Relations on the Apostol Type (p, q)-frobenius-euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems

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1 Tish Joal of Aalysis ad Nmbe Theoy 27 Vol 5 No Available olie a hp://pbssciepbcom/ja/5/4/2 Sciece ad Edcaio Pblishig DOI:269/ja Relaios o he Aposol Type (p -Fobeis-Ele Polyomials ad Geealizaios of he Sivasava-Pié Addiio Theoems Ba K * Depame of Mahemaics Facly of Edcaios Uivesiy of Adeiz *Coespodig aho: ba@adeized Received Decembe 22 26; Revised Apil 9 27; Acceped Je 4 27 Absac I his wo we defie ad iodce a ew id of he Aposol ype Fobeis-Ele polyomials based o he (p -calcls ad ivesigae hei some popeies ecece elaioships ad so o We give some ideiies a his polyomial Moeove we ge (p -exesio of Caliz s mai esl i [] Keywods: Geeaig fcio Fobeis-Ele polyomials ad mbes (p -calcls (p -Fobeis- Ele polyomials Aposol-Beolli mbe ad polyomials geealized -Beolli polyomials geealized -Ele polyomials Cie This Aicle: Ba K Relaios o he Aposol Type (p -Fobeis-Ele Polyomials ad Geealizaios of he Sivasava-Pié Addiio Theoems Tish Joal of Aalysis ad Nmbe Theoy vol 5 o 4 (27: 26-3 doi: 269/ja Iodcio Defiiios ad Noaios Thogho his pape we always mae se of he followig oaio; deoes he se of aal mbes deoes he se of oegaive ieges deoes he se of eal mbes ad deoes he se of complex mbes The (p -mbes ae defied by p [ ] p which is aal geealizaio of he -mbe sch ha p [ ] [ ] Noe ha (p -mbe is symmeic: ha is [ ] [ ] p The (p -deivaive of a fcio f is defied by ( ( ( ( p x f px f x Dx : f ( x : D f ( x x The (p -Gass Biomial fomla is defied by ( 2 ( 2 ( x+ y p x y whee he oaios ((p -Gass Biomial coefficies ad [ ] ((p -facoial ae defied by [ ] [ ] [ ] [ ] [ ] 2 [ ] [ ] [ ] ( z ad The (p -expoeial fcios ( ( E z ae defied by ad [ ] ( 2 ( x p x [ ] ( 2 E ( x x I his wo we iodce Aposol ype (p -Fobeis-Ele polyomials We give some ew ideiies fo he Aposol ype (p -Fobeis-Ele polyomials Also we pove some explici expessios Fom his fom we easily see ha e E ( z Defiiio Le The (p -Beolli ad polyomials ( mbes xy ae defied by meas of he geeaig fcios i [6]: < 2 π [ ] e (

2 Tish Joal of Aalysis ad Nmbe Theoy 27 ( xy [ ] ( x E ( y < 2 π ( Defiiio 2 Le The (p -Ele mbes xy ae defied by meas of he geeaig fcios i [6]: ad polyomials ( [ ] [ 2] ( [ 2] ( < π + ( xy [ ] ( x E ( y < π + Defiiio 3 Le The (p -Beolli mbes ( ad polyomials ( ( xy i x y of ode ae defied by meas of he geeaig fcios i [6]: ( < 2 π ( [ ] e ( ( xy [ ] ( x E ( y < 2 π ( (2 Defiiio 4 Le The (p -Ele mbes ( ad polyomials ( ( xy i x y of ode ae defied by meas of he geeaig fcios i [6]: [ ] ( 2 < π (3 [ ] e ( + ( ( xy [ ] [ 2] ( x E ( y < π ( + ( ( (4 Classical Fobeis-Ele polyomials x ; of ode is defied by he followig elaio [7] ( x ( x ; e e (5 whee algebaic mbe Similaly Fobeis-Ele polyomials ( ( x ; ; of ode is defied by he followig elaio ([7] ( x ( x ; ; e e (6 Defiiio 5 The Aposol ype -Fobeis-Ele polyomials ( ( xy ; ; of ode i x y ad Aposol ype -Fobeis-Ele mbe ; ; of ode i [9] especively ( ( ( ( x y; ; e( x E( y e ( ( ( ; ; e ( p Defiiio 6 Le ad < < We defie he Aposol ype (p -Beolli polyomials ( ( xy ; ; of ode i x y ad he Aposol ype (p -Beolli mbes ( ( ; ; of ode i x y especively ( ( xy ; ; [ ] e x E y ( ( ( ( ( ; ; [ ] e ( p Defiiio 7 Le ad < < We defie he Aposol ype (p -Ele polyomials ( ( xy ; ; of ode i x y ad he Aposol ype (p -Ele mbes ( ( ; ; of ode i x y especively ( ( xy ; ; [ ] [ 2] e x E y ( + ( ( ( [ 2] ( ; ; [ ] e ( + Defiiio 8 We dee Aposol ype (p -Fobeis- Ele polyomials ( ( xy ; ; of ode i x y

