SOME DOUBLE BINOMIAL SUMS RELATED WITH THE FIBONACCI, PELL AND GENERALIZED ORDER-k FIBONACCI NUMBERS

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43 Number SOME DOUBLE BINOMIAL SUMS RELATED WITH THE FIBONACCI PELL AND GENERALIZED ORDER-k FIBONACCI NUMBERS EMRAH KILIÇ AND HELMUT PRODINGER ABSTRACT We cosider some double biomial sums related to the Fiboacci ad Pell umbers ad a multiple biomial sum related to the geeralized order-k Fiboacci umbers The Lagrage-Bürma formula ad other well-kow techiques are used to prove them 1 Itroductio The geeratig fuctio of the Fiboacci umbers F is F x x = 1 x x =0 Similarl the geeratig fuctio of the Pell umbers P is P x = =0 x 1 x x The geeralized order-k Fiboacci umbers f k are defied b f k = k i=1 f k i for >k with iitial coditios f k j = j 1 for 1 j k For example whe k = 3 the geeralized Fiboacci umbers f 3 reduced to the Triboacci umbers T defied b 010 AMS Mathematics subject classificatio Primar 05C38 15A15 Secodar 05A15 15A18 Kewords ad phrases Fiboacci umbers geeratig fuctios Lagrage- Bürma formula Received b the editors o April ad i revised form o October DOI:10116/RMJ are Copright c 013 Rock Moutai Mathematics Cosortium 975

2 976 EMRAH KILIÇ AND HELMUT PRODINGER T = T 1 + T + T 3 with T 1 =1T =adt 3 =4for>3 For these umber sequeces we recall the combiatorial represetatios due to [ 3 5]: i=1 i = F i 1 1/ i = P i +1 i=0 i j = F +3 j i 0 1 j Amog the formulas the last formula seems to be differet from the first above sice it icludes double sums see [] The authors of the above-cited papers use a combiatorial approach to prove these results For ma similar idetities we refer to [6] I this paper we shall derive some ew double biomial sums related with the Fiboacci Pell ad geeralized order-k Fiboacci umbers ad the use the Lagrage-Bürma formula ad other well-kow techiques to prove them The Lagrage-Bürma formula is a ver useful tool if oe kows a series expasio for x but would like to obtai the series for x i terms of We recall the formula for details see [1 4]: Suppose a series for i powers of x is required whe = xφ Assume that Φ is aaltic i a eighborhood of = 0 with Φ0 0 The x = /Φ = a a 1 0 =1 The the two equivalet versios of the Lagrage-Bürma iversio formula ca be writte as F =F 0 + =1 x! [ d 1 d 1 F Φ ] x=0

3 SOME DOUBLE BINOMIAL SUMS 977 or F 1 xφ = =0 x [ d F Φ! d ] x=0 We would like to rephrase this usig the otatio of the coefficiet-of operator: F 1 xφ = [ ] F Φ x ; =0 we will use it i this form Double biomial sums We start with a result related to Fiboacci umbers: Theorem 1 For >0 F 4 1 = 0 ij + i + j j i Proof Westartfrom + i [ j ]1 + +i = j ad compute + j S = 1 + +i i i=0 = + j i/ i i i 0 [ = 1+ j+ + 1 ] j+ 1 + ; here the desired sum takes the form: [ [ j ] 1+ j+ + 1 ] j+ 1 + j=0

4 978 EMRAH KILIÇ AND HELMUT PRODINGER = [ j ] 1+ j [ j ] 1 j+ 1 + = [ j ] 1+ j/ j + [ j ] 1 j+ 1 + Let us cosider the first sum: [ j ] 1+ j/+ 1 + This is of the form with F = [ j ]F Φ j ad Φ = 1+ The Lagrage-Bürma formula ca ow be applied to this sum The geeral formula is give b: [ j ]F Φ j x j = F 1 xφ We eed the istace x = 1 here ad the variables x ad are liked via = xφ Notice that Φ must be a power series i with a costat term differet from zero Therefore b the solutio of =Φ we fid = α =1+ 5/ ad so = 1+ 5 Fα = Φ α = Φ α =

5 SOME DOUBLE BINOMIAL SUMS 979 So our evaluatio is The secod term is [ j ] 1+ j/ j This is the istace x = 1 which traslates to = 1 ad so the secod term is F 1 1+Φ 1 =0 The last sum is [ j ] 1 j+ 1 + = j+ 1 [ j ] j+ 1 + = j+ 1 [ j ] 1 + This is agai of the form [ j ]F Φ j with F = ad Φ = 1

6 980 EMRAH KILIÇ AND HELMUT PRODINGER We eed the istace x = 1 here ad the lik is 1 = x B the solutio of the last equatio we fid = β where β = 1 5/ ad so we write ad = β = F β = 1 1 Φ α = So our evaluatio is Altogether [ ] = α4 1 β 4 1 = F as desired Theorem For >0 F 4+1 = 1 ij +1 + i + j j 1 i 1 Proof Sice [ j 1 ] 1 + +i + i = j 1

