Some results and conjectures about recurrence relations for certain sequences of binomial sums.

Transcription

1 Soe results ad coectures about recurrece relatos for certa sequeces of boal sus Joha Cgler Faultät für Matheat Uverstät We A-9 We Nordbergstraße 5 Joha Cgler@uveacat Abstract I a prevous paper [] I have obtaed soe results about recurreces of certa boal sus The a was to fd a eplaato of the rearable dettes ( ) + 5+ = F ad ( ) + 5 = F + for Fboacc ubers F by puttg these dettes to a ore geeral cotet These dettes have frst bee obtaed by I Schur [4] hs proof of the faous Rogers-Raaua dettes I the eate coputer eperets have led to several coectures about the cocrete for of these recurreces soe of whch are proved ths paper Soe geeral results We recall the followg ow fact (cf []): Theore Let be tegers ad The sequeces a ( z ) = z zz + + () satsfy lear recurreces of order wth costat coeffcets More precsely there est uquely detered polyoals ad ( ) ( ) = p s = + c s ()

2 p ( s ) = c( ) s () ( ) =+ wth teger coeffcets such that ( ) z p ( E ) a ( z ) = + ( ) z a ( z ) = z (4) where E deotes the shft operator defed by E f( ) = f( + ) Setch of the proof We observe as [] that the recurrece relatos for the boal coeffcets ply that t ( ) = + satsfes the recurrece relato t ( ) = t ( + ) + t ( + ) (5) If we troduce the shft operator K defed by K f( ) = f( ) the (5) ca be wrtte the for t ( ) = Et ( ) = ( K + K ) t ( ) = ( K + K ) t ( ) (6) Ths ples p E t = p K + K t (7) ( )( ) ( )( ) It turs out that order to prove (4) t suffces to show ( ) ( ) ( ) p ( + ) = + ( ) (8) = For the we get fro (7) ( ) p ( E ) t ( ( ) ) = t ( + ) + ( ) t ( ( ) ) (9) = for all

3 Ths eas that p E = ( ) ( ) + + ( ) = ( ) + ( ) + If we replace by ultply each sde by ( ) = + = z z ad su over all we get z p( E ) a ( z ) ( ) z a ( z ) Idetty (8) s equvalet wth Ths follows fro ( ) p ( + ) = + ( ) () = p( + ) = c( ) + = = ( ) = c ( + ) = p + = + = = For = we set ( ) ( ) ( ) p = ( s) ( ) The () reas true for = It s easy to verfy that the polyoals p ( s ) of the for () whch satsfy () are uquely detered Let ow b = c for ( ) ( ) ( ) ( ) ( ) < The () ay be reforulated as ( ) ( ) () ( ) + ( ) b ( )( + ) = ( + ) = ( ) =+

4 If we cosder the hooorphs whch seds + to a prtve root of uty ζ of order the t has bee show [] that () ples b ( ) = ( ζ ( ζ ) ) () = = Let s be the ' s eleetary syetrc fucto of ζ ( ζ ) < ad π the power su of those ubers Let (od ) be the reader odulo wth (od ) < (od ) The π = ( ) ζ = ( ) = = (od ) Newto s forula gves ( ) π s= ( ) s = Sce s ( ) = b we get ' s Theore Let d ( ) = ( ) ad let b ( ) be defed by (od ) b ( ) = d ( b ) ( ) wth b ( ) = ( (od )) = The for we have ( ) ( ) = + ( ) = ( ) ( ) = () p s c s b s ad ( ) ( ) = ( ) = ( ) ( ) ( ) =+ (4) p s c s b s 4

5 Cocrete evaluatos Now we wat to obta ore cocrete forato about the polyoals p ( s ) Frst I derve soe ow results (cf []) fro the preset pot of vew The case = For = forula () tells us that there are polyoals p ( ) s of the for c s whch satsfy It s easy to see that the sequece ( ) p ( ) begs wth Everyoe falar wth the Lucas polyoals edately guess that 5 p ( + ) = + (5) = wll L ( s) = s = (6) p ( ) = ( ) ad thus p ( ) ( ) s = L s holds It s well ow that the Lucas polyoals satsfy the recurrece L ( s) = L ( s) + sl ( s) wth tal values L ( s ) = ad L ( s) = Therefore t reas oly to show that p ( ) ( ) s = L s satsfes (5) e L ( + ) = + (7) Ths s of course a well-ow result It ay be proved the followg way: The polyoals L ( + ) satsfy the recurrece ( ) L ( + ) ( + ) L ( + ) + L ( + ) = E ( + ) E+ L ( + ) = + + If pe ( ) f( ) = we call p( z ) the characterstc polyoal of the sequece f ( ) Thus the characterstc polyoal of the sequece L ( + ) s z ( + ) z+ = ( z )( z ) Ths ples that the costat sequece ad the sequece ( ) satsfy the sae recurrece as ( L ( ) + ) Therefore we get E ( + ) E+ L ( + ) = ( )( ) wth tal values L + = = ad ( ) L ( + ) = ( + ) =

