ON THE VOLKENBORN INTEGRAL OF THE q-extension OF THE p-adic GAMMA FUNCTION

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1 Joural of Mathematical Aalysis ISSN: , URL: wwwiliriascom/jma Volume 8 Issue 2 (2017), Pages ON THE VOLKENBORN INTEGRAL OF THE -EXTENSION OF THE -ADIC GAMMA FUNCTION ÖGE ÇOLAKOĞLU HAVARE, HAMA MENKEN Abstract I the reset w we cosider the -etesio of the -adic gamma fuctio We derive the Voleb itegral of the -etesio of the -adic gamma fuctio by usig its Mahler easio Meover, we give a ew reresetatio f the -etesio of the -adic Euler costat 1 Itroductio The -Calculus aeared i the 18th cetury ad it cotiues to develo raidly The -calculus has a great iterest ad has bee studied by Euler, Gauss who discovered -biomial fmula ad others The systematic develomet of the - calculus bega with FH Jacso i the early 20th cetury Although the -calculus has bee studied f over a cetury, -aalogue of secial umbers ad olyomials are still of iterest [1] The letter stads f uatum, ad the -biomial coefficiets lay a imtat role i uatum calculus similar to that of the diary biomial coefficiets i diary calculus Also, the biomial coefficiets are also ow combiatios combiatial umbers I costructig the roerties ad idetities of some secial umbers, biomial coefficiets have great iterest The -aalogue of the biomial coefficiets a imtat role lay i develoig the they of the -aalogue of these secial umbers [5] The -biomial coefficiets Gaussia olyomials aear i may idetities o -series I additio, they are studied i several combiatial eviromets as artitios of itegers F a easy hadlig of the -biomial coefficiets i combiaty it is essetial to be familiar with the basic combiatial structures that admit those coefficiets as geeratig olyomials [9] The adic umbers itroduced by the Germa mathematicia Kurt Hesel ( ), are widely used i mathematics: i umber they, algebraic geometry, reresetatio they, algebraic ad arithmetical dyamics, ad crytograhy The adic umbers have bee used alyig fields with successfully alyig i 2000 Mathematics Subject Classificatio 11S80,11E95, 11S23 Key wds ad hrases -etesio of -adic gamma fuctio; Voleb itegral; - etesio of -adic Euler costat; -biomial coefficiet c 2017 Ilirias Research Istitute, Prishtië, Kosovë Submitted November 25, 2016 Published Aril 1, 2017 Commuicated by Farruh Muhamedov 64

2 VOLKENBORN INTEGRAL OF THE -EXTENSION OF THE -ADIC GAMMA FUNCTION65 suerfield they of adic umbers by VS Vladimirov ad I V Volovich I additio, the adic model of the uiverse, the adic uatum they, the adic strig they such as areas occurred i hysics (f detail see [24],[25]) Throughout this aer, is a fied odd rime umber ad by, Q ad C we deote the rig of -adic itegers, the field of -adic umbers ad the comletio of the algebraic closure of Q, resectively The aalogue of called the followig otatio, () , 1 f ay oegative iteger As 1; () The biomial coefficiet is a aalogue f the biomial coefficiet, also called a Gaussia coefficiet a Gaussia olyomial A -biomial coefficiet, is defied by ()! ( )! ()! where ()! deotes the factial which give by (0)! 1 ad ()! () ( 1) (2) (1) whe > 0 Aother eressio of the -biomial coefficiet f oegative itegers, with is defied by ( 1)( 1 1)( 1 1) ( 1)( 1 (11) 1)( 1) It is clearly that whe 1; the biomial coefficiets reduce to the usual biomial coefficiets The -biomial coefficiets have -Pascal rule; where 1 1 ([6], [11]) 1 1 1, (12) I 1975 Y Mita [17] defied the -adic gamma fuctio Γ by the fmula Γ () lim ( 1) j 1 j< (,j)1 f, where aroaches through ositive itegers The -adic gamma fuctio Γ () has a great iterest ad has bee studied by J Diamod (1977) [7], D Barsy (1977) [2], M Boyarsy (1980) [3], B Dw (1983) [8], T Kim (1997) [12] ad others The relatioshi betwee some secial fuctios ad the -adic gamma fuctio Γ () was ivestigated by B Gross ad N Koblitz (1979) [10], H Cohe ad E Friedma (2008) [4] ad I Shairo (2012) [21] The -etesio of the -adic gamma fuctio Γ, () is defied by N Koblitz [14] as follows:

3 66 Ö ÇOLAKOĞLU HAVARE, H MENKEN Let C, 1 < 1, 1The -etesio of the -adic gamma fuctio Γ, () is defied by fmula Γ, () lim ( 1) j< (,j)1 1 j 1 f, where aroaches through ositive itegers We recall that lim Γ, Γ 1 N Koblitz determied the relatio betwee the derivative of Γ, () with the -etesio of the -adic Euler costat γ, is defied by the fmula ( Γ,(0) 1 1 ) γ, (13) The -etesio of the -adic gamma fuctio Γ, () was studied by N Koblitz (1980, 1982) [14], [15], H Naazato (1988) [18],Y S Kim (1998) [13] ad others I 1958, Mahler itroduced a easio f cotiuous fuctios of a adic variable usig secial olyomials as biomial coefficiet olyomial [16] Cad studied a aalogue of Mahler easios f cotiuous fuctios i adic aalysis, relacig biomial coefficiet olyomials ( ( ) with a aalogue ) f a adic variable C with 1 < 1 [6] The Mahler coefficiets Γ, are determied by the followig roositio: Proositio 11 [6]Let Γ, ( 1) τ, () 0 ( ) (14) be the Mahler series of Γ, with 1 < 1 1 ad C where τ, () be th -Mahler coefficiet of the seuece Γ, ( 1) The followig euality ( 1) τ, () X ()! 1 X 1 X E 1\ () E (15) () ad 0 E 1\ () E ( () ) b, () X ()! 0 where E (X) is the -etesio of the eoetial series ad also may deote a slightly differet series holds The Voleb itegral was itroduced by A Voleb i his PhD dissertatio ad subseuetly i the set of twi aers [22], [23]; a me recet treatmet of the subject ca be foud i [20] The idefiite sum of a cotiuous fuctio f : C is the cotiuous fuctio Sf iterolatig 1 f(j) ( N) Istead of Sf() ( ) we j0 ca write 1 1 f(j) lim f(j) see [19] ad [20] The ifite sum of cotiuous j0 j0 fuctio has below roerty

4 VOLKENBORN INTEGRAL OF THE -EXTENSION OF THE -ADIC GAMMA FUNCTION67 Sf( 1) Sf() f () (16) Let f be a fuctio from C 1 ( C ) The Voleb itegral of f o is defied by the fmula 1 f()d : lim f(j) (Sf) (0) (17) j0 F ay f C 1 ( C ), the relatio holds: f( 1)d f()d f (0) (18) 2 Mai Results To evaluate Voleb itegral of the -etesio of the -adic gamma fuctio, the followig eualities ca be eressed: Lemma 21 Let 1 < 1 with C, N ad f C ( C ) such that the -Mahler easio of f, f() τ, () The, the idefiite of f is give by Sf() 0 τ, () 1 0 Proof Let Let 1 < 1 with C, N, Assume that Sf() σ, () From (16), we get 1 σ, () 0 By usig (12), we have ( σ, () So we have 0 0 σ, () 0 σ, () 1 1 σ, ( 1) 1 We ow that N Thus, we ca write σ, ( 1) 0 τ, () 0 ) τ, () 0 τ, () 0 τ, () 0 τ, () 0

