Fibonacci polynomials, generalized Stirling numbers, and Bernoulli, Genocchi and tangent numbers

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1 Fiboacci polyomials, geeralied Stirlig umbers, ad Beroulli, Geocchi ad taget umbers Joha Cigler Abstract We study matrices hich trasform the sequece of Fiboacci or Lucas polyomials ith eve idex to those ith odd idex ad vice versa. They tur out to be itimately related to geeralied Stirlig umbers ad to Beroulli, Geocchi ad taget umbers ad give rise to various idetities betee these umbers. There is also a close coectio ith the Aiyama-Taigaa algorithm. Sice such umbers have bee extesively studied it is possible that some of these results are already hidde i the literature. I ould be very grateful for such iformatio.. Itroductio Let ( F () s ) be the sequece of Fiboacci polyomials. The geeratig fuctios of the subsequeces of polyomials ith eve ad odd idex are related by F () s e F () s = (+ )! e + (+ )! If e associate ith each sequece x = ( x ) the sequece y ( y ) y e x + + = (+ )! e + (+ )! e obtai a ifiite triagular matrix A ( a (, )) = such that. = defied by y = ax (, ). = The matrix A begis ith

2 The above metioed idetity of geeratig fuctios implies that ad that if e deote by i+ i A= [ i] [ i] + a (, ) = ( ) G + G the positive Geocchi umbers. + The startig poit of this paper as the observatio that the eigevectors of A are sequeces of cetral factorial umbers T (, ). More precisely the eigevector correspodig to the eigevalue +,, is ( T( +, + ) ). I other ords A ca be ritte i the form A= T( i+, + ) D T( i+, + ), here D deotes the diagoal matrix ith etries,, 3,. This fact is equivalet ith the idetity e T( +, + ) + T( +, + ) + ( ). = + e + (+ )! (+ )! If e defie liear fuctioals ϕ + o the vector space of polyomials i s by the ad ϕ + F + () s = T( +, + ) ϕ + F + ( s) = ( + ) T( +, + ) ( ϕ ) (, ), + s = LS here LS(, ) are the Legedre-Stirlig umbers hich have bee studied by G.E. Adres ad Littleoh [] ad by Y. Gelieau ad J. Zeg [8]. These facts lead to some iterestig idetities such as = ( ) ( )! T(, ) = G

3 or = (! ) T B ( ) ( +, + ) = ( + ). + To all these formulae the Aiyama-Taigaa algorithm is applicable. Aother property of A is the fact that it ca be ritte as i + i + A= [ i] [ i]. i i + This leads amog other thigs to e proofs of Seidel's idetity for Geocchi umbers ad of Kaeo's idetity. The aim of the preset paper is to give a systematic accout of these ad related results.. Some ell-o facts We cosider (a variat of) the Fiboacci polyomials defied by F () s = s. = (.) They satisfy the recursio F( s) = F ( s) + sf ( s) ith iitial values F ( s ) = ad F() s =. The first terms are,,,,, 3,43,56, 6 4,. The correspodig Lucas polyomials are defied by L () s = s = (.) ad satisfy the same recurrece as the Fiboacci polyomials, but ith iitial values L ( s ) = ad L. s = The first terms of this sequece are,,, 3, 4,55,69,. 3

4 Let + + 4s α = (.3) ad + 4s β = (.4) be the roots of the equatio s =. The for s the ell-o Biet formulae give 4 α β F () s = α β (.5) ad L () s = α + β. (.6) For s = e have istead 4 F = 4. (.7) It is also ell o that L( s) = F+ ( s) + sf ( s). This follos immediately from s = αβ ad + + α β αβ α β = α α β + β α β Sice deg F ( s) = deg F ( s) = deg L ( s) = deg L ( s) = each of the sets { F ( s) },{ F ( s) },{ L ( s) },{ L ( s) } is a basis for the vector space of polyomials i s. The Geocchi umbers G. =,,3,7,55,73,387,99569, (cf. OEIS A5 ) ad their relatives (,,,,, 3,,7,, 55, ) g their expoetial geeratig fuctio = (cf. OEIS A36968) are defied by e = + = g = + ( ) G, e e!! + + (.8) 4