3 28 Tish Joal of Aalysis ad Nmbe Theoy ad Aposol ype (p -Fobeis-Ele mbes ( ( ; ; of ode especively ( ( xy ; ; [ ] ( x E ( y ( (7 ( ( ; ; (8 [ ] e ( Leig p i (7 we have ( xy ; ; ( [ ] p x E y p ( ( ( e ( ( x y; ; e ( x E ( y [ ] ( [9] Pig ad i (7 we have whee Usig ( ( ; ; ( xy xy ( ( xy is ( p ( -Ele polyomials of ode i las eaio we have ( ( xy ( ( xy ; ; ( ( whee xy is -Ele polyomials of ode Leig i las eaio we have ( ( xy ( ( xy ; ; ( ( whee xy is Hemie based Ele polyomials of ode 2 Some Basic Popeies fo he Aposol Type -Fobeis-Ele Polyomials Poposiio Aposol ype Fobeis-Ele polyomials saisfy he followig elaios ( + β ( xy ; ; ( ( (2 β ; ; ; ; ( xy ( ( xy ; ; ( xy ; ; (22 ( ( x y + ( β ( xy ; ; ( ( (23 β x; ; y ; ; ( ( Theoem Fo ad x y a he followig elaioships hold e: ( x+ ay ; ; ( ( 2 (24 s s y ; ; p xa s ( ( s ( ( xy + a ; ; ( ( ( 2 s s x; ; p ya s s Poof Usig Defiiio ( ( x+ ay ; ; [ ] E ( y (( x + a ( ( l l ( y ; ; l [ l ] ( 2 ( p x+ a [ ] ( ( y ; ; [ ] ( 2 ( p x+ a ( ( 2 ( y ; ; p [ ] s s ya s s Compaig he coefficies of [ ] we have (24 Similaly he ohe eaio is bee calclaio Theoem 2 Thee is he followig elaio fo he geealized Aposol ype -Fobeis-Ele polyomials ( 2 ( ; ; ( x y; ; (25 ( ( xy ; ; xy ; ;

4 Tish Joal of Aalysis ad Nmbe Theoy 29 Poof By sig he ideiy 2 ( ( ( ( ( ( ( ( ( ( e ( ( ( x E ( y 2 ( ( ( ( ( ( ( x E ( y ( ( ( x ( ( E ( y e ( ( ( 2 ( ; ; [ ] ( ; ; [ ] ( xy ; ; [ ] ; ; ( ( xy [ ] Compaig he coefficie of [ ] Rema Fo ( xy p we pove (25 ; ; Sbsiig y i (25 We have Caliz esl ([] eaio 29 Theoem 3 Thee is he followig elaio fo he geealized Aposol ype (p -Fobeis-Ele polyomial ( xy ; ; ( xy ; ; ( ( x y + Poof By sig he ideiy e ( E ( 2 ( ( ( ( e e We wie as ( ( ( ( ( ( ( ( ( ( e x E y ( e ( ( ( ( ( x E y x E y (26 ( xy ; ; [ ] e ( ( xy ; ; [ ] ( ( ( x E y ( xy ; ; [ ] ( xy ; ; [ ] [ ] ( ( x+ y [ ] Compaig he coefficies of ( xy ; ; [ ] we have ( xy ; ; ( ( x y + 3 Explici Relaio fo he Aposol Type (p -Fobeis-Ele Polyomials Theoem 4 Thee is he followig elaio fo he Aposol ype (p -Fobeis-Ele poly-omials ( ( xy ; ; ( y ; ; ( y ; ; ( x; ; Poof Sice (7; ( ( ( xy ; ; [ ] ( x E ( y ( ( E ( y ( ( x ( (3