5 68 Ö ÇOLAKOĞLU HAVARE, H MENKEN Hece, we have The, we ca write σ, ( 1) τ, () N σ, () 1 τ, ( 1) N Sf() By some comutig, we obtai Sf() 1 τ, ( 1) 0 τ, () 1 0 If we tae f (), we obtai the followig results; Collary 22 If ad C with 1 < 1 the S 1 Lemma 23 F, s, N ad C with 1 < 1, the followig idetity: s s l j 1 ( j s 1 ) s j 1 s 1 1 ( 1 1) ( 1) j is valid Proof From (11), we have 0 s 1 [ ( s 1)( s ] 1) ( 1 1)( 1) By alyig the defiitio of the derivative, we ca write s l s ( s 1 1)( s 1) l s ( s 1)( s 1 1) 1 ( 1 1)( 1) We easily obtai the followig relatio with a little rearragig s l j 1 ( s j s 1 ) s j 1 1 ( 1 1) ( 1) j 0 s 1 s Theem 24 F, s, N ad C with 1 < 1, the Voleb itegral of -biomial coefficiet is s l s s 1 d 1 s j ( s 1) 1 1 ( s j 1)

6 VOLKENBORN INTEGRAL OF THE -EXTENSION OF THE -ADIC GAMMA FUNCTION69 Proof From (17) ad Collary 22, we easily obtai the followig: ( ) s s d s (0) 1 s d ( s l s 1 s ( s 1 By usig Lemma 23 ad (17) ad we ca rewrite (21) s j 1 d l s ( s j s 1 ) s j 1 ( 1 1) ( 1) j 0 Usig (11) we easily discover s l s d s s d l s 1 1 s 1 ) ) (0) (21) j 1 ( s j s 1 ) s j 1 ( 1) j 0 s j ( s 1) ( s j 1) I the case s 0 i Lemma 24, we obtai followig collary:: s 1 s s 1 ( 1 1) Collary 25 F, N ad C with 1 < 1, the relatio holds: ( 1) l d ( 1) 2 ( 1 1) Theem 26 F all, N ad 1 < 1 1 followig idetity holds: ( 1) l Γ, ( 1)d 0 where τ, () is defied by Proositio 11 τ, () ( 1) 2 ( 1 1) with C, the Proof From Proositio 11 ad Collary 25 we have Γ, ( 1)d τ, () d τ, () d (22) 0 0 ( 1) l Γ, ( 1)d τ, () 0 ( 1) 2 ( 1 1) s 1

7 70 Ö ÇOLAKOĞLU HAVARE, H MENKEN Theem 27 Let, s F 1 < 1 1 with C the followig idetity: l s1 s 2 ( Γ, ( s)d τ, () 1 s1 1 ) ( s1 j 1) is true where τ, () is defied by Proositio 11 Proof Usig Proositio 11, we have s 1 Γ, ( s)d τ, () d 0 s 1 τ, () 0 By Theem 24, we ca write l s1 s 2 Γ, ( s)d τ, () d ( s1 j s1 1 ) ( s1 j 1) As a result, we idicate the -etesio of the -adic Euler costat with - Mahler coefficiet of the seuece Γ, Theem 28 The euality holds: γ, τ, () ( 1) l 1 (1 ) ( 1) 2 ( 1 1) 1 1 (2)( 1) 2 ( 1) 0 f, 1 < with C where τ, () is defied by Proositio Proof From (18) we have Γ, ( 1)d Γ, ()d Γ,(0) (23) I the case s 0 i Theem 27, we get l 1 2 Γ, ()d τ, () Γ, ()d l 1 ( 1) ( 1 1 ) τ, () (3) 2 ( 1) 0 Γ, ()d τ, () ( 1) l 1 1 (3) 2 ( 1) 0 1 j ( 1) ( 1 j 1) 1 j ( 1) ( 1 j 1) 1 j ( 1) ( 1 j (24) 1) 1 j ( 1) ( 1 j 1)

