In this paper, we consider the q-extensions of Boole polynomials. From those
polynomials, we derive some new and interesting properties and identities
related to special polynomials.Comment: 11 page

Kim, Dae San

Kim, Taekyun

Seo, Jong Jin

English

arXiv

In this paper, we consider the q-extensions of Boole polynomials. From those polynomials, we derive some new and interesting properties and identities related to special polynomials

San Kim, Dae

Jin Seo, Jong

Arab Journals Platform

Appl. Math. Inf. Sci.9, No. 6, 3153-3158 (2015) 3153Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.12785/amis/090644A Note on q-Analogue of Boole PolynomialsDae San Kim1, Taekyun Kim2,∗ and Jong Jin Seo31 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea3 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of KoreaReceived: 8 Mar. 2015, Revised: 6 May 2015, Accepted: 7 May 2015Published online: 1 Nov. 2015Abstract: In this paper, we consider theq-extensions of Boole polynomials. From those polynomials,we derive some new andinteresting properties and identities related to special polynomials.Keywords: q-Boole number,q-Boole polynomial,q-Euler number,q-Euler polynomial1 IntroductionLet p be a fixed odd prime number. Throughout thispaper,Zp,Qp and Cp will denote the ring of p-adicintegers, the field ofp-adic numbers and the completionof algebraic closure ofQp. The p-adic norm | · |p isnormalized as|p|p = 1/p. The space of continuousfunctions onZp is denoted byC(Zp). Let q be anindeterminate inCp with |1 − q|p < p−1/p−1. Theq-number of x is defined by [x]q =1−qx1−q . Note thatlimq→1 [x]q = x. For f ∈ C(Zp), the fermionic p-adicq-integral onZp is defined by Kim to beI−q( f ) =∫Zpf (x)dµ−q(x) = limN→∞1[pN]−qpN−1∑x=0f (x)(−1)x,where[x]−q =1− (−q)x1+q(see[1−9]).(1.1)From (1.1), we note thatqnI−q( fn)+ (−1)n−1I−q( f ) = [2]qn−1∑l=0(−1)n−1−lql f (l),where fn(x) = f (x+n),(n≥ 1) (see[4]).(1.2)In particular, forn=1,qI−q( f1)+ I−q( f ) = [2]q f (0). (1.3)As is well known, the Boole polynomials are defined bythe generating function to be∞∑n=0Bln(x|λ )tnn!=11+(1+ t)λ(1+ t)x, (see[2,12]).(1.4)When λ = 1,2Bln(x|1) = Chn(x) are Changheepolynomials which are defined by2t +2(1+ t)x =∞∑n=0Chn(x)tnn!(see[2,3,13,14]). (1.5)The Euler polynomials of orderα are defined by thegenerating function to be(2et +1)αext =∞∑n=0E(α)n (x)tnn!, (see[2,11]). (1.6)Whenx= 0,E(α)n = E(α)n (0) are called the Euler numbersof orderα.In particular, for α = 1,En(x) = E(1)n (x) are called theordinary Euler polynomials.The Stirling number of the first kind is given by thegenerating function to belog(1+ t)m= m!∞∑l=mS1(l ,m)t ll !,(m≥ 0), (1.7)and the Stirling number of the second kind is defined bythe generating function to be(et −1)m = m!