The additive analogues of Pseudo-Smarandache, Smarandache-simple functions
and their duals have been recently studied by J. Sandor. In this note, we obtain q-analogues of Sandor's theorems

Taekyun, Kim

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Scientia MagnaVol. 2 (2006), No. 1, 30-34A note on q-nanlogue of Sa´ndor’s functionsTaekyun KimDepartment of Mathematics Education Kongju National UniversityKongju 314- 701 , South KoreaC. Adiga and Jung Hun HanDepartment of Studies in Mathematics University of MysoreManasagangotri Mysore 570006, IndiaAbstract The additive analogues of Pseudo-Smarandache, Smarandache-simple functionsand their duals have been recently studied by J. Sa´ndor. In this note, we obtain q-analoguesof Sa´ndor’s theorems [6].Keywords q-gamma function; Pseudo-Smarandache function; Smarandache-simple func-tion; Asymtotic formula.Dedicated to Sun-Yi Park on 90th birthday§1. IntroductionThe additive analogues of Smarandache functions S and S∗ have been introduced by Sa´ndor[5] as follows:S(x) = min{m ∈ N : x ≤ m!}, x ∈ (1,∞),andS∗(x) = max{m ∈ N : m! ≤ x}, x ∈ [1,∞),He has studied many important properties of S∗ relating to continuity, differentiability andRiemann integrability and also p roved the following theorems:Theorem 1.1.S∗ ∼ log xlog log x (x→∞).Theorem 1.2. The series∞∑n=11n(S∗(n))α,is convergent for α > 1 and divergent for α ≤ 1.Vol. 2 A note on q-nanlogue of Sa´ndor’s functions 31In [1], Adiga and Kim have obtained generalizations of Theorems 1.1 and 1.2 by the use ofEuler’s gamma function. Recently Adiga-Kim-Somashekara-Fathima [2] have established a q-analogues of these results on employing analogues of Pseudo-Smarandache, Smarandache-simplefunctions and their duals as follows: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,∞),P∗(x) = max{m ∈ N : m! ≤ px}, p > 1, x ∈ [1,∞).He has also proved the following theorems:Theorem 1.3.Z∗ ∼ 12√8x+ 1 (x→∞).Theorem 1.4. The series ∞∑n=11(Z∗(n))α,is convergent for α > 2 and divergent for α ≤ 2. The series∞∑n=11n(Z∗(n))α,is convergent for all α > 0.Theorem 1.5.logP∗(x) ∼ log x (x→∞),Theorem 1.6. The series ∞∑n=11n(log log nlogP∗(n))αis convergent for all α > 1 and divergent for α ≤ 1.The main purpose of this note is to obtain q-analogues of Sa´ndor’s Theorems 1.3 and 1.5.In what follows, we make u se of the following notations and definitions. F. H. Jackson defineda q-analogues 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 the gammafunction asΓq(x) =(q; q)∞(qx; q)∞(1− q)1−x, 0 < q < 1,32 T. Kim, C. Adiga and J. H. Han No. 1andΓq(x) =(q−1; q−1)∞(q−x; q−1)∞(1− q)1−xq(x2), 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 TheoremsWe now defined the q-analogues of Z and Z∗ as follows:Zq(x) = min{1− qm1− q : x ≤Γq(m+ 2)2Γq(m)}, m ∈ N, x ∈ (0,∞),andZ∗q (x) = max{1− qm1− q :Γq(m+ 2)2Γq(m)≤ x}, m ∈ N, x ∈[Γq(m+ 2)2Γq(1),∞),where 0 < q < 1. Clearly, Zq(x) → Z(x) and Z∗q (x) → Z∗(x) as q → 1−. From the definitionsof Zq and Z∗q , it is clear thatZq(x) = 1, if x ∈(0, Γq(3)2Γq(1)]1−qm1−q , ifx ∈(Γq(m+1)2Γq(m−1) ,Γq(m+2)2Γq(m)],m ≥ 2, (1)andZ∗q =1− qm1− q if x ∈[Γq(m+ 2)2Γq(m),Γq(m+ 3)2Γq(m+ 1)). (2)Since1− qm−11− q ≤1− qm1− q =1− qm−11− q + qm−1 ≤ 1− qm−11− q + 1,(1) and (2) imply that for x > Γq(3)2Γq(1) ,Z∗q ≤ Zq ≤ Z∗q + 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 ≤√1 + 8xq − 12q, x ≥ Γq(3)2Γq(1).Proof. IfΓq(k + 2)2Γq(k)≤ x < Γq(k + 3)2Γq(k + 1), (3)Vol. 2 A note on q-nanlogue of Sa´ndor’s functions 33thenZ∗q =1− qk1− qand(1− qk)(1− qk+1)− 2x(1− q)2 ≤ 0 < (1− qk+1)(1− qk+2)− 2x(1− q)2. (4)Consider the functions f and g defined byf(y) = (1− y)(1− yq)− 2x(1− q)2andg(y) = (1− yq)(1− yq2)− 2x(1− q)2.Note that f is monotonically decreasing for y ≤ 1+q2q and g is strictly decreasing for y ≤ 1+q2q2 .Also f(y1) = 0 = g(y2) wherey1 =(1 + q)− (1− q)√1 + 8xq2q,y2 =(q + q2)− q(1− q)√1 + 8xq2q3.Since y1 ≤ 1+q2q , y2 ≤ 1+q2q2 and qk < 1+q2q < 1+q2q2 , from (4), it follows thatf(qk) ≤ f(y1) = 0 = g(y2) < g(qk).Thus y1 < qk < y2 and hence1− y21− q <1− qk1− q <1− y11− q .i.e. √1 + 8xq − (1 + 2q)2q2< Z∗q ≤√1 + 8xq − 12q.This completes the proof.Remark. Letting q − 1− in the above theorem, we obtain Sa´ndor’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,∞),andP ∗q (x) = max{m ∈ N : Γq(m+ 1) ≤ px}, p > 1, x ∈ [1,∞),where 0 < q < 1. Clearly, Pq(x) → P (x) and P ∗q → P∗(x) as q → 1−. From the definitions ofPq and P ∗q , we haveP ∗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, thenP∗(x) ∼ x log plog(11−q) (x→∞).34 T. Kim, C. Adiga and J. H. Han No. 1Proof. If Γq(n+ 1) ≤ px < Γq(n+ 2), thenP ∗q (x) = nandlog Γq(n+ 1) ≤ log px < log Γq(n+ 2). (5)But by the q-analogue of Stirling’s formula established by Moak [4], we havelog Γq(n+ 1) ∼(n+12)log(qn+1q − 1)∼ n log(11− q). (6)Dividing (5) throughout by n log(11−q), we obtainlog Γq(n+ 1)n log(11−q) ≤ x log pP ∗q (x) log(11−q) < log Γq(n+ 2)n log(11−q) . (7)Using (6) and (7), we deducelimx→∞x log pP ∗q (x) log(11−q) = 1.This completes the proof.References[1] C. Adiga and T. Kim, On a generalization of Sa´ndor’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 Sa´ndor’sfunction, J. Inequal. Pure and Appl. Math., 4(2003), 1-5.[3] T. Kim, Non-archimedean q-integrals 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. Sa´ndor, On an additive analogue of the function S, Note Numb. Th. Discr. Math.,7(2001), 91-95.[6] J. Sa´ndor, On an additive analogue of certain arithmetic function, J. SmarandacheNotions, 14(2004), 128-133.

A note on q-nanlogue of Sandor's functions

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