8 VOLKENBORN INTEGRAL OF THE -EXTENSION OF THE -ADIC GAMMA FUNCTION71 The, we rewrite the euality (23) usig the euality (24) ad Theem 26 Γ,(0) τ, ()( 1) l 1 2 ( 1 1) j ( 1) 2 ( 1) ( 1 j 1) 0 ( 1) (3) From the euality (13), we obtai a ew reresetatio f the -etesio of the -adic Euler costat γ, with the coefficiets of the ower series γ, τ, () ( 1) l 1 (1 ) ( 1) 2 ( 1 1) j ( 1) (2)( 1) 2 ( 1) ( 1 j 1) 0 Acowledgmets The auths would lie to tha the aoymous reviewer f his/her careful readig ad may valuable suggestios that heled us imrove this article Refereces [1] S Araci ad M Açıgöz, A ote o the values of weighted -Berstei olyomials ad weighted -Geocchi umbers, Advaces i Differeces Euatios, (2015) 1-9 [2] D Barsy, O Mita s -adic Gamma Fuctio, Groue d Etude d Aalyse Ultramétriue, 5e aée (1977/78), 3, (1978),1-6 [3] M Boyarsy, -adic Gamma Fuctios ad Dw Cohomology, Tras Amer Math Soc 257(2), (1980) [4] H Cohe ad E Friedma, Raabe s fmula f -adic gamma ad zeta fuctios, A Ist Fourier (Greoble) 58(1), (2008), [5] RB Ccio, O ; -Biomial Coefficiets, Itegers: Electroic Joural of Combiatial Number They, 8, (2008), #29 [6] K Cad, A -Aalogue of Mahler Easios I, Advaces i Mathematics, 153(2), (2000) [7] J Diamod, The -adic log gamma fuctio ad -adic Euler costat, Tras Amer Math Soc 233, (1977) Diamod J (1977) [8] B Dw, A ote o -adic gamma fuctio, Groue de travail d aalyse ultrametriue, 9(3), (1983),J1-J10 [9] D Foata ad G Ha, The -series i combiotics; Permütatio Statistics (2004) Lecture Note [10] Gross, B H ad Koblitz, N, Gauss Sums ad the -adic Γ-fuctio, The Aals of Mathematics, Secod Series, 109(3), (1979), [11] V Kac ad P Cheug, Quatum Calculus, Sriger, 1943 [12] T Kim, A ote o aalogue of gamma fuctios, ProcCofer o 5th Trascedetal Number They 5, No 1, Gaushi Uiv Toyo Jaa, (1997), [13] Y S Kim, -aalogues of -adic gamma fuctios ad -adic Euler costats, Commuicatios of the Kea Mathematical Society, 13 (4), (1998), [14] N Koblitz, -etesio of the -adic gamma fuctio, Trasactios of the America Mathematical Society, 260(2), (1980), [15] N Koblitz, -etesio of the -adic gamma fuctio II, Trasactios of the America Mathematical Society, 273(1), (1982), [16] K Mahler, A Iterolatio Series f Cotiuous Fuctios of a -adic Variable, Joural für die reie ud agewadte Mathemati, 199, (1958), [17] Y Mita, A -adic aalogue of the Γ-fuctio, J Fac Sciece Uiv, Toyo, 22(2), (1975) [18] H Naazato, The -aalogue of the -adic gamma fuctio, Kodai Math J, 11(1), (1988),

9 72 Ö ÇOLAKOĞLU HAVARE, H MENKEN [19] A M Robert, A Course i -adic Aalysis, Graduate Tets i Mathematics 198, Sriger- Verlag New Y, 2000 [20] W H Schihof, Ultrametric Calculus: A Itroductio to -adic Aalysis, Cambridge Uiversity Pres, 1984 [21] I Shairo, Frobeius ma ad the -adic gamma fuctio, J Number They, 132(8), (2012), [22] A Voleb, Ei -adisches Itegral ud seie Aweduge I, Mauscrita Math 7(4), (1972), [23] A Voleb, Ei -adisches Itegral ud seie Aweduge II, Mauscrita Math 12, (1974), [24] I V Volovich, Number they as the ultimate hysical they, Prerit No TH 4781/87, CERN, Geeva, (1987) [25] V S Vladimirov ad I V Volovich, Sueraalysis I Differetial calculus, The Math Phys 59,(1984) Özge Çolaoğlu Havare Mersi Uiversity, Sciece ad Arts Faculty, Mathematics Deartmet, 33343, Mersi- Turey address: ozgecolaoglu@mersiedutr Hamza Mee Mersi Uiversity, Sciece ad Arts Faculty, Mathematics Deartmet, 33343, Mersi- Turey address: hmee@mersiedutr