On Some Identities and Generating Functions for Mersenne Numbers and Polynomials

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1 Turish Joural of Aalysis ad Number Theory, 8, Vol 6, No, 9-97 Available olie at htt://ubsscieubcom/tjat/6//5 Sciece ad Educatio Publishig DOI:69/tjat-6--5 O Some Idetities ad Geeratig Fuctios for Mersee Numbers ad Polyomials Ali Boussayoud, Mourad Chelgham,*, Souhila Boughaba LMAM Laboratory ad Deartmet of Mathematics, Mohamed Seddi Be Yahia Uiversity, Jijel, Algeria LMAM Laboratory ad Deartemet of Mathematics, Freres Metouri Costatie Uiversity, Algeria Deartmet of Mathematics, Mohamed Seddi Be Yahia Uiversity, Jijel, Algeri *Corresodig author: chelghamm@yahoofr Received Jauary, 8; Revised March 7, 8; Acceted Jue, 8 Abstract I this aer, we itroduce a ew oerator i order to derive some roerties of homogeeous symmetric fuctios By maig use of the roosed oerator, we give some ew geeratig fuctios for Mersee umbers, Mersee umbers ad roduct of sequeces ad Chebychev olyomials of secod id Keywords: Mersee umbers, Geeratig fuctios, Symmetric fuctios Mathematics Subject Classificatio Primary 5E5; Secodary B9 Cite This Article: Ali Boussayoud, Mourad Chelgham, ad Souhila Boughaba, O Some Idetities ad Geeratig Fuctios for Mersee Numbers ad Polyomials Turish Joural of Aalysis ad Number Theory, vol 6, o (8): 9-97 doi: 69/tjat-6--5 Itroductio I this aer we cosider oe of the sequeces of ositive itegers satisfyig a recurrece relatio ad we give some well-ow idetities for this tye of sequeces [9] Oe of the sequeces of ositive itegers (also defied recursively) that have bee studied over several years is the well-ow Fiboacci (ad Lucas) sequece May aers are dedicated to Fiboacci sequece, such as the wors of Hoggatt i [], ad Koshy i [], amog others Others sequeces satisfyig a secod-order recurrece relatios are the mai toic of the research for several authors, such as the studies of the sequeces { J } N ad { j } N of Jacobsthal ad Jacobsthal-Lucas umbers, resectively I this aer we do ot have such id of geeraliatio, but we will follow closely some of these studies About the Mersee sequece, also some studies about this sequece have bee ublished, such as the wor of Koshy [], where the authors ivestigate some divisibility roerties of Catala umbers with Mersee umbers M as their subscrits, develoig their wor i [] I umber theory, recall that a Mersee umber of order, deoted by M, is a umber of the form, where is a oegative umber This idetity is called as the Biet formula for Mersee sequece ad it comes from the fact that the Mersee umbers ca also be defied recursively by M+ M +, () with iitial coditios M, M Sice this recurrece is ihomogeeous, substitutig by +, we obtai the ew form M+ M+ +, () Subtractig () to (), we have that M+ M+ M+ + (M + ) ad the M+ M+ M, other form for the recurrece relatio of Mersee sequece, with iitial coditios M, M The roots of the resective characteristic equatio are r r+, r ad r, ad we easily get the Biet formula M The mai urose of this aer is to reset some results ivolvig the Mersee umbers usig defie a ew useful oerator deoted by δ for which we ca formulate, exted ad rove ew results based o our revious oes [,,4] I order to determie geeratig fuctios of the roduct of Mersee umbers ad Chebychev olyomials of first ad secod id, we combie betwee our idicated ast techiques ad these reseted olishig aroaches

