q-fibonacci polynomials and q-catalan numbers Johann Cigler [ ] (4) I don t know who has observed this well-known fact for the first time.

Transcription

1 -Fiboacci polyoials ad -Catala ubers Joha Cigler The Fiboacci polyoials satisfy the recurrece F ( x s) = s x = () F ( x s) = xf ( x s) + sf ( x s) () with iitial values F ( x s ) = ad F( x s ) = These polyoials are itiately related to the Catala ubers C = + : The polyoials F + ( x ) are a basis of the vector space of polyoials If we defie the liear fuctioal L by LF ( + ( x )) = [ = ] (3) we get + Lx = ad Lx = C (4) I do t ow who has observed this well-ow fact for the first tie I this ote I wat to give a overview about soe ore or less ow geeralizatios of this result The -Fiboacci polyoials of Leoard Carlitz A well-ow aalogue of the Fiboacci polyoials which apparetly has first bee itroduced by L Carlitz [4] ad has further bee studied i [6] ad [8] is F ( x s ) = s x = (5) It satisfies the recurrece F( xs ) = xf 3 ( xs ) + sf ( xs ) F( xs ) = F( xs ) = (6) The polyoials F + ( x ) are a basis of the vector space [ x ] of all polyoials i x whose coefficiets are ratioal fuctios i We ca therefore defie a liear fuctioal L o [ x ] by LF ( + ( x )) = = (7)

2 We defie coefficiets a ( ) by Fro x = af ( ) + ( x ) (8) = + + ( ) + ( ) ( )) + af ( ) ( x ) = x x = a ( ) xf ( x ) = a F x + F x = F ( x ) a( ) + a( + ) we get a( ) = [ = ] a ( ) = a ( ) a( ) = a( ) + a( + ) (9) Fro this we get a ( ) = Lx If we set the we have ˆ = ˆ ˆ + + xf ( x) F F Fˆ ( x) = F ( x ) () Defie ow the ubers a ˆ = L( x F + ) They satisfy a = = a = a + a + where a = if < Therefore we have a = a( ) i other words a ( ) = LxF ( ˆ + ) () These ubers have a obvious cobiatorial iterpretatio: Cosider all oegative lattice paths i which start i () with upward steps () ud dowward steps ( ) We associate to each upward step edig o the height the weight ad to each dowward step edig o the height the weight The weight of the path is the product of the weights of all steps of the path The a ( ) is the weight of all lattice paths fro () to ( )

3 It is clear that a(+ ) = Let a( ) = C The we have = C = C C () To obtai this recurrece decopose each lattice path fro () to ( ) ito the first path which returs to the x axis ad the rest path The first path goes fro () to ( + ) ad cosists of a risig seget followed by a path fro () to ( ) (but oe level higher) ad a fallig seget The weight of each of the dowward steps o this higher level is ties the correspodig weight of the steps of the oral level This shows that C( ) is a aalogue of the Catala ubers C = + which satisfy the recurrece C = C C (3) = This aalogue of the Catala ubers has bee itroduced by Carlitz ad Riorda [3] It is well-ow that i the classical case = all a ( ) are give by a closed forula More precisely we have + a( ) = + + (4) ad + + a(+ + ) = + + (5) This is euivalet with + a ( ) = = + (6) For o such explicit forulas are ow For sall values of it is easy to copute a ( ) I geeral this is a polyoial i [ ] of order But there are o ow forulae for a ( ) Probably these seueces ad especially ( C ) are ot -holooic but I do t ow if this has already bee proved If we write F ( x ) = p( ) x (7) + = the we have p( + ) = p(+ ) = ad + p( ) = ( ) (8) ad 3

4 + + p(+ + ) = ( ) Fro (8) we see that the atrix (9) ( ai ) ie ( ) i = is the iverse of ( pi ) ( ) i = ai ( ) p ( ) = I () i = i = There is also aother coectio with Fiboacci polyoials Defie the geeratig fuctio A () s = a( ) s () The Carlitz ad Riorda [3] have show that A s F+ s A s F s () = ( ) () ( ) () This ca easily be proved by iductio First we observe that fro a( ) = a( ) + a( + ) = a(( ) ( ) ) + a( ( + ) + ) we get A() s = sa () s + A+ () s or A s A s sa s () = () () (3) For sall values of forula () is obviously true The we get + + = ( ) = + F ( s ) sf( s ) A( s) F( s ) sf A () s A () s sa () s A () s s A () s = F ( s ) A ( s) F ( s ) s( F ( s ) A ( s) F ( s ) ) = ( + ) ( s ) = F ( s ) A ( s) F ( s ) + + Forula () ca be stated i the followig for ( ) s a( ) s s C = + = = 4