∞∑l=mS2(l ,m)t ll !, (see[11,12]). (1.8)∗ Corresponding author e-mail:tkkim@kw.ac.krc© 2015 NSPNatural Sciences Publishing Cor.3154 D. S. Kim et. al. : A Note onq-Analogue of Boole PolynomialsIn this paper, we consider theq-extensions of Boolepolynomials. From those polynomials, we derive new andinteresting properties and identities related to specialpolynomials.2 q-analogue of Boole polynomialsIn this section, we assume thatt ∈Cp with |t|p < p−1p−1 andλ ∈ Zp with λ 6= 0. From (1.3), we note that∫Zp(1+ t)x+λ ydµ−q(y) =1+q1+q(1+ t)λ(1+ t)x=∞∑n=0[2]qBln,q(x|λ )tnn!,(2.1)whereBln,q(x|λ ) are theq-Boole polynomials which aredefined by11+q(1+ t)λ(1+ t)x =∞∑n=0Bln,q(x|λ )tnn!. (2.2)From (2.1), we can derive the following equation :∫Zp(x+λyn)dµ−q(y) =[2]qn!Bln,q(x|λ ). (2.3)Whenx= 0,Bln,q(λ ) = Bln,q(0|λ ) are called theq-Boolenumbers.Now, we observe that(1+ t)x+λ y = e(x+λ y) log(1+t)=∞∑m=0(x+λy)mm!(log(1+ t))m=∞∑m=0(x+λy)mm!m!∞∑n=mS1(n,m)tnn!=∞∑n=0{n∑m=0(x+λy)mS1(n,m)}tnn!.(2.4)The q-Euler polynomials are defined by the generatingfunction to be[2]qqet +1ext =∞∑n=0En,q(x)tnn!. (2.5)Note that limq→1En,q(x) = En(x).Whenx= 0,En,q =En,q(0) are called theq-Euler numbers.By (1.3), we easily get∫Zpe(x+y)tdµ−q(y) =[2]qqet +1ext=∞∑n=0En,q(x)tnn!.(2.6)Thus, by (2.6), we get∫Zp(x+ y)ndµ−q(y) = En,q(x),(n≥ 0). (2.7)From (2.1), (2.4) and (2.7), we have∫Zp(1+ t)x+λ ydµ−q(y)=∞∑n=0{n∑m=0∫Zp(x+λy)mdµ−q(y)S1(n,m)}tnn!=∞∑n=0{n∑m=0λ mEm,q( xλ)S1(n,m)}tnn!.(2.8)Therefore, by (2.1), (2.3) and (2.8), we obtain thefollowing theorem.Theorem 1For n≥ 0, we haveBln,q(x|λ ) =1[2]qn∑m=0λ mEm,q( xλ)S1(n,m),and∫Zp(x+λyn)dµ−q(y) =[2]qn!Bln,q(x|λ ).From (2.3), we note thatBln,q(x|λ ) =1[2]q∫Zp(x+λy)ndµ−q(y).Whenλ = 1, we haveBln,q(x|1) =1[2]q∫Zp(x+ y)ndµ−q(y). (2.9)As is known,q-Changhee polynomials are defined by thegenerating function to be[2]q[2]q+qt(1+ t)x =∞∑n=0Chn,q(x)tnn!. (2.10)Thus, by (2.10), we get∫Zp(1+t)x+ydµ−q(y)=[2]q[2]q+qt(1+t)x=∞∑n=0Chn,q(x)tnn!.(2.11)From (2.11), we have∫Zp(x+ y)ndµ−q(y) =Chn,q(x),where(x)n = x(x−1) · · ·(x−n+1).(2.12)By (2.9) and (2.12), we getBln,q(x|1) =1[2]qChn,q(x). (2.13)c© 2015 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.9, No. 6, 3153-3158 (2015) /www.naturalspublishing.com/Journals.asp 3155By replacingt by et −1 in (2.2), we see that[2]qqeλ t +1ext = [2]q∞∑n=0Bln,q(x|λ )1n!(et −1)n= [2]q∞∑n=0Bln,q(x|λ )∞∑m=nS2(m,n)tmm!=∞∑m=0m∑n=0[2]qBln,q(x|λ )S2(m,n)tmm!,(2.14)and[2]qqeλ t +1ext =[2]qqeλ t +1e(xλ )λ t=∞∑m=0Em,q( xλ)λ mtmm!.(2.15)Therefore, by (2.14) and (2.15), we obtain the followingtheorem.Theorem 2For m≥ 0, we havem∑n=0Bln,q(x|λ )S2(m,n) =1[2]qEm,q( xλ)λ m.Let us define theq-Boole numbers of the first kind withorderk(∈ N) as follows :[2]kqBl(k)n,q(λ )=∫Zp· · ·∫Zp(λ (x1+ · · ·+ xk))ndµ−q(x1) · · ·dµ−q(xk),(n≥ 0).(2.16)Thus, by (2.16), we see that[2]kq∞∑n=0Bl(k)n,q(λ )tnn!=∫Zp· · ·∫Zp∞∑n=0(λ (x1+ · · ·+ xk)n)tndµ−q(x1) · · ·dµ−q(xk)=∫Zp· · ·∫Zp(1+ t)λ (x1+···+xk)dµ−q(x1) · · ·dµ−q(xk)=(1+q1+q(1+ t)λ)k=[2]kq∞∑n=0(∑l1+···+lk=n(nl1, · · · , lk)Bll1,q · · ·Bllk,q)tnn!.