5 ad the taget umbers are defied by Note that by comparig coefficiets e get T + =,,6,7,7936,35379,36856, (cf. OEIS A8) e e = ( ). (.9)! + + T + + g+ G+ T+ ( ) = = (.) It is also ell-o that umbers defied by G = ( ) ( ) B, here ( B ) is the sequece of Beroulli B = for > ith B =. The list of Beroulli umbers begis ith,,,,,,,,,,,,,,,,, B (.) = For later uses let us also recall the geeratig fuctio of the Beroulli umbers B =. (.)! e. Coectio costats The folloig theorem gives a explicit computatio of some basis trasformatios. Theorem. The bases ( F + ) ad F + are coected by G + + F+ () s = () F+, = + (.) ad + B F+ () s = F+ (). s = + + (.) 5

6 The bases cosistig of Lucas polyomials are coected by T + + G + + L+ () s = () L () s () L() s + = = = + (.3) ad B L() s = L + (). s = + (.4) Proof ) Sice α+ β = e have This gives or α β ( β) ( α) α β e e = e e = e e e. (.5) F() s F() s = e ( )!! (.6) + e!! + F() s = F+ () s (.7) + This reduces our tas to fidig the matrix hich correspods to the operator "multiplicatio by + e " of the correspodig expoetial geeratig fuctios. Comparig coefficiets e see that for x = ( x) ad y = ( y) is equivalet ith y= Mx + e x = y!! (.8) here M ( mi (, ) ) >. = ith m (, ) = ( ) 6 for <, m (, ) = ad m (, ) = for

7 The iverse of (.8) is ( ) g x = y = g + y = y! e! ( )!!! =, thus Thus (.7) implies F () s () + = x g F + g + = ( ) y. = + (.9) ad e get as special case g + + G + + F+ () s = F () F, + + = + + = + i.e. (.). From α β ( β) ( α) α β e + e = e + e = e e + e (.) e get i the same ay as before I this case (.9) implies (.3). + e!! L() s = L () s. (.) ) Aother cosequece of (.5) is No F () s F () s!! ( e ) =. (.) ( e ) x = ( ) x = y!! =! is equivalet ith 7

8 y + = ( ) x. = (.3) The iverse is b x y y! = ( e )! =!! here b = B for ad b () =. This follos from e b = B + = + = =.!! e e e This implies x b = y. = + (.4) From (.) e see that ith x = F( s) ad y = F( s), y =, e get + + b + + b + b F + () s = y() y F, = + = = + = = + + i.e. (.). L () s L () s.! (+ )! + + (.) implies ( e ) = Therefore e get from (.) ith x = L( s) ad y = L+ ( s), y+ =, b B L () s = y() L (), s = + = = + + i.e. (.4). The matrix correspodig to "multiplicatio by ( e ) " of the geeratig fuctios is i i + C = ( ) [ i]. i, = (.5) Its iverse is give by 8

9 i bi ( ) = [ ]. + C i i, = (.6) Here [ P ] = of property P is true ad [ P ] = else. If e apply C to the colum vector y ith etries y+ = F+ ( s), y =, the the etries x of x = C y are x = F ( s). For eve this is trivial because i x b y F F = + = = = + + the oly o-vaishig term occurs for =. For odd e get the idetities (.). If e apply C to the colum vector y ith etries y = L( s), y + =, the the etries x of x C y = are x = L ( s). For eve e get (.4). For odd the correspodig idetity is agai trivial. The matrices (.5) ad (.6) are itimately coected ith Stirlig umbers. 3. Stirlig umbers ad cetral factorial umbers 3.. I order to mae the paper self-cotaied I shall first state some ell-o facts about Stirlig umbers ad some geeraliatios (cf. [4]). Let ( ) = be a icreasig sequece of positive umbers. Defie the Stirlig umbers of the secod id S (, ) by S (, ) = S (, ) + S (, ) (3.) ith iitial values S (, ) = [ = ] ad S (,) = () ad the Stirlig umbers of the first id s (, ) by s (, ) = s (, ) ( ) s (, ) (3.) ith s (, ) = [ = ] ad s (,) = ( ). = This is equivalet ith 9