5 Coparig coefficiets we get for = C ( ) = = (4) This is also a iediate coseuece of () if we iterpret the left had side as LxF ( ( x )) Rear + Carlitz [4] has show forula (8) i the euivalet for = ( ) + ( ) for = x c F x soe polyoials c It is easy to verify that c = a( ) He coputed the first values but apparetly did ot otice that c = a( ) = C although soe te years earlier he ad Riorda [3] already had studied C( ) The -Catala ubers of George Adrews George Adrews [] has foud a aalogue of the Catala ubers where the a ( ) have a closed forula ad where there exists a aalogue of the recursio (3) too There exist correspodig Fiboacci polyoials (cf[6]) Let + 4 t = (5) + + ( + )( + ) Defie polyoials by the recurrece f ( xs ) = xf ( xs ) t ( 3) sf ( xs ) f ( x s) = f ( x s) = (6) It ca easily be verified that 4 s f x s x ( + )( + ) + ( ) = = + = (7) 5

6 We also eed the polyoials f ( xs ) which satisfy They are give by f ( xs ) = xf ( xs ) t ( ) sf ( xs ) f ( x s) = f ( x s) = (8) + 4 s f ( x s) = x (9) = + + ( + )( + ) = The the polyoials f+ ( x) are a basis of the vector space [ x ] of all polyoials i x whose coefficiets are ratioal fuctios i We ca therefore defie a liear fuctioal L o [ x ] by Now we defie coefficiets a ( ) by L( f+ ( x)) = = (3) x = a ( ) f ( x) (3) = + Fro a ( ) f ( x) = x x = a ( ) xf ( x) = a ( ) f ( x) + t ( ) f( x)) + = f ( x) a( ) + t a( + ) we get a( ) = [ = ] a ( ) = t() a ( ) a ( ) = a ( ) + ta ( + ) (3) Fro this we get a ( ) = Lx Let ˆ f ( ) x f x = t() t() t( ) The we have xfˆ ( x) = t( ) fˆ + fˆ + 6

7 Defie ow the ubers a ˆ = L( x f + ( x)) They satisfy a = [ = ] a = a + t a + where a = if < Therefore we get a ( ) = Lx ( fˆ ( x)) (33) + Fro (3) it ca easily be verified that + [ + ] + a( ) = + + [ + + ] + + ( + )( + ) [ ] + = 4 ( ) = + ( ) = (34) ad [ ] a(+ + ) = ( ) == = ( )( ) = (35) If we write f ( x) p( ) x + = (36) = the we have p( + ) = p(+ ) = ad p( ) = = = = + (37) ad = p(+ + ) = ( + )( + ) (38) 7

8 = Fro (3) we have agai that ( ai ) ( ) i is the iverse of ( pi ) ( ) i = Therefore we ow that a( ) p( ) = [ = ] (39) = This eas + [ + ] = [ + + ] ( ) + ( + ) = + = + or [ ] [ ] ( ) ( ) = [ = ] = Replacig by + we get + [ + + ] + + ( ) ( ) = [ = ] = [ ] If we let we get the euivalet forula + [ + + ] + + ( ) = [ = ] = [ ] or [ + ]! [ + + ] [ ]! [ ]!! + = = = [ + + +! ] Therefore (39) is euivalet with ( ) = [ = ] = [ + ]! [ ] + + ( ) ( ) = [ = ]! (4) Let ow + c = a( ) = = + [ + ] [ ] + + ( + ) ( + ) = = (4) 8

9 This ca also be writte i the for c = ( ) ( + ) + (4) The Vaderode forula gives ( ) ( ) = = Defiig hz = z (43) we coclude that hzhz = + z (44) Therefore the geeratig fuctio f ( z) = c z satisfies + + f ( z) = 4z = h( 4 z) 4z 4z + ( + ) ( + ) + (45) 4 z For = this reduces to the well-ow forula Cz = z Fro 4z 4 z f ( z) f ( z) 4z = + + we get f( z) + f( z) 4 = + zf ( z) f ( z) + ( + ) (46) Coparig coefficiets we obtai the recurrece relatio 4 c c c (47) + + = + ( ) = with iitial value c ( ) = 9

10 Defie ow the geeratig fuctio A () s = a( ) s The we get as above a( ) = a( ) + t a( + ) = a(( ) ( ) ) + t a( ( + ) + ) ad therefore A() s = sa () s + t A+ () s or t ( ) A( s) = A ( s) sa ( s) (48) This iplies t() t() t( ) A ( s) = f ( s) A ( s) f ( s) (49) + This relatio obviously holds for sall I geeral we have t()() t t ( )( ta ) ( s) = t()() t t ( ) A( s) sa ( s) + f ( s) A( s) f ( s) t( ) s( f( s) A( s) f + ( s) ) = = f ( s) t( ) sf ( s) A ( s) f ( s) t( ) sf ( s) = f ( s) A ( s) f ( s) Coparig the coefficiet of s we get c t ( + )( + ) + 4 = = + = = (5) This is also a coseuece of (33) 3 Aother -aalogue of the Fiboacci polyoials Fially we cosider the aalogue studied i [7] ad [9] Fib ( x s) = s x = which has bee If we set Fib ( x ) p( ) x = =

