ࡱ> q` 1 bjbjqPqP .::1L L L L $p < ~ ~ ~ ~ ~ ~ ~ $>h^~ ~ ~ ~ .~ ~ ~ j*L A. 0<o.~  hx T G~ ~ ~  ~ ~ ~ < Int. J. of Pure and Appl. Mathematics, 53(1) (2009), 69 - 81 ON GENERALIZED DIFFERENCE OPERATOR OF THIRD KIND AND ITS APPLICATIONS IN NUMBER THEORY M.Maria Susai Manuel EMBED Unknown , G.Britto Antony Xavier EMBED Unknown , V.Chandrasekar EMBED Unknown , R.Pugalarasu EMBED Unknown  and S.Elizabeth EMBED Unknown   EMBED Unknown Department of Mathematics, Sacred Heart College, Tirupattur - 635 601, TamilNadu, S.India e-mail:manuelmsm_03@yahoo.co.in  EMBED Unknown Department of Mathematics, Auxilium College, Vellore, Tamil Nadu, S.India. Abstract In this paper, the authors extend the theory of the generalized difference operator  EMBED Unknown  to the Generalized difference operator of the second kind  EMBED Unknown  and third kind  EMBED Unknown  for the positive reals  EMBED Unknown  and  EMBED Unknown . We also present the discrete version of Leibnitz Theorem, Binomial Theorem, Newton's formula with reference to  EMBED Unknown . Also by defining its inverse, we establish a few formulae for the sum of first and second partial sums of the higher powers of arithmetic progression in number theory. AMS Classification [2000]: 39A Key words: Generalized difference operator, generalized polynomial factorial, Partial sums. >   ) * ܹܔo]J$j˻GN h*h*B*UVphf"jh*h*B*EHUphf$jʻGN h*h*B*UVphf"jh*h*B*EHUphf$jɻGN h*h*B*UVphf"jh*h*B*EHUphf$jȻGN h*h*B*UVphfjh*h*B*Uphfh*h*B*phf h*5\h*h*5B*\ph3fh*=>P ~  ! A B C L  / 0 $dha$gd*`gd*dhgd*$ hdh^ha$gd*gd* $dha$gd*dhgd*dhgd*1 * + , - 1 < = L M N O P Q ` a b c ˻ހm[ހL<jh*h*B*Uphh*h*B*mHphfsH"j h*h*B*EHUphf$jλGN h*h*B*UVphfh*h*B*phf"j2 hph*B*EHUph$jͻGN hph*B*UVphhph*B*phjhph*B*UphhphpB*phhph*jh*h*B*Uphf"jh*h*B*EHUphf " A C L      # $ % & > ? N Ͽ|qg\Rjh*EHUjһGN h*UVjh*EHUjѻGN h*UVjh*EHUjлGN h*UVjh*Uh*h*h*B*phh*mHsHh*h*B*mHphsHjh*h*B*Uph"jlh*h*B*EHUph$jϻGN h*h*B*UVphh*h*B*phN O P Q V W f g h i 0 1 ޾ޭޟޛh_h*h*5B*\ph h*5\jv h*EHUjջGN h*UVjh*EHUjԻGN h*UVh*jh*Ujh*EHUjӻGN h*UV0 1 ,1h/ =!"#$% Ddx,F  3 A?"?2iBaӦ6`>T +?Rce1ν$n~;9a$8ZG%q%Zb?qQ0Bh~P|b<!$|N"ATYyX9.D=.(0Ku/wp+眶UU<^}_t}P"EDX $ҹcoR*Ӆlq vfn7TmNqϸRTvjerjԅlQ766z57,3KNyoL3e9ni)zI& XmOZi+Ԣ`ih1 i԰@$UDp'+R?.!P4!3NTUS"r_+*<F9{Dd,F  3 A?"?2ŒvrE* K`!ŒvrE* K4x]QJA=3I<`P +$`j Ho,B%zg*80gνܝs/. yqu%d ?y>b` BkS d%fH5Ʌ f?Uh;&\1ypD+5jE5m盏 mUǏ~J۠lz!Q$Jz~{WO0x\lZC3fvmY%j#3+fBfUZ"PQ睖8fZjK%): 2cގCiD:΀3t%Bh;=5$YEZ粘:xF1-or~$R; H.e}}clwd7yDd+F  3 A?"?2`2D(*Y^`!`2D(*Y4]Z5x]QJA=3I<`P fWP+ <&!eUyH$`#GXXI%zg\2g}=x BgE;#s !FhyB<<&%|AnTys\X*E=.4IuO:w0duӬ <^Tw:z)li"DHMY=]ϖ [mZlW3hYmD|{F<~}YPmU"͗Ye")-2628EA V2t8@Fʶ=kY#gOWB0yh0SUAqM.Y|[0&^Ώ$KvǪ=R Y"j_^vTE< czHDd+F  3 A?"?2I2{#ELuGv `!I2{#ELuG4]Z5x]QJA=3I<`P fWP+ nd->] 1q:xM"8KITKW0&NJ.UzTe}}blwxd7'zDd,F  3 A?"?2E=Plb `!E=Plbdtx]QAOAfJQ&R%bCmiRnlVYmj8/$GN/38HNDͰք}7߼'0 8Zq"BBas4_\J^ %Z40Pci2h?}r(*oԇ@CABDEFHGILJKOM]PQ\STUVWXYZ[^`abcdefghiRoot EntryS F*GData #WordDocumentR.ObjectPoolUj**_1313323976Fj*j*Ole CompObjiObjInfo !"#&)*+,-./013458;<=>?@ABCEFGJMNOPQRSTUWXY\_`abcdefghjklmpstuvwxyz{}~ FMathType 5.0 Equation MathType EFEquation.DSMT49q4 u.  &N & MathType  $ Times New Roman؟