10 s (, ) x = ( x ())( x ()) ( x ( )), (3.3) = S (, ) ( x ) = x (3.4) = = ad S x (, ) x =. ( () x)( () x) ( x) (3.5) Let i, S = S ( i, ) (3.6) ad i, s = s ( i, ). (3.7) The it is clear that s = S. (3.8) The same holds for the ifiite matrices S = ( S (, i ) ) ad s = i, ( s i ) i, (, ). 3.. For S (, ) of the secod id. = e get the Stirlig umbers s (, ) of the first id ad the Stirlig umbers Note that the matrices ( Si (, ) ) ad i, (, ) i, ( s (, i ) ) i, respectively for = +. (, i ) si are the same as i, S ad Cosider the ifiite matrix C ( c(, i ) ) i, = ith for > i, hich has bee defied i (.5). i ci (, ) = ( ) i + for i ad ci (, ) =

11 Theorem 3. The matrix C ca be factored i the folloig ay: i+ i C = [ i] [ i] i, i, = ( +, + ) ( + )([ = ] ( +, + ). ( Si ) ( i i ) ( si ) i, i, i, (3.9) Proof This theorem ca be derived from the folloig simple idetities. The idetity i i+ C[ i] = [ i] i, i, (3.) is aother formulatio of the trivial result ( ) = + = = = for. The idetity i [ i] S( i, ) = S( i+, + ) i, i, i, (3.) is a ell-o property of the Stirlig umbers ad easily proved by iductio: S(, ) S(, ) S(, ) S(, ) S(, ) = + = = = = = = S (, + ) + S(, ) S(, ) = + = = S (, + ) + S (, ) + S (, + ) = S ( +, + ). Fially e have

12 i + [ i] ( S( i, ) ) = ( S( i+, + )( + ) ). i, i, i, (3.) This also is easily proved by iductio: + S (, ) = S (, ) + S (, ) = S ( +, + ) + S ( +, ) = = = = = S ( +, + ) + S(, ) S(, ) = + = = S ( +, + ) + S ( +, ) S (, ) + S ( +, + ) S( (, )) = ( + ) S( +, + ) + S( +, ) S(, ) S(, ) = ( + ) S( +, + ). To derive (3.9) observe that (3.) gives i+ i C = [ i] [ i], i, i, (3.) gives i, i i + [ i] = Si ( +, + )( + ) Si (, ) ad (3.) gives i,, i [ i] = ( S( i, ) ) (( S( i+, + ) ),, ). i i i, Combiig these idetities gives (3.9). Aother cosequece of (3.) ad (3.) is i i+ [ i] [ i] = ( S( i, ) ) (( i+ )([ i = ] ) ( s( i, ) ). i, i, i, i, i, (3.3)

13 3.3. If e choose = e get the cetral factorial umbers t (, ) = s (, ) of the first id ad the cetral factorial umbers T(, ) = S (, ) of the secod id respectively. These umbers have bee itroduced i [] ith a differet otatio. Further results ca be foud i [4], Exercise 5.8. The folloig tables sho the upper part of the matrices of cetral factorial umbers. See also OEIS A Ti = = ( (, )) 6 i, ti = = ( (, )) 6 i, 3.4 For = ( + ) e get the Legedre-Stirlig umbers LS(, ) ad ls(, ) studied i [] ad [8] (cf. OEIS A795 ad A9467). LS i = = ( (, )) 6 i, ( ls(, i )) 6 = i, =

14 There are some iterestig relatios betee cetral factorial umbers ad Legedre-Stirlig umbers hich are aalogous to the correspodig results about Stirlig umbers. Theorem 3. ad Proof LS (, ) = T ( +, + ) = (3.4) + LS (, ) = ( + ) T ( +, + ) = (3.5) For < + all terms vaish. For = + e have = ad = + + LS(, ) = = ( + ) = ( + ) T ( +, + ) LS(, ) = = = T ( +, + ). No assume that (3.4) ad (3.5) are already o up to. The LS(, ) LS(, ) = + = = ( ) ( ) + = LS(, ) LS(, ) = + + = ( ) ( ) ( ) + = LS(, ) + ( + ) LS( = =, ) + LS(, ) = = T (, ) + ( + ) T (, + ) + ( + ) T(, + ) = T ( +, + ). I the same ay e get 4

