A note on the p-adic gamma function and q-changhee polynomials

Transcription

1 Available olie at wwwisr-publicatioscom/jmcs J Math Computer Sci, 18 (2018, Research Article Joural Homepage: wwwtjmcscom - wwwisr-publicatioscom/jmcs A ote o the p-adic gamma fuctio ad q-chaghee polyomials Özge Çolakoğlu Havare, Hamza Meke Mathematics Departmet, Sciece ad Arts Faculty, Mersi Uiversity, 33343, Mersi-Turkey Abstract I the preset work, we cosider the fermioic p-adic q-itegral of p-adic gamma fuctio ad the derivative of p-adic gamma fuctio by usig their Mahler expasios The relatioship betwee the p-adic gamma fuctio ad q-chaghee umbers is obtaied A ew represetatio is give for the p-adic Euler costat Also, we study o the relatioship betwee q- Chaghee polyomials ad p-adic Euler costat usig the fermioic p-adic q-itegral techiques the idea that the q-chaghee polyomial Keywords: p-adic umber, p-adic gamma fuctio, the fermioic p-adic q-itegral, Mahler coefficiets, p-adic Euler costat, q-chaghee Polyomials 2010 MSC: 11S80, 05A30, 33E50, 05A19, 26C05 c 2018 All rights reserved 1 Itroductio Let p be a fixed odd prime umber Throughout this paper by, Q p ad C p we deote the rig of p- adic itegers, the field of p-adic umbers ad the completio of the algebraic closure of Q p, respectively Let q be idetermiate with 1 q p < p p 1 1 Recetly, the q-calculus (Quatum Calculus has a great iterest ad has bee studied by may scietists May geeralizatios of special fuctios with a q- parameter recetly were obtaied usig p-adic q-itegral o (see, eg, [1, 8, 9, 11] For f C( C p, the fermioic p-adic q-itegral o is defied by Kim to be I q (f = N f(xdµ q (x := lim 1 [p N ] q p N 1 where [x] q = 1 ( qx 1+q (see [5, 7, 6] For ay f C( C p, by (11, the relatio where [x] q = 1 qx 1 q f(jq j ( 1 j, (11 pn 1 q I q (f (x + + ( 1 1 I q (f (x = [2] q f(jq j ( 1 1 j, (12 ad N holds Correspodig author addresses: ozgecolakoglu@mersiedutr (Özge Çolakoğlu Havare, hmeke@mersiedutr (Hamza Meke doi: /jmcs Received j=0 j=0

2 Ö Çolakoğlu Havare, H Meke, J Math Computer Sci, 18 (2018, The Chaghee umbers ad polyomials which are derived umbral calculus are defied by Kim et al as the geeratig fuctio to be 2 t + 2 (1 + tx = Ch (x t Whe x = 0, Ch (0 = Ch are called Chaghee umbers see [10] for a summary I [11], Kim et al defied the degeerate Chaghee polyomials ad i [12], Kim et al cosidered q-chaghee polyomials, Ch,q (x, which are give by the geeratig fuctio to be 1 + q q (1 + t + 1 (1 + tx = Ch,q (x t for t p < p 1 p 1 Whe q = 1, Ch,1 (x = Ch (x Whe x = 0, Ch,q (0 = Ch,q are called q-chaghee umbers ad whe q = 1 ad x = 0, Ch,1 (0 = Ch Kim also itroduced the q-chaghee umbers of the secod kid by Ĉh,q = ( x dµ q (x, 0, (see [12], ad the q-chaghee umbers of the secod kid by Ĉh,q (x = ( x y dµ q (y, 0 The geeratig fuctio for such polyomials is give by Ĉh,q (x t = 1 + q 1 + q + t (1 + t1 x (13 I [12], Kim et al obtaied the followig theorems Theorem 11 Let (x = x (x 1 (x + 1 For 0, (x dµ q (x = Ch,q Theorem 12 Let (x = x (x 1 (x + 1 For 0, the followig relatio holds: (x + y dµ q (y = Ch,q (x Theorem 13 For 0, Theorem 14 For 0 Theorem 15 For 0 ( q dµ q (x = 1 + q Ĉh,q = ( q (1 + q Ĉh,q (x = Ch,q (1 x Note that ( x (x =