ww wf- OlePres000 \Equation Native _1313323977 Fj*j*Ole 2 1yu&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   1 & "Systemf !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   1 FMathType 5.0 Equation MathType EFEquation.DSMT49q4 u.  &NCompObj iObjInfoOlePres000 \Equation Native   & MathType  = Times New Roman؟ww w f- 2 2yu&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   2 & "System f !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   2 FMathType 5.0 Equation MathType EFEquation.DSMT49q_1313323978Fj*j*Ole $CompObj%iObjInfo'OlePres000(\Equation Native 2_1313323979 .Fj*j*Ole 64 u.  &N & MathType  6 Times New Roman؟ww w f- 2 3yu&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   3 & "System f !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   3 FMathType 5.0 Equation MathTyCompObj7iObjInfo9OlePres000:\Equation Native Dpe EFEquation.DSMT49q4 u.  &N & MathType  = Times New Roman؟ww wv f- 2 4yu&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   4 & "Systemv f !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   4_1313323981Fj*j*Ole HCompObj IiObjInfoK FMathType 5.0 Equation MathType EFEquation.DSMT49q4 u.  &N & MathType  6 Times New Roman؟ww w fI- OlePres000!L\Equation Native V_1313323982($Fj*j*Ole Z2 5yu&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   5 & "System fI !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   5 FMathType 5.0 Equation MathType EFEquation.DSMT49q+d 2.  &NCompObj#&[iObjInfo]OlePres000%'^Equation Native i & MathType  $ Times New Roman؟ww w f^-2 1,2,3,4hBy;rB&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   1,2,3,4 & "System f^ !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   1,2,3,4 FMathType 5.0 Equation MathTy_1313323983*Fj*j*Ole nCompObj),oiObjInfoqpe EFEquation.DSMT49q4 u.  &N & MathType  6 Times New Roman؟ww w f- 2 5yu&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'OlePres000+-r\Equation Native |_1313323984":0Fj*j*Ole _!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   5 & "System f !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   5 FMathType 5.0 Equation MathType EFEquation.DSMT49q{ Py.  @&N & MathType` :Symbol >؟ww w f- CompObj/2iObjInfoOlePres00013Equation Native 2 Dy5 MT Extra h؟ww w f- 2 lyy&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l & "System f !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l FMathType 5.0 Equation MathType EFEquation.DSMT49q_13133239856Fj*j*Ole CompObj58iObjInfoOlePres00079Equation Native &_13133239864F<Fj*j*Ole  .  ``&N  & MathTypep `Times New Roman؟ww w f - 2 12@ Times New Roman؟ww w f - 2 ,2:Symbol ؟ww w f - 2 D25 MT Extra Ԙ؟ww w f - 2 ll&MathTypeUU DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l 1 ,!l 2  & "System  f !-  DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l 1 ,!l 2  FMathType 5.0 Equation MathType EFEquation.DSMT49qCompObj;>iObjInfoOlePres000=?Equation Native C .  `&N@ & MathTypep `Times New Roman؟ww w f- 2 123s!@ Times New Roman؟ww w f- 2 ,,&:Symbol E؟ww w f- 2 D,5 MT Extra L؟ww w f- 2 lllE&&3MathTypeUU'DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l 1 ,!l 2 ,!l 3  & "System f !-' DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l 1 ,!l 2 ,!l 3 _1313323987BFj*j*Ole CompObjADiObjInfo FMathType 5.0 Equation MathType EFEquation.DSMT49q{$ .  @`&N  & MathType`  Times New Roman؟ww w fm- 2 12gTimes New RomaOlePres000CELEquation Native _1313323988@LHFj*j*Ole n؟ww w fm- 2 ,24MT Extra ؟ww w fm- 2 ll&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  !l 1 ,!l 2 & "Systemm fm !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  !l 1 ,!l 2 FMathType 5.0 Equation MathTyCompObjGJiObjInfoOlePres000IKEquation Native pe EFEquation.DSMT49q{ Ox.  @&N` & MathType`  Times New Roman؟ww w f- 2 3y4MT Extra ؟ww w f- 2 lyx&MathTypeUUDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  !l 3 & "System f !- DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  !l 3 FMathType 5.0 Equation MathType EFEquation.DSMT49qR )  '."System'_1313323989NFj*j*Ole CompObjMPiObjInfoOlePres000OQzEquation Native C1Table_SummaryInformation(T     /DF9 - @Times New Roman-&,--.  @Times New Roman-. jTimes New Roman- 2 j123ILS~Times New Roman- 2 ~,,MtdSymbol- 2 dD'~PMT Extra- 2 ~PlllEMt"System------'"System|8OZ9-'''- 2 d,& '--' DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  D !l 1 ,!l 2 ,!l 3 Oh+'0h 2aUN'@˼S `!aUN'@˼S 4]Z5x]QJA=3I<`P fWP+ <&!eUyHFFďVɪగ9sk ~gG8c.bq6؂O\1BS `&+*tBțrTY/qf-|qՑK&/pZjꩻͺGLU?{7)&f@$Tuޕ| >-) WmZlW2Ch[mD|{F<~}Y#WmUp"͗Ye"-66689EWA V2t8@Fʎ=oY#gOWB`ih1zӨH8SR?.&U4i˥~cU "rGoQLN6{Dd,hF  3 A?"?2,ćt3M7fA`!ćt3M7f۠@$|xuRMOQ=MeCb@3&JIHظ N?Gɔή7(&,F~ ~ _aGʸ2qo:%Fܼλ=>$T  MeDID"8zD q^Z<gd2֐aCL]OҨzbI'dtK1Dw\e)ejAyi'.{Mg/o8!g",jʈ -шیnۅm ׋$~\8JPEV/ nKZo4VAVIfbU!"si-<:ZXqpn֮S·Q(23? 5,5VCFX 6LT{zs ۼ#~MTΝF-MxwW :]sMf|xzaDd|F   3 A ?"?2mR ,MOe1zv^I`!AR ,MOe1zv^``t;xڝS;oAfC@&R*.# ;+1sY`!QNȝn>+1s`xڝSMo@Yp,9 D)F&SĂ"HNSA! !*-'38U 3qZ(G;;;o[Y ?L (w DZŖ'-N0/ HQddk )ڑWPwk]`WVz3x}[<D9HM/07yլi~fWiNc^M J̡k8FQ`'_hvrE܏Qx( T{ͻ-"I9b.Qʁ2a7&\3ǝvtS13d6 8%TTQk;A^u߫zՓL+1Zl=j76'!8:jf契k ]-\+YÆ uQa`i(@Qfx.P|-NȿmH@AuFJQ O(ا}~+SBZ֍~m݋|-X6]heH[ڣ}(E%pV᠊XCj\_jZ=_gi $nƣߑ'jսn XۍmCTVdgjz@9Q<#҅ȵ5iG gv|t7stel6XD `'U1K< 7( 5ܓHMQ\=ZfΘ:ډ߷J^:LdqXev&Ñ"T\=fK Ϣ?DdhF   3 A ?"? 2!+2kEy`!+2kEy@4[}xuROP9)D8?ZUDJ-E"ׄ.$b4nqK$TIP-#j0vkW0gvBm`&@ժ'3aP6BZ] M%K2=b,:9}aUE[qv2!t8`Rp$+펶٨[{8|9c״ad湚a1&dk=!vB$m7-cןMP!;~7D QZ.VVmpU{o%B*7Rv_͚ݔHELc\_z__X3 ;kG=+&GI5 }8s+s,x11 l1MkӼAA)K[. IoOnm1)d\0j_iv7W "{Dd|F   3 A ?"? 2}Nȝn>+1sY `!QNȝn>+1s`xڝSMo@Yp,9 D)F&SĂ"HNSA! !*-'38U 3qZ(G;;;o[Y ?L (w DZŖ'-N0/ HQddk )ڑWPwk]`WVz3x}[<D9HM/07yլi~fWiNc^M J̡k8FQ`'_hvrE܏Qx( T{ͻ-"I9b.Qʁ2a7&\3ǝvtS13d6 8%TTQk;A^u߫zՓL+1Zl=j76'!8:jf契k ]-\+YÆ uQa`i(@Qfx.P|-NȿmH@AuFJ*DA@D aDefault Paragraph FontRi@R  Table Normal4 l4a (k@(aNo List ROR aStyle1$ dha$B*CJOJQJph>P`> * Body Text 2$dha$DQ`D * Body Text 3$dha$5\1=>P~!ABCL/03000000@0@0@0000000000000=>P~!ABCL/03K00K00K00K00K00K00K00K00K00K00K00K00K00K00K00K00I00I00I00I00 001* N 1  0 1  1  )+<LNP`b#%>NPVfh1::::::::::::: hO<iOļjO|"kOlO!mOnO|oOpOg"qOt3 3 =*urn:schemas-microsoft-com:office:smarttags PlaceType=*urn:schemas-microsoft-com:office:smarttags PlaceName8 *urn:schemas-microsoft-com:office:smarttagsCity9 *urn:schemas-microsoft-com:office:smarttagsplace hJ   "@3"@3P~033)0:9a;F? p[ ap?{)n"*>9,Q7k=Q'AHI'bINR_3j%r%z%l~*mX6l~ Hod.9nsmz_"+ ox_(##P@Ic3w_I$@OO