(2.17)Therefore, by (2.17), we obtain the following corollary.Corollary 3For n≥ 0, we haveBl(k)n,q = ∑l1+···+lk=n(nl1, · · · , lk)Bll1,q · · ·Bllk,q.Theq-Euler polynomials of orderk are defined by thegenerating function to be∫Zp· · ·∫Zpe(x1+···+xk+x)tdµ−q(x1) · · ·dµ−q(xk)=([2]qqet +1)kext =∞∑n=0E(k)n,q(x)tnn!.(2.18)Thus, by (2.18), we get∫Zp· · ·∫Zp(x1+ · · ·+ xk+ x)ndµ−q(x1) · · ·dµ−q(xk) = E(k)n,q(x).Whenx= 0,E(k)n,q = E(k)n,q(0) are called theq-Euler numbersof orderk.From (2.16), we note that[2]kqBl(k)n,q(λ )=∫Zp· · ·∫Zp(λ (x1+ · · ·+ xk))ndµ−q(x1) · · ·dµ−q(xk)=n∑l=0S1(n, l)∫Zp· · ·∫Zpλ l (x1+ · · ·+ xk)l dµ−q(x1) · · ·dµ−q(xk)=n∑l=0S1(n, l)λ l E(k)l ,q .(2.19)Therefore, by (2.19), we obtain the following theorem.Theorem 4For n≥ 0, we haveBl(k)n,q(λ ) =1[2]kqn∑l=0S1(n, l)λ l E(k)l ,q .By replacingt by et −1 in (2.17), we get[2]kq∞∑n=0Bl(k)n,q(λ )1n!(et −1)n =([2]qqeλ t +1)k=∞∑m=0E(k)m,qλ mtmm!,(2.20)and[2]kq∞∑n=0Bl(k)n,q(λ )1n!(et −1)n = [2]kq∞∑n=0Bl(k)n,q(λ )∞∑m=nS2(m,n)tmm!= [2]kq∞∑m=0{m∑n=0Bl(k)n,q(λ )S2(m,n)}tmm!.(2.21)Therefore, by (2.20) and (2.21), we obtain the followingtheorem.Theorem 5For m≥ 0, we havem∑n=0Bl(k)n,q(λ )S2(m,n) =1[2]kqE(k)m,qλ m.Let us define the higher-orderq-Boole polynomials ofthe first kind asfollows :[2]kqBl(k)n,q(x|λ )=∫Zp· · ·∫Zp(λx1+ · · ·+λxk+ x)ndµ−q(x1) · · ·dµ−q(xk),wheren≥ 0 andk∈ N.(2.22)c© 2015 NSPNatural Sciences Publishing Cor.3156 D. S. Kim et. al. : A Note onq-Analogue of Boole PolynomialsFrom (2.22), we can derive the generating function of thehigher-orderq-Boole polynomials of the first kind asfollows :[2]kq∞∑n=0Bl(k)n,q(x|λ )tnn!=∫Zp· · ·∫Zp(1+ t)λ x1+···+λ xk+xdµ−q(x1) · · ·dµ−q(xk)=([2]q1+q(1+ t)λ)k(1+ t)x(2.23)By (2.17), we easily get([2]q1+q(1+ t)λ)k(1+ t)x=[2]kq(∞∑l=0Bl(k)l ,q (λ )t ll !)(∞∑m=0m!(xm)tmm!)=[2]kq∞∑n=0(n∑m=0m!(xm)n!m!(n−m)!Bl(k)n−m,q(λ ))tnn!=[2]kq∞∑n=0(n∑m=0m!(xm)(nm)Bl(k)n−m,q(λ ))tnn!.(2.24)Therefore, by (2.23) and (2.24), we obtain the followingtheorem.Theorem 6For n≥ 0, we haveBl(k)n,q(x|λ ) =n∑m=0(nm)Bl(k)n−m,q(λ )(x)m.Replacingt by et −1 in (2.23), we have[2]kq∞∑n=0Bl(k)n,q(x|λ )(et −1)nn!=([2]q1+qeλ t)kext=∞∑m=0E(k)m,q( xλ)λ mtmm!,(2.25)and[2]kq∞∑n=0Bl(k)n,q(x|λ )(et −1)nn!=[2]kq∞∑m=0(m∑n=0Bl(k)n,q(x|λ )S2(m,n))tmm!.(2.26)Thus, from (2.25) and (2.26), we have the followingtheorem.Theorem 7For m≥ 0 and k∈N, we havem∑n=0Bl(k)n,q(x|λ )S2(m,n) =1[2]kqλ mE(k)m,q( xλ).From (2.22), we have[2]kqBl(k)n,q(x|λ )=∫Zp· · ·∫Zp(λx1+ · · ·+λxk+ x)ndµ−q(x1) · · ·dµ−q(xk)=n∑l=0S1(n, l)×∫Zp· · ·∫Zp(λx1+ · · ·+λxk+ x)l dµ−q(x1) · · ·dµ−q(xk)=n∑l=0S1(n, l)λ l E(k)l ,q( xλ).(2.27)Therefore, by (2.27), we obtain the following theorem.Theorem 8For n≥ 0, k∈N, we haveBl(k)n,q(x|λ ) =1[2]kqn∑l=0S1(n, l)λ l E(k)l ,q( xλ).Now, we consider the q-analogue of Boolepolynomials of the second kind as follows :B̂ln,q(x|λ ) =1[2]q∫Zp(−λy+ x)ndµ−q(y),(n≥ 0).