor [6].

To prove our main theorems we make use of the followingq-analogue of Stirling’s formula established by D.S. Moak [5]:

(1.1) log Γq(z)∼

z− 1 2

log

qz−1 q−1

+ 1

logq

Z −zlognq

logq

udu eu −1 +Cq+

X

k=1

B2k (2k)!

logq qz −1

2k−1

qz P2k−1(qz),

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whereCqis a constant depending uponq, andPn(z)is a polynomial of degreensatisfying, Pn(z) = (z−z2)Pn−10 (z) + (nz+ 1)Pn−1(z), P0 = 1, n ≥1.

2. SOMEPROPERTIES OFSq ANDSq From the definitions ofSqandSq, it is clear that

(2.1) Sq(x) =m ifx∈(Γq(m),Γq(m+ 1)], form≥2, and

(2.2) Sq(x) = m ifx∈[Γq(m+ 1),Γq(m+ 2)), form≥1.

(2.1) and (2.2) imply Sq(x) =

Sq(x) + 1, ifx∈(Γq(k+ 1),Γq(k+ 2)), Sq(x), ifx= Γq(k+ 2).

Thus

Sq(x)≤Sq(x)≤Sq(x) + 1.

Hence it suffices to study the functionSq. The following are the simple properties ofSq. (1) Sq is surjective and monotonically increasing.

(2) Sq is continuous for allx∈[1,∞)\A, whereA={Γq(k+ 1) :k ≥2}.Since

x→Γlimq(k+1)+Sq(x) =k and lim

x→Γq(k+1)Sq(x) = (k−1), (k ≥2),

Sq is continuous from the right atx = Γq(k+ 1), k ≥ 2,but it is not continuous from the left.

(3) Sq is differentiable on(1,∞)\Aand since lim

x→Γq(k+1)+

Sq(x)−Sqq(k+ 1)) x−Γq(k+ 1) = 0 fork≥1,it has a right derivative inA∪ {1}.

(4) Sq is Riemann integrable on[a, b],whereΓq(k+ 1)≤a < b, k ≥1.

(i) If[a, b]⊂[Γq(k+ 1),Γq(k+ 2)], k ≥1,then Z b

a

Sq(x)dx = Z b

a

kdx=k(b−a).

(ii) Forn > k, we have Z Γq(n+1)

Γq(k+1)

Sq(x)dx=

(n−k)

X

m=1

Z Γq(k+m+1) Γq(k+m)

Sq(x)dx

=

(n−k)

X

m=1

(k+m−1)[Γq(k+m+ 1)−Γq(k+m)]

=

(n−k)

X

m=1

(k+m−1)Γq(k+m)[q+q2+· · ·+qk+m−1].

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(iii) Ifa∈[Γq(k+ 1),Γq(k+ 2))andb∈[Γq(n),Γq(n+ 1))then Z b

a

Sq(x)dx=

Z Γq(k+2) a

Sq(x)dx+

Z Γq(n) Γq(k+2)

Sq(x)dx+ Z b

Γq(n)

Sq(x)dx

=k[Γq(k+ 2)−a] +

n−k−2

X

m=1

(k+m)Γq(k+m+ 1)

×(q+q2+...+qk+m) + (n−1)[b−Γq(n)], by (ii).

3. MAINTHEOREMS

We now prove our main theorems.

Theorem 3.1. If0< q <1, then

Sq(x)∼ logx log

1 1−q

.

Proof. IfΓq(n+ 1) ≤x <Γq(n+ 2), then

(3.1) log Γq(n+ 1)≤logx <log Γq(n+ 2).

By (1.1) we have

log Γq(n+ 1)∼

n+ 1 2

(3.2)

log

qn+1−1 q−1

∼nlog 1

1−q

.

Dividing (3.1) throughout bynlog

1 1−q

, we obtain

(3.3) log Γq(n+ 1)

nlog 1

1−q

≤ logx Sq(x) log

1 1−q

< log Γq(n+ 2) nlog

1 1−q

. Using (3.2) in (3.3) we deduce

n→∞lim

logx Sq(x) log

1 1−q

= 1.

This completes the proof.

Theorem 3.2. The series (3.4)

X

n=1

1 n(Sq(n))α is convergent forα >1and divergent forα≤1.

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Proof. Since

Sq(x)∼ logx log

1 1−q

, we have

A logn log

1 1−q

< Sq(n)< B logn log

1 1−q

,

for alln ≥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(logn)α . By the integral test,P 1

n(logn)α converges forα >1and diverges for0≤α≤1.Ifα <0,then

1

n(logn)α > 1n forn ≥3.HenceP 1

(nlogn)α diverges by the comparison test.

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, Theq-gamma andq-beta functions, Applicable Analysis, 8 (1978), 125–141.

[3] T. KIM, Non-archimedeanq-integrals associated with multiple Changheeq-Bernoulli polynomials, Russian J. Math. Phys., 10 (2003), 91–98.

[4] T. KIM, An anotherp-adicq-L-functions and sums of powers, Proc. Jangjeon Math. Soc., 2 (2001), 35–43.

[5] D.S. MOAK, Theq-analogue of Stirlings formula, Rocky Mountain J. Math., 14 (1984), 403–413.

[6] J. SÁNDOR, On an additive analogue of the functionS, Notes Numb. Th. Discr. Math., 7 (2001), 91–95.

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