15 + LS(, ) LS(, ) = + = = ( ) + = LS(, ) LS(, ) = + + = ( ) + ( ) + = LS(, ) + ( + ) LS(, ) + = = T( +, + ) = T + + T + + T + + = + T + + (, ) (, ) (, ) (, ). Corollary 3.3 [ ] i i LS i, = T( i+, + ) ( ) (3.6) ad [ ] i + i ( LS( i, ) ) = ( T( i+, + ) )([ = i]( + ) ). (3.7) Corollary 3.4 i + i () [ i] [ i] = T( i +, + ) [ i = ]( + ) T( i +, + ). (3.8) ad i i + i i = + ( LS i )( i ) ( LS i ) [ ] [ ] (, ) [ ] (, ). (3.9) Proof i+ i [ i] [ i] () = T ( i +, + ) [ i = ]( + ) LS( i, ) LS( i, ) T ( i +, + ) I the same ay e get (3.9). Aother iterestig result is 5

16 Theorem 3.5 The folloig idetities hold: + T( +, + ) = LS( +, + ) = (3.) ad + T( +, + ) = ( + ) LS( +, + ). = + (3.) Equivaletly this meas i + [ i] ( T( i+, + ) ) = ( LS( i+, + ) ) i i i i,,, (3.) ad i + [ i] ( T( i+, + ) ) = ( LS( i+, + ) ) ([ i = ]( + ) ). i, i, i, i + i, (3.3) Proof For < both sides vaish. For = e get ad + respectively. + T(, ) T(, ) T(, ) + + = = = = = T(, ) T(, ) = ( ) = ( ) = T( +, + ) + ( + ) LS(, = ( ) + ) = T(, ) ( ) T(, ) ( ) LS(, ) ( ) ( ) = = = LS(, ) + (( + ) + ( + )) LS(, + ) = LS( +, + ). For the secod sum e get 6

17 + T(, ) T(, ) T(, ) + + = = + = + = = T(, ) T(, ) = ( ) = ( ) = T( +, + ) + = ( ) + T( +, + ) = ( ) = T(, ) ( ) T(, ) ( ) ( ) + + = + = + + T(, ) ( ) T(, ) = ( ) = ( ) + + = LS (, ) + ( + ) ( + ) LS (, + ) + LS (, ) + ( + ) LS (, + ) = ( + ) LS (, ) + ( + )( + ) LS (, + ) = ( + ) LS ( +, + ). Corollary 3.6 i + i + [ i] [ i] i i + = i, i, ( Ti ( +, + ) ) ([ i= ]( + ) ) ( Ti ( +, + ) ) i, i, i, (3.4) ad i + i + [ i] [ i] i + i = i, i, ( LS i + + ) ( i = + ) ( LS i + + ) (, ) [ ] (, ). i, i, i, (3.5) Proof The left-had side of (3.4) equals ( ) LS( i+, + ) [ i = ]( + ) T( i+, + ) T( i+, + ) LS( i+, + ). i, i, i, i, i, I the same ay the left-had side of (3.5) equals 7

18 ( ) Ti ( +, + ) LSi ( +, + ) LSi ( +, + ) [ i= ]( + ) Ti ( +, + ). i, i, i, i, i, Comparig (3.4) ith (3.8) e see that i + i i + i + [ i] [ i] = [ i] [ i] i i + i, i,. (3.6) This implies i + i + i + i [ i] [ i] = [ i] [ i] i i + i, i, ( LS i ) ( i ) ( LS i ) = ( +, + ) [ = ]( + ) (, ). i, i, i, (3.7) Remar Michael Schlosser has sho me a simple direct proof of the first idetity i (3.7). It suffices to sho that =. + = = This is equivalet ith = + + = +. This follos from the Chu-Vadermode formula: ( ) = + + i i = ( ). i= i i = = = + i= + i i + 8

19 3.5. We later eed the folloig result. Lemma 3.7 Let () = ad ˆ = ( + ). The ˆ S (, ) = S ( +, + ) S (, + ) (3.8) ad If for some fuctio F() the ˆ s = s + = (, ) (, ). (3.9) F(, ) = S (, ) F s (, ) = F() + S (, ) F s (, ) (3.3) = = ˆ S F s F F = = ˆ ( + ) = ( + ) (, ) (, ) (, ) (, ). (3.3) Proof The first assertios follo from S ˆ (, ) x + x ( x) x = = ( () x)( () x) ( ( + ) x) x ( () x)( () x) ( ( + ) x) = + = ( x) S (, ) x S (, ) S (, ) x ad ˆ s (, ) x = ( x ()) ( x ( )) = ( x ) s (, ) x = = or + ˆ s (, ) x = s ( +, ) x. x = = This implies 9