2 94 Turish Joural of Aalysis ad Number Theory Defiitios ad Some Proerties I order to reder the wor self-cotaied we give the ecessary relimiaries tools; we recall some defiitios ad results Defiitio [6] Let B ad P be ay two alhabets We defie S ( B P) by the followig form Πε P( t) S ( B Pt ), () Πbε B( bt) with the coditio S ( B P) for < Equatio () ca be rewritte i the followig form where, S B Pt S Bt S Pt S( B P) S j( P) Sj( B) j () We ow that the olyomial whose roots are P is writte as S( x P) S j( P) x, with card( P) j O the other had, if B has cardiality equal to, ie, B { x}, the () ca be rewritte as follows [6]: S ( x Pt ) P t) xt) S ( x P) + + S ( x Pt ) + t, xt) where S+ ( x P) x S( x P) for all The summatio is actually limited to a fiite umber of terms sice S have for all > I articular, we x S ( x P) P S Px S Px S Px ( ) + ( ) + ( ) +, where S ( B) are the coefficiets of the olyomials S ( x P ) for These coefficiets are ero for > For examle, if all P are equal, ie, P, the we have S ( x ) ( x ) By choosig P ie, P,,, we obtai + S( ) ( ) ad S () By combiig () ad (), we obtai the followig exressio S( B x) S( B) S ( B) x + S ( Bx ) + ( ) x Defiitio [4] Give a fuctio f o R, the divided differece oerator is defied as follows f(,, i, i+, ) f(, i, i, i, i ) i i ( f ) i i+ ee Defiitio The symmetriig oerator δ is defied by δ g g ( g) for all N (4) Proositio [5] Let P {, } a alhabet, we defie the oerator δ as follows ee ( ) ( ) ( ) g S + g + g ( ), for all N δ O the Geeratig Fuctios I our mai result, we will combie all these results i a uified way such that they ca be cosidered as a secial case of the followig Theorem Theorem Let A ad P be two alhabets, resectively, { a, a } ad { b, b }, the we have S( AS ) + ( P ) S ( + ) ( a+ a) δ ( ) aa δ ( ) S( A ) S( A ) () for all Proof By alyig the oerator to the series + f S ( A), we have + S A + + S( a+ a) S( a+ a)

3 Turish Joural of Aalysis ad Number Theory S ( a+ a) O the other had, S AS P + S ( A ) S A S A S( A ) S( A ) ( ) S( A ) S( A ) ( a+ a)( ) aa ( ) ( ) S( A ) S( A ) ( a+ a) aa S ( A ) S ( A ) S ( + ) a+ a δ ( ) aa δ ( ) S( A ) S( A ) Thus, this comletes the roof We ow derive ew geeratig fuctios of the roducts of some well-ow olyomials Ideed, we cosider Theorem i order to derive Mersee umbers ad Tchebychev olyomials of secod id ad the symmetric fuctios If ad A{,} we deduce the followig lemma Lemma [] Give a alhabet P {, }, we have S ( + ) () ( ) Relacig by ( ) i (), we obtai S ( [ ] ) () + ( )( + ) Choosig ad such that,, ad substitutig i () we ed u with S + [ ] +, which reresets a geeratig fuctio for Mersee umbers, such that M S ( + [ ] ) If, ad A { a, a} we deduce the followig theorems Theorem [7] Give two alhabets A { a, a} ad P {, } we have S( AS ) ( P ) ( a+ a) aa( + ) S( A ) S( A ) Theorem [8] Give two alhabets A { a, a} P {, } we have S( AS ) ( P ) aa S( A ) S( A ) From (5) we ca deduce S ( AS ) ( P ) aa S( A ) S( A ) Case : Relacig by ( ) ad ad (6) yields S( a+ [ a]) S ( + [ ]) a by ( a a) + aa( ) ( a)( a )( a )( a ) aa S ( a [ a ]) S ( [ ]) + + ( a)( a )( a )( a ) (4) ad (5) (6) a i (4) (7) (8)

