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volume 4, issue 5, article 84, 2003.

Received 19 September, 2003;

accepted 29 September, 2003.

Communicated by:J. Sándor

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Journal of Inequalities in Pure and Applied Mathematics

ON AQ-ANALOGUE OF SÁNDOR’S FUNCTION

1C. ADIGA, ²T. KIM, D. D. SOMASHEKARA AND SYEDA NOOR FATHIMA

1Department of Studies in Mathematics Manasa Gangothri, University of Mysore Mysore-570 006, INDIA.

2Institute of Science Education Kongju National University, Kongju 314-701

S. KOREA.

EMail:tkim@kongju.ac.kr

c

2000Victoria University ISSN (electronic): 1443-5756 132-03

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On aq-Analogue of Sándor’s Function

C. Adiga, T. Kim, D. D.

Somashekara and Syeda Noor Fathima

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J. Ineq. Pure and Appl. Math. 4(5) Art. 84, 2003

Abstract

In this paper we obtain aq-analogue of J. Sándor’s theorems [6], on employing theq-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.

This paper was supported by Korea Research Foundation Grant (KRF-2002-050- C00001).

Dedicated to Professor Katsumi Shiratani on the occasion of his 71^stbirthday

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+q²)· · ·(1 +q+...+qⁿ⁻¹), cf. [3,4],

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

Γ_q(x) = (q;q)∞

(q^x;q)∞

(1−q)^1−x, 0< q <1, 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 that Γ_q(x) → Γ(x) as q → 1, where Γ(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 functionsSandS_∗by S(x) = min{m∈N :x≤m!}, x∈(1,∞),

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)∼ 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 Theorems1.1 and1.2.

We now define theq-analogues ofS andS∗as follows:

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

S_q^∗(x) = max{m ∈N : Γ_q(m+ 1) ≤x}, x∈[1,∞), where0< q <1.

ClearlyS_q(x)→S(x)andS_q^∗(x)→S∗(x)asq→1⁻.

In Section2of this paper we study some properties ofS_qandS_q^∗, which are similar to those ofS andS∗ studied by Sándor [6]. In Section3we prove two theorems which are theq-analogues of Theorems1.1and1.2of 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 Γ_q(z)∼

z− 1 2

log

q^z −1 q−1

+ 1

logq

Z −zlogⁿq

−logq

udu e^u−1 +Cq+

∞

X

k=1

B_2k (2k)!

logq q^z−1

2k−1

q^z P2k−1(q^z),

whereC_qis a constant depending uponq, and P_n(z)is a polynomial of degree nsatisfying,

Pn(z) = (z−z²)P_n−1⁰ (z) + (nz+ 1)Pn−1(z), P0 = 1, n≥1.

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2. Some Properties of S

_q

and S

_q^∗

From the definitions ofS_qandS_q^∗, it is clear that

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

(2.2) S_q^∗(x) = m ifx∈[Γ_q(m+ 1),Γ_q(m+ 2)), form≥1.

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







S_q^∗(x) + 1, ifx∈(Γ_q(k+ 1),Γ_q(k+ 2)), S_q^∗(x), ifx= Γ_q(k+ 2).

Thus

S_q^∗(x)≤S_q(x)≤S_q^∗(x) + 1.

Hence it suffices to study the functionS_q^∗. The following are the simple properties ofS_q^∗.

(1) S_q^∗is surjective and monotonically increasing.

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

Since

x→Γlim_q(k+1)⁺S_q^∗(x) = k and lim

x→Γ_q(k+1)⁻S_q^∗(x) = (k−1), (k ≥2), S_q^∗ is continuous from the right at x = Γ_q(k + 1), k ≥ 2, but it is not continuous from the left.

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(3) S_q^∗is differentiable on(1,∞)\Aand since lim

x→Γq(k+1)⁺

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

(4) S_q^∗is Riemann integrable on[a, b],whereΓ_q(k+ 1)≤a < b, k ≥1.

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

a

S_q^∗(x)dx= Z b

a

kdx=k(b−a).

(ii) Forn > k, we have Z Γq(n+1)

Γq(k+1)

S_q^∗(x)dx

=

(n−k)

X

m=1

Z Γq(k+m+1) Γq(k+m)

S_q^∗(x)dx

=

(n−k)

X

m=1

(k+m−1)[Γ_q(k+m+ 1)−Γ_q(k+m)]

=

(n−k)

X

m=1

(k+m−1)Γ_q(k+m)[q+q²+· · ·+q^k+m−1].

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

a

S_q^∗(x)dx

=

Z Γq(k+2) a

S_q^∗(x)dx+

Z Γq(n) Γq(k+2)

S_q^∗(x)dx+ Z b

Γq(n)

S_q^∗(x)dx

=k[Γ_q(k+ 2)−a] +

n−k−2

X

m=1

(k+m)Γ_q(k+m+ 1)

×(q+q²+...+q^k+m) + (n−1)[b−Γ_q(n)], by (ii).

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3. Main Theorems

We now prove our main theorems.

Theorem 3.1. If0< q <1, then

S_q^∗(x)∼ logx log

1 1−q

. Proof. IfΓ_q(n+ 1)≤x <Γ_q(n+ 2), then

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

By (1.1) we have

log Γ_q(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 S_q^∗(x) log

1 1−q

< log Γq(n+ 2) nlog

1 1−q

. Using (3.2) in (3.3) we deduce

n→∞lim

logx S_q^∗(x) log

1 1−q

= 1.

This completes the proof.

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Theorem 3.2. The series (3.4)

∞

X

n=1

1 n(S_q^∗(n))^α is convergent forα >1and divergent forα≤1.

Proof. Since

S_q^∗(x)∼ logx log

1 1−q

, we have

A logn log

1 1−q

< S_q^∗(n)< B logn log

1 1−q

,

for alln ≥ N > 1, A, B > 0.Therefore to examine the convergence or diver- gence 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.

If α < 0, then _n(log¹_n)α > _n¹ for n ≥ 3. Hence P ₁

(nlogn)^α diverges by the comparison test.

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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 anotherp-adicq-L-functions and sums of powers, Proc. Jang- jeon 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.