3 Ö Çolakoğlu Havare, H Meke, J Math Computer Sci, 18 (2018, p-adic umbers itroduced by the Germa mathematicia Kurt Hesel ( , are widely used i mathematics: i umber theory, algebraic geometry, represetatio theory, algebraic ad arithmetical dyamics, ad cryptography p-adic umbers have bee used i applied fields with successfully applyig i superfield theory of p-adic umbers by Vladimirov ad Volovich I additio, p-adic model of the uiverse, p-adic quatum theory, p-adic strig theory such as areas occurred i physics (for detail see [18, 17] I 1975, Morita [14] defied the p-adic gamma fuctio p by the formula p (x = lim x ( 1 1 j< (p,j=1 for x, where approaches x through positive itegers The p-adic gamma fuctio p is aalytic o ad satisfies the fuctioal relatio { xp (x, x p (x + 1 = p = 1, p (x, x p < 1 The p-adic Euler costat γ p is defied by the formula j, γ p := p(1 p (1 = p(1 = p(0 (14 The p-adic gamma fuctio p (x has a great iterest ad has bee studied by Barsky (1981 [2], Diamod (1977 [3], Dwork (1983 [4] ad others For x, the symbol ( ( x is defied by x 0 = 1 ad ( x := x(x 1 (x + 1, ( = 1, 2, The fuctios x ( x (x Zp, N form a orthoormal base of the space C( C p with respect the orm This orthoormal base has the followig property: ( x = 1 ( 1 j 1 j=0 j, (see, [16, p168] (15 j I 1958, Mahler itroduced a expasio for cotiuous fuctios of a p-adic variable usig special polyomials as biomial coefficiet polyomial [13] It meas that for ay f C( C p, there exist uique elemets a 0, a 1, of C p such that f(x = a (x The base {( } : N is called Mahler base of the space C(Zp C p, ad the elemets {a : N} i f(x = ( a x are called Mahler coefficiets of f C(Zp C p The Mahler expasio of the p-adic gamma fuctio p ad its Mahler coefficiets are determied by the followig propositio Propositio 16 ([15, 16] Let p (x + 1 = a, (x,

4 Ö Çolakoğlu Havare, H Meke, J Math Computer Sci, 18 (2018, ad ( exp x + xp 1 x p p 1 x = b x, (x E The, a = ( 1 +1 b for all, where E is the regio of covergece of the power series x 2 Mai results I this paper, we cosider p-adic gamma fuctio with the fermioic p-adic q-itegral We derive the relatioship betwee q-chagee polyomials ad p-adic gamma fuctio We obtai the fermioic p-adic q-itegral of p-adic gamma fuctio ad the derivative of p-adic gamma fuctio For the p-adic Euler costat A ew represetatio is obtaied Also, we study o the Chaghee polyomials ad p-adic Euler costat I what follows, we idicate the fermioic p-adic q-itegral with Mahler coefficiets of p-adic gamma fuctio Theorem 21 For x, the followig equality holds: where a is defied by Propositio 16 Proof Let x We have p (x + 1dµ q (x = p (x + 1dµ q (x = Z p Note that ( x (x = From Theorem 11, we get Ch,q a, a dµ q (x = p (x + 1dµ q (x = a a Ch,q Usig Theorem 13 we ca rewrite (21 ad we have the followig corollary Corollary 22 For N, where a is defied by Propositio 16 p (x + 1dµ q (x = =1 j=0 ( q a, 1 + q dµ q (x (21 Uder coditio of Propositio 16 ad usig (15, derivative of p-adic gamma fuctios, p is obtaied as 1 p(x ( 1 j = a, (22 j j where a is defied by Propositio 16 Theorem 23 The relatioship betwee the q-chaghee polyomials ad the p-adic Euler costat is give by γ p = 1 ( 1 j ( qchj,q + Ch j,q ( 1 a ( j j! [2] q =1 j=0

5 Ö Çolakoğlu Havare, H Meke, J Math Computer Sci, 18 (2018, Proof Whe f (x = p (x ad = 1 i (12, we get q p (x + 1 dµ q (x + p (x dµ q (x = [2] q p (0 =1 j=0 From (22 ad (14, we ca write 1 ( 1 j 1 ( x q a j j dµ q (x + 1 ( 1 j 1 1 a dµ q (x = [2] j j q γ p (23 =1 j=0 Usig Theorem 11 ad Theorem 12 we ca rewrite (23 as 1 ( 1 j 1 ( a qchj,q + Ch j,q ( 1 = [2] ( j j! q γ p =1 j=0 Theorem 24 The p-adic Euler costat has the expasio ( 1 ( 1 q j+1 j γ p = a ( j [2] q (1 + q j + 1 (1 + q i, (j i! =1 j=0 where a is defied by Propositio 16 Proof Firstly, we compute Ch,q ( 1 From Theorem 15, we have Ĉh,q (2 = Ch,q (1 2 From (13 Ĉh,q (x t = 1 + q ( 1 x t = ( 1 t ( 1 x 1 + q + t (1 + q t, or Ĉh,q (x t = ( 1 i ( 1 x t 1 + q i The, we have Note that x = ( 1 ( x or or Ĉh,q (x = Ĉh,q (x = Ĉh,q (x = Ch,q (1 x = Ĉh,q (x = Whe x = 2 i (24 we have Ch,q ( 1 = ( 1 i ( 1 x 1 + q i ( 1 i (1 x i (1 + q i, ( i! ( 1 i ( 1 + x (1 + q i ( i! ( 1 (1+q i ( i! of Ch,q ( 1 we ca rewrite Theorem 23 as the followig ( γ p = 1 ( 1 j a ( j j! =1 j=0 q i ( 1 i ( 1 ( 1 + x (1 + q i ( i! ( j q j 1+q j! + j! The proof of theorem is completed with a little calculatios, i (24 Usig Theorem 13, Theorem 11 ad value ( 1 j (1+q i (j i! [2] q