5 3 Tish Joal of Aalysis ad Nmbe Theoy E y ( e ( ( ( ( ( x; ; [ l ] ( y ; ; [ ] ( x E ( y ( x ( ( ( y ; ; [ ] l l p l ( ( ; ; x [ l ] l l Compaig he coefficies of l [ ] we have (3 Theoem 5 Thee is he followig elaio bewee Aposol ype (p -Fobeis-Ele polyomials ad he geealized Aposol (p -Beolli polyomials Poof ( ( xy ; ; + + [ ] + + ( x; + + ( x; + ( ( y ; ; x E y ( ( ( ( ( y ; ; ( x; ( y ; ; ( x ; [ ] (32 ( E ( y ( x ( ( ( + + [ ] + ( ( y [ ] ( x; ; ; Compaig he coefficies of [ ] ( x; we have (32 Coollay Thee is he followig elaio bewee Aposol ype (p -Fobeis-Ele polyomials ad he geealized Aposol (p -Ele polyomials ( ( xy ; ; 2 Acowledgemes ( x ( x; ( ; ( y ; ; The pese ivesigaio was sppoed by he Scieific Reseach Pojec Admiisaio of Adeiz Uivesiy Refeeces [] Caliz L Eleia mbes ad polyomials Mah Mag 32( [2] Caliz L -Beolli mbes ad polyomials De Mah J 5( [3] Caliz L -Beolli ad Eleia mbes Tas Ame Mah Soc 76( [4] Ceci M Ca M ad K V -exesios of Geocchi mbes J Koea Mah Soc 43( [5] Cheo G S A oe o he Beolli ad Ele polyomials Appl Mah Lee 6( [6] Da U Acigoz M ad Aaci S O (p -Beolli (p -Ele ad (p -Geocchi polyomials 26 sbmied [7] Kim T Ideiies ivolvig Fobeis-Ele polyomials aisig fom o-liea diffeeial eaio J Nmbe Theoy 32( [8] Kim T Some fomlae fo he -Beolli ad Ele polyomials of highe ode J Mah Aaly Appl 273( [9] K B A Noe o he Aposol ype -Fobeis-Ele Polyomials ad Geealizaios of he Sivasava-Pie Addiio Theoems Filoma 26 3( [] K B ad Simse Y Fobeis-Ele ype polyomials elaed o Hemie-Beolli polyomials Nmeical Aalysis ad Appl Mah ICNAAM 2 Cof Poc 389( [] K B ad Simse Y O he geealized Aposol ype Fobeis-Ele polyomials Adv i Diff E [2] Kac V ad Cheg P Qam Calcls Spige (22

6 Tish Joal of Aalysis ad Nmbe Theoy 3 [3] Lo Q-M ad Sivasava H M Some elaioships bewee he Aposol-Beolli ad Aposol-Ele polyomials Comp Mah App 5( [4] Lo Q-M Some esls fo he -Beolli ad -Ele polyomials J Mah Aal Appl 363 (2 7-8 [5] Mahmdov N I -aaloges of he Beolli ad Geocchi polyomials ad he Sivasava-Pié addiio heoems Discee Dyamics i Nae ad Soc Aicle mbe [6] Mahmdov N I O a class of -Beolli ad -Ele polyomials Adv i Diff Ea 23 [7] Simse Y Geeaig fcios fo -Aposol ype Fobeis- Ele mbes ad polyomials Axioms ( [8] Simse Y Geeaig fcios fo geealized Silig ype mbes Aay ype polyomials Eleia ype polyomials ad hei applied Axiv: 3848v2 [9] Sivasava H M Some geealizaio ad basic (o - exesios of he Beolli Ele ad Geocchi polyomials Appl Mah Ifom Sci 5( [2] Sivasava H M K B ad Simse Y Some families of Geocchi ype polyomials ad hei iepolaio fcio Iegal Tas ad Special fc 23(22 [2] Sivasava H M Gag M ad Chodhay S A ew gealizaio of he Beolli ad elaed polyomials Rssia J Mah Phys 7( [22] Sivasava H M Gag M ad Chodhay S Some ew families of he geealized Ele ad Geocchi polyomials Taiwaese J Mah 5( [23] Sivasava H M ad Choi J Seies associaed wih he zea ad elaed fcios Klwe Academic Pblish Lodo (2 [24] Sivasava H M ad Pié A Remas o some elaioships bewee he Beolli ad Ele polyomials Appl Mah Lee 7( [25] Tembley R Gaboy S ad Fgée B J A ew class of geealized Aposol-Beolli polyomials ad some aaloges of he Sivasava-Pié addiio heoems Appl Mah Lee 24(

Spectrum of The Direct Sum of Operators. 1. Introduction

Spectrum of The Direct Sum of Operators. 1. Introduction Specu of The Diec Su of Opeaos by E.OTKUN ÇEVİK ad Z.I.ISMILOV Kaadeiz Techical Uivesiy, Faculy of Scieces, Depae of Maheaics 6080 Tabzo, TURKEY e-ail adess : zaeddi@yahoo.co bsac: I his wok, a coecio