20 ˆ ˆ S (, ) F + s (, ) = S ( +, + ) S (, + ) F + s ( +, ) = = = = S ( +, + ) F + s ( +, ) + S (, + ) F + s ( +, ) = = = = + = S ( +, ) F s (, ) + S (, ) F s (, ) = = = = = ( F F ) = (, ) ( +, ). 4. Some iterestig matrices 4.. We o already that F( s) F( s) = e.!! (4.) This ca also be ritte i the form F() s e F+ () s + =. (4.)! e + (+ )! If e defie a liear fuctioal λ o the vector space of polyomials i s by e get λ F [ ], + s = = (4.3) e λ( F ( s)) =.! e + (4.4) Usig (.8) this gives (cf. [5],[3]) λ( F ( s)) = ( ) G. (4.5) By comparig coefficiets e get agai (cf. [3]) F () s = a(, ) F () s (4.6) + + = ith

21 + a (, ) = ( ) G +. + (4.7) We call the ifiite matrix A= ( a(, i ) ) i, ad the fiite parts A a i i = matrices. E.g. = (, ) Geocchi, A5 = (4.8) It is clear that A is the matrix versio of "multiplicatio ith geeratig fuctios. More precisely e have e e " of a certai id of + Propositio 4. Let x x x, = x y y y = y ad A= ( a i ) i, (, ). The y = Ax is equivalet ith For later uses e ote that + e x + y =. (4.9) (+ )! e + (+ )! This follos from (4.6) for s =. a (, ) =. (4.) = For special values of s (4.6) gives some iterestig idetities. Cosider for example s =. ( F ( s)) =,,,,,, is periodic ith period 6. Here

22 This gives for example a(3+,3 ) = a(3+,3 + ). = = For s = e get F = 4 4. Therefore = 4 ( + ) a(, ) = +. By comparig coefficiets of s e get + a (, ) = = for. This is equivalet ith the matrix idetity i+ i A= [ i] [ i]. (4.) If e recall that F s F 3 i [ i] s F5 = i, 3 s F7 ad F s F 4 i+ [ i] s F6 = i, 3 s F8 e see that (4.) is the same as (4.6). By (3.8) e get the folloig represetatio By (3.6) e also have A ( T i )( i )(( T i )) = ( +, + ) [ = ]( + ) ( +, + ). (4.) i+ i+ A = [ i] [ i] i i + i, i,. (4.3) Thus e get

23 Theorem 4. The Geocchi matrix A has the folloig represetatios i+ i A= [ i] [ i], i+ i+ A = [ i] [ i] i i + i, i, (4.4) (4.5) ad A= T( i+, + ) [ i = ]( i+ ) t( i+, + ), (4.6) i, i, i, here ti (, ) are the cetral factorial umbers of the first id ad Tithe (, ) cetral factorial umbers of the secod id. Note the aalogy ith (3.9) ad (3.). (4.5) implies i+ i+ [ i] A = [ i] i i + i, i,. By cosiderig the first colum of these matrices e get = + ( ) G + = [ = ]. Slightly reformulated this is Seidel's idetity for Geocchi umbers ([3]) = = G [ ]. = (4.7) i The matrix [ i] has bee evaluated by Dumot ad Zeg [6]. We do't eed this result. We are oly iterested i the first colum. Let media Geocchi umbers (cf. OEIS A5439). H + =,,, 8, 56, 68, be the 3

24 Lemma 4.3 The elemets of the first colum of i [ i] are the umbers ( ) H +. The first to colums of i i+ [ i] [ i] are. The umbers of the secod colum are 8 r() = H + =, r() = H + =, r() = H =, r(3) = H = 8, r(4) = H = 56, ad i geeral for For example r = λ( s) = ( ) H+. i 3 [ i] = i, i i+ 3 [ i] [ i] = i, i, To prove this lemma e eed the fact (cf. [3],[5]) hich ca serve as defiitio of the media Geocchi umbers that G +. (4.8) = = H + 4