6 Ths ples that L ( + ) = for all as asserted Aother way to derve ths result s by cosderg the geeratg fucto ( + ) z p ( + ) z = ( + ) z = + = z z ( + ) z+ z are uquely detered we get the geeratg fucto of the Lucas polyoals z p ( s) z = z + sz Sce the polyoals We have thus show that L ( E ) a( z) = z+ a( z) (8) z holds Eg for = 5 we get L E = E E + E Therefore ( ) ( E 5E + 5E+ ) a( 5 ) = Now we have 5 ( E 5E + 5E+ ) = ( E+ )( E E ) The Fboacc ubers satsfy the sae recurrece Cosderg the tal values we get ( ) + 5+ = F ad ( ) + 5 = F + I [] we have also obtaed the geeratg fucto + F( ) + F ( ) ( ) z a z = (9) L ( ) z+ z where F ( s) = s = are the Fboacc polyoals As show [] ad [] for z =± there are always spler recurreces The forulas are dfferet depedg whether s odd or eve For eaple F ( ) + F ( ) F ( ) ( ) + a + = = + L+ ( ) + F+ F ( ) ( ) 6

7 Rear It should be oted that a ( + ) has a sple cobatoral terpretato (cf []) It s the uber of the set A ( ) of all lattce paths of legth whch are cotaed the strp < y< start at the org ad cosst of ortheast steps () ad + southeast steps ( ) Defe a pea as a verte preceded by a ortheast step ad followed by a southeast step ad a valley as a verte preceded by a southeast step ad followed by a ortheast step The heght of a verte s ts y coordate The peas wth heght at least ad the valleys wth heght at ost are called etreal pots Defe the d( v) weght of a path v by wvt () = t where dv () s the uber of etreal pots of the path v The weght of a set of paths s the su of the weghts of all paths of the set It has bee show [] that f we set = for < the the weght of the set A( ) s + ( ) + ( ) wa ( ( ) t) ( ) = t + For t = ths reduces by Vaderode s forula to ( ) + (+ ) For eaple wa ( ( ) t) = t = F + ( t) = As show [] wth other ethods for the polyoal wa ( ( ) t ) satsfes the recurrece relato ( ) F ( E + t E ) ( + E) F ( E + t E ) + EF ( E + t E ) w( A( ) t) = of order whch for t = reduces to the recurrece relato ( ) ( ) F ( E ) F ( E ) w( A( )) = F ( E ) F ( E ) a( + ) = + + of order It should also be oted that ( a ( 5 ) ) ( a ( 7 ) ) ad ( a ( 9 ) ) are the sequeces A45 A8495 ad A655 of The O-Le Ecyclopeda of Iteger Sequeces 7

8 The case = I [] I have obtaed eplct forulas for the recurrece relatos of the sus a ( z) = + + ) z Fro Theore we ow that there are polyoals v ( ) ad w ( ) such that v( E) zw( E) + ( ) z a( z) = z Coputer eperets led to the table v ( ) ( ) w A specto of these polyoals led to the coecture that they satsfy the recurreces v ( ) = v ( ) v ( ) () wth tal values ad wth tal values v v v ( ) = ( ) = ( ) = () w ( ) = w ( ) + w ( ) () w ( ) = w( ) = w ( ) = () It s ot dffcult to detere these polyoals eplctly: For > we have ad v ( ) = ( ) (4) w ( ) = (5) For the recurreces are easly verfed ad the tal values cocde for = 8

9 Thus we have eperetal evdece for Theore For the sequece a ( z): = + + z (6) satsfes the recurrece for all v E zw E a z + z a z = (7) z ( ( ) ( )) ( ) ( ( ) ) ( ) Rear I [] ufortuately soe typos corrupted the forulato of ths theore Wth the otatos troduced above we have oly to verfy that the polyoals p ( s ) are gve by v ( s ) ad w ( s ) e p ( s) = v ( s) = ( ) s (8) ad p( s) = w ( s) = s (9) Note that v ( ) = v ( ) ad w ( ) = w ( ) Wth other words we have to show that v ( + ) w ( + ) + ( ) = () These polyoals satsfy the recurreces v ( s) = v ( s) sv ( ) ad w ( s) = sw ( ) + s w ( ) for all f we set v ( s) = w ( s) = The sequeces v ( + ) ad the costat sequece satsfy the recurrece ( E ( ) E ) v ( ) = ad ( E ( ) E ) + + = ad the sequeces 9