(2.28)Thus, by (2.28), we getB̂ln,q(x|λ ) =1[2]qn∑l=0S1(n, l)(−1)l λ l∫Zp(−xλ+ y)ldµ−q(y)=1[2]qn∑l=0S1(n, l)(−1)l λ l El ,q(−xλ).(2.29)Whenx= 0, B̂ln,q(λ ) = B̂ln,q(0|λ ) are called theq-Boolenumbers of the second kind. From (2.28), we can derivethe generating function of̂Bln,q(x|λ ) as follows:∞∑n=0B̂ln,q(x|λ )tnn!=1[2]q∫Zp(1+ t)−λ y+xdµ−q(y)=(1+ t)λq+(1+ t)λ(1+ t)x.(2.30)By replacingt by et −1 in (2.30), we get∞∑n=0B̂ln,q(x|λ )(et −1)nn!=eλ tq+eλ text=1qe−λ t +1ext=1[2]q∞∑m=0(−1)mλ mEm,q(−xλ)tmm!,(2.31)and∞∑n=0B̂ln,q(x|λ )(et −1)nn!=∞∑m=0(m∑n=0B̂ln,q(x|λ )S2(m,n))tmm!.(2.32)c© 2015 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.9, No. 6, 3153-3158 (2015) /www.naturalspublishing.com/Journals.asp 3157Therefore, by (2.31) and (2.32), we obtain the followingtheorem.Theorem 9For m≥ 0, we have(−1)mλ m[2]qEm,q(−xλ) =m∑n=0B̂ln,q(x|λ )S2(m,n),andB̂lm,q(x|λ ) =1[2]qm∑l=0S1(m, l)(−1)l λ l El ,q(−xλ).Fork∈N, let us define theq-Boole polynomials of thesecond kind with orderk as follows :B̂l(k)n,q(x|λ )=1[2]kq∫Zp· · ·∫Zp(−(λx1+ · · ·+λxk)+ x)n×dµ−q(x1) · · ·dµ−q(xk).(2.33)Then we have[2]kqB̂l(k)n,q(x|λ ) =n∑l=0S1(n, l)λ l (−1)l El ,q(−xλ).From (2.33), we can derive the generating function ofB̂l(k)n,q(x|λ ) as follows :∞∑n=0B̂l(k)n,q(x|λ )tnn!=1[2]kq∫Zp· · ·∫Zp(1+ t)−(λ x1+···+λ xk)+xdµ−q(x1) · · ·dµ−q(xk)=((1+ t)λq+(1+ t)λ)k(1+ t)x=(1q(1+ t)−λ +1)k(1+ t)x=∞∑n=0Bl(k)n,q(x|−λ )tnn!.(2.34)Thus, by (2.34), we getB̂l(k)n,q(x|λ ) = Bl(k)n,q(x|−λ ),(n≥ 0). (2.35)Indeed,(−1)n[2]qBln,q(x|λ )n!= (−1)n∫Zp(x+λyn)dµ−q(y)=∫Zp(−yλ − x+n−1n)dµ−q(y)=n∑m=0(n−1n−m)∫Zp(−yλ − xm)dµ−q(y)=n∑m=1(n−1m−1)m!m!∫Zp(−yλ − xm)dµ−q(y)= [2]qn∑m=1(n−1m−1)B̂lm,q(−x|λ )m!,and(−1)n[2]qB̂ln,q(x|λ )n!=n∑m=0(n−1m−1)∫Zp(−x+ yλm)dµ−q(y)= [2]qn∑m=1(n−1m−1)B̂lm,q(−x|λ )m!.References[1] A. Bayad, T. Kim, Identities involving values of Bernstein,q-Bernoulli, and q-Euler polynomials,Russ. J. Math. Phys.18 (2011), no. 2, 133-143.[2] D. S. Kim, T. Kim, Integral Transforms Spec. Funct.25(2014), no. 8, 627-633.[3] D. S. Kim, T. Kim, J. J. Seo, A Note on ChangheePolynomials and Numbers,Adv. Studies Theor. Phys.7(2013), no. 20, 993–1003.[4] T. Kim, Identities on the weighted q-Euler numbersand q-Bernstein polynomials,Adv. Stud. Contemp. Math.(Kyungshang)22 (2012), no. 1, 7-12.[5] T. Kim, A study on the q-Euler numbers and thefermionic q-integral of the product of several type q-Bernstein polynomials onZp, Adv. Stud. Contemp. Math.(Kyungshang)23 (2013), no. 1, 5-11.[6] T. Kim, J. Choi, Y.-H. Kim, On extended Carlitz’s typeq-Euler numbers and polynomials,Adv. Stud. Contemp.Math. (Kyungshang)20 (2010), no. 4, 499-505.[7] T. Kim, A note on p-adic q-integral onZp associated withq-Euler numbers,Adv. Stud. Contemp. Math. (Kyungshang)15 (2007), no. 2, 133-137.