7 SOME DOUBLE BINOMIAL SUMS 981 ad +1 + j S = 1 + +i i 1 i=1 = + j i+1/ i i i 0 [ = 1+ j+ 1 ] j / ; here the desired sum takes the form: +1 [ [ j 1 ] 1+ j+ 1 ] j / j=1 = j 1[ j 1 ] 1+ j / j 1[ j 1 ] 1 j / = [ [ j ] 1+ ] j/++1/ / 1 1 j j 1[ j 1 ] 1 j / Let us start with oe term i the above sum: [ j ] 1+ j/++1/ / This is of the form: with F = [ j ]F Φ j 1+ +1/ /

8 98 EMRAH KILIÇ AND HELMUT PRODINGER ad Φ = 1+ This is the istace x =1whichbα =1+ 5/ traslates to ad = 1+ 5 Fα =α 4+ Φ α = So our evaluatio is: 1 1 Φ α = α 4+ 5 The secod term is: [ j ] 1+ j/++1/ / 1 j This is the istace x = 1 which traslates to = 1 adsothe secod term is: F 1 1+Φ 1 =0 Fiall the last term is of the form: [ j 1 ] 1 j / j 1 = j+ 1 [ j 1 ] j / j 1 = j+ 1 [ j ] / This is of the form: [ j ]F Φ j

9 SOME DOUBLE BINOMIAL SUMS 983 with F = ad Φ = 1 This is the istace x = 1 which traslates to = β =1 5/ Thus F β = β 4+ Φ β = F β 1 Φ β = 1+ 1 β 4+ 5 So our evaluatio is: [ 1 1 α ] 1 β = F 4+1 as claimed Theorem 3 For >0 F 4 = F 4 3 = + i + j j 1 i + i + j j +1 i +1 i=0 j=0 i=0 j=0 Agai b usig the Lagrage-Bürma formula Theorem 3 ca be similarl proved

10 984 EMRAH KILIÇ AND HELMUT PRODINGER Theorem 4 For >0 F + + F +1 = 0 ij i j j i Proof First we replace i b i ad get i j j i j 0 j i Now we compute the geeratig fuctio of it: i j z j i j 0 0 j i = i z i j 1 z i+1 j 0 j i = z j 1 z j 1 z 1+j = 1 z 1 z z 1 3z + z = z z z z which is the geeratig fuctio of the umbers F + + F +1 / The followig results are similar: Theorem 5 For >0 F = F 1 = i j j 1 i 1 i i j j i=1 j=1 0 j i Theorem 6 For >0 1 F +1= F i 1 = i=0 0 i j i j j i

11 SOME DOUBLE BINOMIAL SUMS 985 Proof Multiplig the right had side of 1 b z ad summig over weget S = i j z j i 0 0 i j = h + j j z h+i+j j i 0 i j h 0 = j z h + j i+j z h i j 0 i j h 0 = j z i+j 1 i 1 z j+1 0 i j = i 0 = z 3i 1 z i+1 = 1 z 1 z1 3z + z z 1 3z + z z which is the geeratig fuctio of the umbers F +1 For the Pell umbers we give the followig result: Theorem 7 For 0 P +1 = 0 i j i j j i Proof Multiplig the right had side of b z ad summig over weget S = 0 = z 0 i j h 0 i j j i h + j j z h+i+j j i 0 i j

12 986 EMRAH KILIÇ AND HELMUT PRODINGER = 0 i j = 0 i j = 0 i j j i j i z h + j i+j z h j h 0 z i+j 1 z j 1 z j+1 1 z j+1 j i z i = z j 1 + zj 1 z j+1 = z 1 z1 + z/1 z = 1 1 z z This is the geeratig fuctio of the umbers P +1 Now we give a double sum for the Triboacci umbers: Theorem 8 For 0 T = 0 j i i i j i j j Proof Cosider T z = 0 0 j i = 0 j i i i j z i j j i j h z i z h j i j h 0 = i j z i z i j j 1 z i j+1 0 j i = h z h+j z h j 1 z h+1 h 0 Let t = z /1 z ad we cotiue with T z = 1 z h j t h = 1 1 z j 1 z 0 0 j h 0 0 j z j t j 1 t j+1

13 SOME DOUBLE BINOMIAL SUMS 987 = 1 1 z t 1 zt/1 t = 1 1 z 1 1 t zt = z 1 z /1 z z 3 /1 z = 1 1 z z z 3 which is the geeratig fuctio of the Triboacci umbers as expected So the proof is complete B usig the same proof method as i Theorem 8 we get a more geeral result: Theorem 9 For >0 f k = 0 i k i 1 i1 i1 i ik 1 i k i 1 i i i 3 i k where f k is the th geeralized order-k Fiboacci umber REFERENCES 1 GE Adrews R Aske ad R Ro Special fuctios Ec Math Appl 71 Cambridge Uiversit Press Cambridge 1999 AT Bejami ad JJ Qui Proofs that reall cout Mathematical Associatio of America Washigto DC J Ercolao Matrix geerator of Pell sequece FiboQuart P Herici Applied ad computatioal complex aalsis Vol 1 Joh Wile & Sos Ic New York AF Horadam Pell idetities Fibo Quart R Kott Fiboacci ad golde ratio formulae uk/hostedsites/rkott/fiboacci/ 7 J Riorda Combiatorial idetities Joh Wile & Sos Ic New York 1968 TOBB Uiversit of Ecoomics ad Techolog Mathematics Departmet Akara Turke address: ekilic@etuedutr Departmet of Mathematics Uiversit of Stellebosch 760 Stellebosch South Africa address: hprodig@suacza