10 w ( ) + ad ( ) satsfy ( E ( + ) E ) w ( + ) = ad ( E ( + ) E )( ) = Therefore f ( ) = v ( + ) w ( + ) + ( ) satsfes the recurrece ( E ( + ) E + )( E ( + ) E ) f ( ) = of order 6 Sce the frst 6 tal values are we get f( ) = for all It s aga structve to cosder the geeratg fuctos z p ( s) z = ad z + sz sz p( s) z = sz s z ( + ) z z Ths gves p ( + ) z = = + ( + ) z+ z z z z ad ( + ) z z + = = p ( ) z ( + ) z z + z z z These aga ply ( p( + ) + p( + )) z = = ( + ( ) ) z z + z I addto to Theore we copute the geeratg fucto b( ) + c( ) z+ ( ) z a ( z ) = () z v( ) w( ) + ( ) z z Coputer eperets suggest that ( ) b satsfes the sae recurrece as ( ) b( ) = b ( ) b ( ) The tal values are b( ) = b( ) = + b( ) = + + The geeratg fucto of these polyoals s + t+ t b + ( ) t = t + t v e Let = r ( ) t The t s easly verfed that r ( ) = t+ t = The frst values of the sequece ( r ( ) ) are Ths gves b ( ) = r( ) + r ( ) + r ( ) +

11 w The polyoals c ( ) satsfy the sae recurrece as c( ) = c ( ) + c ( ) The tal values are c ( ) = c ( ) = c ( ) = + The geeratg fucto of these polyoals s + ( ) t+ t c+ ( ) t = t t ( ) e ε ( ) ( ) = = Let = ( ) s t The s t t ε ( ) where ε ( ) (od) wth ε ( ) { } Ths ca easly be verfed by cosderg each of the cases s( ) s ( ) s+ ( ) separately The frst values of the sequece ( s( ) ) are Aother represetato s s( ) = [ + (od) ] = Fally we get c ( ) = s ( ) + ( ) s ( ) + s ( ) + I order to prove () we ote frst that z v( ) w( ) ( ) z a( z) + z s a polyoal d ( z ) of degree < For a polyoal p( ) P( p( )) = p = Therefore z d( z) = P v( ) w( ) a( z) = p let Sce o factor cotas egatve powers of z we see that the su of the frst ters of P v a( ) = d ( ) ( ) = a ( z) = a( z) = Let

12 The a ( z) = z z () = = Ths ples ( ( ) ( )) d ( ) d ( ) + d ( ) = P v ( ) a ( ) a ( ) + v ( ) a ( ) a ( ) = Sce the tal values cocde we get d ( ) = b( ) Net we detere the coeffcet of z of ( ) d z Fro () we get z v ( ) a( ) = = O the other had we have z ( ) ( )( ) w z z + = + + = z + = Ths ples ( ) z d z = For odd dces we get z + + w ( ) z z ( )( ) z ( ) = + = + = Ths ples ( ) z d z + = As above we verfy that p( ) = P w( ) = satsfes the recurrece p ( ) p ( ) p ( ) = e the sae recurrece as w ( ) I the sae way we get that q ( ) = P v( ) + satsfes the = sae recurrece Wrtg ( ) ( ) ( ) ( ) d z b c z = + + z we see therefore that c ( ) c ( ) c ( ) = fro whch equato () follows

13 Soe geeral observatos As a specal case of Theore we see that z + z a ( z ) = + z = z z Fro ( + ) = + ( )( + ) = p c ad the fact that ( ) + p ( + ) = ( ) for we see that () ples that all coeffcets p ( + ) ust vash e P p ( + ) = By these codtos p ( ) s s uquely detered + ( ) Ths observato ples that For ( ) p ( s) = ( ) s = ( ) () ( ) p ( + ) = ( ) ( + ) = ( ) r ad therefore the coeffcet of s gve by r ( ) r r ( ) = ( ) ( ( ) ) ( ( ) r+ ) = ( ) r r! = for all r such that r Let q ( ) = ( ) r = ( )( ) ( r+ ) Ths s a polyoal of degree r r r Wth ths otato the coeffcet of s ( E ) q( ) Sce r! ( E ) = ( + ) = ( ) + the operator E decreases the degree r of a polyoal by Therefore ( E ) q( ) = ad the asserto s proved r!