[8] T. Kim, New approach to q-Euler, Genocchi numbers andtheir interpolation functions,Adv. Stud. Contemp. Math.(Kyungshang)18 (2009), no. 2, 105-112.[9] T. Kim, Note on the q-Euler numbers of higher order,Adv.Stud. Contemp. Math. (Kyungshang)19 (2009), no. 1, 25-29.[10] T. Kim, B. Lee, J. Choi, Y. H. Kim, S. H. Rim,On the q-Euler numbers and weighted q-Bernstein polynomials,Adv.Stud. Contemp. Math. (Kyungshang)21 (2011), no. 1, 13-18.[11] T. Kim, The modified q-Euler numbers and polynomials,Adv. Stud. Contemp. Math. (Kyungshang)(16) (2008), no.2, 161-170.[12] S. Roman, The umbral calculus. Pure and Appliedmathematics,Vol 111. Academic Press, Inc.[ HarcourtBrace Jovanovich, publishers], New York;1984. x+193 pp.ISBN:0-12-594380-6.[13] C. S. Ryoo, T. Kim, R. P. Agarwal,Exploring the multipleChanghee q-Bernoulli polynomials,Int. J. Comput. Math.82 (2005), no. 4, 483–493.[14] Y. Simsek, I. S. Pyung,Barnes’ type multiple Changhee q-zeta functions,Adv. Stud. Contemp. Math. (Kyungshang)10 (2005), no. 2, 121–129.c© 2015 NSPNatural Sciences Publishing Cor.3158 D. S. Kim et. al. : A Note onq-Analogue of Boole PolynomialsDae San Kim receivedhis BS and MS degreesin mathematics from SeoulNational University, Seoul,Korea, in 1978 and 1980,respectively, and the Ph.D.degree in mathematics fromUniversity of Minnesota,Minneapolis, MN, in 1989.He is a full professor in theDepartment of Mathematics at Sogang University, Seoul,Korea. He has been there since 1997, following a positionat Seoul Women’s University. His research interestsinclude number theory (exponential sums, modular forms,zeta functions,p-adic analysis, umbral calculus) andcoding theory. He is a member of AMS (AmericanMathematical Society) and IEEE. He is a referee andeditors for mathematical journals.Taekyun Kimreceived the PhD degreein Mathematics for p-adicanalytic number theoryat Kyushu University. He isa full Professor at Departmentof Mathematics, KwangwoonUniversity, Seoul, Republic ofKorea. His research interestsare in the areas ofp-adicnumber theory, special functions, applied mathematicsand mathematical physics. He has published researcharticles in reputed international journals of mathematicsand mathematical physics. He is a referee and editors ofmathematical journals.Jong-Jin Seo is anassociate Professor atDeprtment of Appliedmathematics, PukyongNational University, Pusan,Republic of Korea. He isworking in the area ofp-adicanalysis and number theory atDepartment of Mathematicsin the Graduate School ofKwangwoon university, Seoul, Republic of Korea. Hissupervisor is Professor Taekyun Kim and his researchinterests are in the areas ofp-adic number theory, specialfunctions, applied mathematics and mathematicalphysics.c© 2015 NSPNatural Sciences Publishing Cor.

A note on q-analogue of Boole polynomials

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