4 96 Turish Joural of Aalysis ad Number Theory This case cosists of four related arts Firstly, the substitutios of i (7) give a a,, ad,, S( a+ [ a]) S ( + [ ]) +, which reresets a ew geeratig fuctio for roduct of Fiboacci umbers with Mersee umbers, such that FM S( a+ [ a]) S ( + [ ]) Secodly, the substitutio of a a,, ad,, i (8) give S ( a+ [ a]) S ( + [ ]) 4, which reresets a ew geeratig fuctio for Mersee umbers of secod order, such that + + M S ( a [ a ]) S ( [ ]) Thirdly, the substitutio of i (8) give a a,, ad,, S ( a+ [ a]) S ( + [ ]) + 4, which reresets a ew geeratig fuctio for roduct of Jacobsthal umbers with Mersee umbers, such that JM S ( a+ [ a]) S ( + [ ]) Fially, the substitutio of i (8) give a a,, ad,, S ( a+ [ a]) S ( + [ ]) which reresets a ew geeratig fuctio for roduct of Pell umbers with Mersee umbers, such that PM S ( a + [ a ]) S ( + [ ]) Case : Relacig by ( ) ad a by a ad a by ( a ) i (4) yields S( a+ [ a]) S ( + [ ]) ( a a) + 4aa( ) ( a)( a )( a )( a ) + + The substitutio of,, 4aa, (9) i (9) ad set for ease o otatios x a a, we reach + + S ( a [ a]) S ( [ ]) MU ( x ) x 6 x + (5 + 8 x) x which corresods to a ew geeratig fuctio for the combied Mersee umbers ad Tchebychev olyomials of the secod id Theorem 4 For, the ew geeratig fuctio of the roduct of Mersee umbers M ad Tchebychev olyomials of first id is give by x + x MT ( x ) 4 6 x + (5 + 8 x) x + 4 Proof We see that MT( x) S ( a+ [ a] ) S ( + [ ] ) xs ( a [ a] ) + S ( + [ ] ) S( a+ [ a] ) x S ( + [ ] ) S ( a+ [ a] ) MU ( x ) x S ( + [ ] )( ( a) ( a) ) ( a+ a) MU ( x ) S ( + [ ] )( a ) x ( a+ a) S ( + [ ] )( a ),

5 Turish Joural of Aalysis ad Number Theory 97 O the other had, we ow that S + [ ] + from which it follows, MT( x) x 4 6 x + (5 + 8 x) x + 4 x a a +, ( a + a) 6a+ 8a + 6a+ 8a Therefore M T x Acowledgemets x + x 6 x + (5 + 8 x) x The authors would lie to tha the aoymous referees for their valuable commets ad suggestios Refereces [] A Boussayoud, M Kerada, N Harrouche, O the -Lucas umbers ad Lucas Polyomials, Turish Joural of Aalysis ad Number5() -5, (7) [] A Boussayoud, M Bolyer, M Kerada, O Some Idetities ad Symmetric Fuctios for lucas ad ell umbers, Electro J Math Aalysis Al 5(), -7, (7) [] A Boussayoud, O some idetities ad geeratig fuctios for Pell-Lucas umbers, OlieJ Aal Comb -, (7) [4] A Boussayoud, N Harrouche, Comlete Symmetric Fuctios ad - Fiboacci Numbers, Commu Al Aal, , (6) [5] A Boussayoud, M Boulyer, M Kerada, A simle ad accurate method for determiatio of some geeralied sequece of umbers, It J Pure Al Math8, 5-5, (6) [6] A Boussayoud, A Abderrea, M Kerada, Some alicatios of symmetric fuctios, Itegers 5, A#48, -7, (5) [7] A Boussayoud, M Kerada, R Sahali, W Rouibah, Some Alicatios o Geeratig Fuctios, J Cocr Al Math, -, (4) [8] A Boussayoud, M Kerada, Symmetric ad Geeratig Fuctios, It Electro J Pure Al Math 7, 95-(4) [9] P Catario, H Camos, P Vasco, O the Mersee sequece, A Math Iform46, 7-5, (6) [] V E, Hoggatt, Fiboacci ad Lucas Numbers A ublicatio of the Fiboacci Associatio Uiversity of Sata Clara, Sata Clara, Houghto Miffli Comay, 969 [] T Koshy, Fiboacci ad Lucas Numbers with Alicatios, Joh Wiley, New Yor, [] T Koshy, Z Gao, Catala umbers with Mersee subscrits, Math Sci 8, 86-9 ()