6 Ö Çolakoğlu Havare, H Meke, J Math Computer Sci, 18 (2018, Theorem 25 The followig relatio holds where a is defied by Propositio 16 Proof Let x We have By usig Theorem 12 we ca write Theorem 26 For x, s, we have p (x + sdµ q (x = p (x + sdµ q (x = Ch,q (s 1 a, a p (x + sdµ q (x = p (x + s dµ q (x = Proof Let x, s From (22, we have p (x + s dµ q (x = By usig Theorem 12 we ca write (x + s 1 dµ q (x Ch,q (s 1 a 1 ( 1 j 1 Ch j,q (s 1 a ( j j! =1 j=0 1 ( 1 j 1 + s 1 a dµ q (x j j =1 j=0 p (x + s dµ q (x = 1 ( 1 j 1 Ch j,q (s 1 a ( j j! =1 j=0 I the case s = 1 i Theorem 26 we obtai the followig coclusio Corollary 27 For x, we have where Ch,q are q-chaghee umbers Ackowledgmet p (x dµ q (x = 1 ( 1 j 1 Ch j,q a, ( j j! =1 j=0 This work is supported by the Scietific ad Techological Research Coucil of Turkey (TÜBİTAK Refereces [1] S Araci, D Erdal, J J Seo, A study o the fermioic p-adic q-itegral represetatio o associated with weighted q-berstei ad q-geocchi polyomials, Abstr Appl Aal, 2011 (2011, 10 pages 1 [2] D Barsky, O Morita s p-adic gamma fuctio, Math Proc Cambridge Philos Soc, 89 (1981, [3] J Diamod, The p-adic log gamma fuctio ad p-adic Euler costats, Tras Amer Math Soc, 233 (1977, [4] B Dwork, A ote o the p-adic gamma fuctio, Study group o ultrametric aalysis, 9th year: 1981/82, Marseille, (1982, Ist Heri Poicaré, Paris, (1983, 10 pages 1

7 Ö Çolakoğlu Havare, H Meke, J Math Computer Sci, 18 (2018, [5] T Kim, q-volkebor itegratio, Russ J Math Phys, 9 (2002, [6] T Kim, q-euler umbers ad polyomials associated with p-adic q-itegrals, J Noliear Math Phys, 14 (2007, [7] T Kim, Some idetities o the q-euler polyomials of higher order ad q-stirlig umbers by the fermioic p-adic itegral o, Russ J Math Phys, 16 (2009, [8] T Kim, Symmetry of power sum polyomials ad multivariate fermioic p-adic ivariat itegral o, Russ J Math Phys, 16 (2009, [9] D S Kim, T Kim, Daehee umbers ad polyomials, Appl Math Sci (Ruse, 7 (2013, [10] T Kim, D S Kim, T Masour, S-H Rim, M Schork, Umbral calculus ad Sheffer sequeces of polyomials, J Math Phys, 54 (2013, 15 pages 1 [11] T Kim, H-I Kwo, J J Seo, Degeerate q-chaghee polyomials, J Noliear Sci Appl, 9 (2016, , 1 [12] T Kim, T Masour, S-H Rim, J J Seo, A ote o q-chaghee Polyomials ad Numbers, Adv Studies Theor Phys, 8 (2014, , 1 [13] K Mahler, A iterpolatio series for cotiuous fuctios of a p-adic variable, J Reie Agew Math, 199 (1958, [14] Y Morita, A p-adic aalogue of the -fuctio, J Fac Sci Uiv Tokyo Sect IA Math, 22 (1975, [15] A M Robert, A course i p-adic aalysis, Graduate Texts i Mathematics, Spriger-Verlag, New York, ( [16] W H Schikhof, Ultrametric calculus, A itroductio to p-adic aalysis, Cambridge Studies i Advaced Mathematics, Cambridge Uiversity Press, Cambridge, ( , 16 [17] V S Vladimirov, I V Volovich, Superaalysis, I, Differetial calculus, (Russia Teoret Mat Fiz, 59 (1984, [18] I V Volovich, Number theory as the ultimate physical theory, p-adic Numbers Ultrametric Aal Appl, 2 (2010,