14 It s easly verfed that () satsfes the recurrece relato ( ) E E s p ( ) s = Now we observe that ( ) ( ) = = ( ) p s s We ow that d ( ) = ( ) (od ) Ths ples that d ( ) = ad d ( ) = for For ths case we have (od ) = ad therefore = ( (od )) Thus also b ( ) = d ( b ) ( ) = = Let ow < Fro b ( ) = d ( b ) ( ) = = d ( b ) ( ) + d ( b ) ( ) + d ( b ) ( ) = =+ we deduce that b = d = (od ) because the frst su vashes sce b ( ) = ad the secod su vashes because d ( ) = ( (od )) ( ) ( ) ( ) Therefore for ( ) ( ) < we get = = ( )(od ) ( ) ( + ( )(od )) + + ( ) ( ) ( ) ( ) b 4

15 Therefore we get ( ) ( ) p ( s) = ( ) (( ) s) ( ) (4) ( ) =+ I order to gve a sple descrpto of these results we troduce polyoals ( ) v ( s) = ( ) s ( ) (5) ad w s s ( ) ( ) = ( ) (6) where as always ths paper = for < These polyoals satsfy the recurrece relatos v ( s ) = v ( s ) sv( s ) (7) + + wth tal values v ( s) = ad ( ) v s = for ad w ( s ) = s w ( s ) + s w( s ) (8) wth tal values w ( s) ( ) ( ) w s = s + + = w ( s ) = for ad Let c ( s z ) be the characterstc polyoal of (the recurrece of) the sequece ( p s ) ( ) so that c ( s E) p ( s ) = The characterstc polyoal of v ( + ) s c z z z z z z z z z (9) ( + ) = ( ) ( ) = ( )( ) ad that of w + s ( ( ) ) 5

16 ( )( ) c z z z z z z z ( + ) = ( + ) ( ) = ( ) It s easly verfed that z c ( + z) = c( ( ) ) + z (4) Ths s accord wth the fact that the rght had sde of () has the characterstc polyoal cz ( ) ( z ) ( z ( ) ( ) = ) satsfes c ( ) = c( z) z z These results ca be forulated as Theore 4 For each we have ad p ( ) ( ) s = v s (4) p s = w s (4) ( ) ( ) ( ( ) ) 4 The case = 4 Now we wat to cosder the case = 4 ore detal We already ow that p ( 4 s) = ( ) s = ad 4 p s s 4 ( 4 ) = ( ) 4 = The polyoals p ( 4 ) s tur out to satsfy a recurso of order 6 stead of order 4 It s gve by p ( 4 s ) = sp( 4 s ) + s p( 4 s ) + s p( 4 4 s ) sp( 6 4 s ) (4) 6

17 wth tal values p (4 s ) = 6 p(4 s ) = p(4 s ) = sp (4 s ) = s p (44 s) = 6 s p (54 s) = 5 s To prove ths we observe that we ow already the geeratg fuctos 4 z 4 s z p ( 4 s) z = ad 4 z + p ( 4 s) z = 4 sz s z + sz Ths gves 4 ( + ) z z z ( 4 + ) = = + 4 p z ( + ) z+ z z z z z ad 4 ( + ) z + z+ z p( 4 + ) z = = + 4 ( + ) z + z z + z+ z z Furtherore we ow that p( 4 + ) = + p( 4 + ) p( 4 + ) Ths gves the geeratg fucto z z + z + z z z z + z+ z z ( 4 + ) = p z z + ( + ) z + z = 4 6 z ( + ) z z + z ad thus also sz + s z + s z p( 4 s) z = 4 6 sz s z s z + s z We dd ot fd a sple epresso for the polyoals p ( 4 s ) theselves 5 The case = 5 For = 5 we have foud the followg characterstc polyoals for p ( 5 + ) : c ( + z) = z ( + ) z + = ( z )( z z z z ) 5 c ( + z ) = z ( + z ) ( + ) z z ( + ) z + ( + z ) = ( z z z z )( z z ( ) z ( ) z ( ) z z ) c ( + z ) = z + ( + z ) ( + ) z + z ( + ) z + ( + z ) = ( z z z z )( z z ( ) z ( ) z ( ) z z ) 7

18 c z z z z z z z z ( + ) = ( + ) = ( + )( + ) Oce foud these dettes ca be verfed as above by provg () Soe ope probles Etesve coputer eperets suggest the followg facts: deg c ( + z) = ; for < the polyoal c ( + z) factors the for c ( + z) = v v where v = z ; for > the factorzato s c ( + z) = v v + ; for = we get c ( + z) = v v + ; deg v = ; for odd we have v = v ; v = z+ ( ) Furtherore t sees that as for = ad = also geeral c ( + z) s the oc polyoal proportoal to c ( + ( ) ) So t would suffce to copute v for < z But tll ow I could ot prove these geeral results Refereces [] J Cgler Recurreces for soe sequeces of boal sus Stzugsber ÖAW () 6-8 (also electrocally avalable [] J Cgler A class of Rogers-Raaua type recursos Stzugsber ÖAW (4) 7-9 (also electrocally avalable [] J Cgler Rears o soe sequeces of boal sus to appear Stzugsber ÖAW [4] I Schur E Betrag zur addtve Zahletheore ud zur Theore der Kettebrüche 97 Gesaelte Abhadluge Bd 7-6 Sprger 97 8