11 the we have p( + ) = p(+ ) = ad + ( ) = ( ) p (5) ad + + ( + + ) = ( ) p (5) Let ow L be the liear fuctioal defied by LFib ( + ( x )) = = (53) Let The we get ad x = a( ) Fib + ( x ) (54) = [ ] [ ] + a( ) = + + [ ] [ ] + + a(+ + ) = + + (55) (56) As a special case we obtai Lx = [ + ] (57) To prove this we observe that (54) iplies that ( ai ) ( ) i = is the iverse of ( pi ) ( ) i = I order to show (55) it thus suffices to show that = or = [ ] [ ] [ ] [ ] ( ) = = ( ) = [ = ] But this is euivalet with (4)

12 The sae arguet proves (56) = = [ ] [ ] [ ] [ ] ( ) = = = = It is easy to verify that (54) (55) ad (56) are euivalet with Fib ( x ) x + = = (58) This idetity follows also fro a iversio forula by L Carlitz [] He uses the fact that + d ( ) = ( ) = = (59) for This is a iediate coseuece fro the recursio = if we write it i the operator for ( U) = where U = If we iterate this we get ( U ) ( U ) = = + Sice the left had side ca be writte i the for ( ) U = we get (59) His iversio forula reads as follows: Let u = v (6) the + = v u (6) To prove this observe that

13 ( ) + v + v v = = ( ) = (6) Now for < + + = = +!! ( ) =![ ]![ ]!! = ( ) = = ad + + = = +![ ]! ( ) =![ ]![ + ]![ ]! = ( ) = = + Therefore all sus ( ) = with > vaish ad (6) is proved If we choose u = x the + + ( ) ad we get v = x = Fib x Fib+ ( x ) = x By coparig coefficiets we see that for all s s Fib+ ( x s) = x This proves (58) Rear I the case of the Catala ubers of Carlitz ad Adrews the correspodig Fiboacci polyoials are orthogoal with respect to the liear fuctioal L This follows iediately fro () ad (33) 3

14 For i the first case we get ˆ LF ( + ( x F ) + ( x )) = L p ( ) xf ( x) + = a ( ) p ( ) = [ = ] = = ad i the secod oe ( ˆ L f+ ( x) f+ ( x)) = L ti px ( ) f+ ( x) = ti a ( ) p ( ) = ti [ = ] i= = i= = i= With respect to the fuctioal L defied i (53) the polyoials Fib ( x ) are ot orthogoal sice eg L( Fib( x ) Fib4( x )) = L( xfib4( x )) = ( ) More geerally we have LxFib ( ( )) x = for This follows fro the recurrece 3 4 Fib( x ) = xfib ( x ) xfib 3( x ) + Fib 4( x ) which has bee proved i [9] For this iplies LxFib ( + ( x)) = LxFib ( ( x)) for There are ay other aalogues of the Catala ubers (cf []) A siple class are the Pólya-Gessel Catala ubers cs ( ) defied by cs ( ) = c ( s ) + s csc ( ) ( s ) = with iitial value c( s ) = For these ubers we defie Fiboacci-lie polyoials ϕ ( xs ) by ϕ( xs ) = xϕ( xs ) t ( 3) ϕ( xs ) with iitial values ϕ ( xs ) = ad ϕ ( xs ) = Here t is defied by t = t(+ ) = s As is show i [5] the orthogoal polyoials Φ ( x s ) with respect to the liear fuctioal L defied by the oets Lx = cs ( ) are Φ ( x s) = ϕ( + x s) = + ( ) x s = = + For = ad s = this reduces to ( ) x = For other aalogues I do t ow if there are aalogues of the Fiboacci polyoials which are i the sae relatio as i the above cases 4

15 Refereces [] GE Adrews Catala ubers -Catala ubers ad hypergeoetric series J Cob Th A 44(987) [] L Carlitz Soe iversio forulas Red Circ Palero (963) [3] L Carlitz ad J Riorda Two eleet lattice perutatio ubers ad their - geeralizatio Due J 3(964) [4] L Carlitz Fiboacci otes 4: -Fiboacci polyoials Fiboacci Quart 3(975) 97- [5] J Cigler Eiige -Aaloga der Catalazahle Sitzugsber ÖAW 9 () [6] J Cigler -Fiboacci polyoials Fiboacci Quarterly 4 (3) 3-4 [7] J Cigler Eiige -Aaloga der Lucas- ud Fiboacci-Polyoe Sitzugsber ÖAW () 3- [8] J Cigler Soe algebraic aspects of Morse code seueces Discrete Matheatics ad Theoretical Coputer Sciece 6 (3) [9] J Cigler A ew class of -Fiboacci polyoials Electroic Joural of Cobiatorics (3) R 9 [] J Fürliger ad J Hofbauer -Catala ubers J Cob Th A 4 (985)