25 + + ( s ) = F+ ( s) = ( ) G+ = by (4.5) e see that Sice λ λ By (4.6) e get λ s H+ = ( ). (4.9) F () s i F3 ( s) s [ i] = F5 () s s (4.) ad therefore by applyig λ i λ( s) [ i]. = λ( s ) To prove the secod assertio e ote that F () + = ad therefore (4.) gives the first colum. To obtai the secod colum e recall that H λ ( F s ) = ( ) + = + = ad therefore (3.9) gives = r = + ( ) =. i i + i i = + ( LS i )( i ) ( LS i ) [ ] [ ] (, ) [ ] (, ). If e delete the first ro ad first colum e get the matrix ( LS i ) ( i ) (( LS i )) ( +, + ) [ = ]( + ) ( +, + ). (4.) i, i, i, 5

26 Theorem 4.4 Defie liear fuctioals ϕ by ϕ (, ) F s = T for. The the folloig formulae hold: ( ϕ ) (, ), + s = LS (4.) ϕ ϕ F () s = LS(, ) = T( +, + ) = (4.3) F () s = LS(, ) = ( + ) T( +, + ). = (4.4) + + Proof This is a immediate cosequece of Theorem 3.. There is a simple Seidel-array for computig LS(, ) i terms of T ( +, + ). Theorem 4.5 Let h( i,, ) = T( i+, + ), h(i+,, ) = ( + ) T( i+, + ), i hi (,, ) = hi (,, ) hi (,, ) if, i hi (,, ) = if >. (4.5) The h(,, ) = h(+,, ) = SL (, ). (4.6) For example for = this array begis ith 6

27 Proof It suffices to sho that h( i,, ) = ϕ ( F i + s s ) ad h(i,, ) ϕ ( F i + s s ) i h( i, i, ) = F( s) s = ϕ ( s ) = LS( i, ) ad i For the e have ϕ i i + = ϕ+ ( ) = ϕ+ = + + h(i, i, ) F( ss ) ( s) LSi (, ). For = this is clear by our defiitio. We have oly to verify that h( i,, ) = h( i,, ) h(i,, ) ad h(i+,, ) = h(i+,, ) h( i,, ) for i. + =. But this follos from F () ss = F () ss F () ss = F () ss F () ss i + i + 3 i + i ( ) + i ( ) ad F () ss = F () ss F () ss = F () ss F () ss. i + i + 4 i + 3 i ( ) + i ( ) + Note that h(+,, ) = h(,, ). This follos from the idetity = F () s = F () s s. + + = Remar If e replace ϕ ( () Fi + s s ) by h( i, ) λ ( F i + s s ) = e get the origial Seidel triagle for the Geocchi umbers i [3], Beilage, hich Seidel called "Differee-Tableau der Beroulli'sche Zähler". Its first terms are 7

28 Here e have h(, ) = ( ) H +. It has the special property that e ca also compute the first colum ithout previous oledge of the Geocchi umbers because h(+,) = h(, ) ad the fact that h(,) = F ( s) = for >. λ + = 4.. For our ext results e eed the square of A. For example 3 4 A 5 = Theorem 4.6 The square of ith A is give by A (, )[ ] = aa i i (4.7) i, = aa(, ) = ( ) G + 4. ( + )( + ) (4.8) 8

29 Proof Let + =. f x ( + )! Sice y equivalet ith = Ax is equivalet ith + e y = f ( ) e see that (+ )! e + = Ay = A x is e e e = y = f ( )! + ( )! + + e e e e e e = f + f e + e + e + (4.9) Sice e G = e + +! + + (4.3) ad 3 e e e G, = = (4.3) e + e + e + ( 3)! e get + 4 f = ( ) x +! = + e G e (4.3) ad e G + 4 f x. = e! = (4.33) From (4.3) ad (4.33) ad usig (4.9) e get immediately (4.8). It oly remais to prove (4.3). This follos from 9

30 e e (+ ) G e = + e = + +!(+ )(+ ) = G +! +. Theorem 4.7 (Further properties of ad for A ) aa(,) = a( +,) (4.34) aa(, ) = a(, ) a( +, ) (4.35) Proof. This follos from (4.8). * 4.3. Itimately coected ith the liear fuctioal λ is the liear fuctioal λ defied by Sice λ( sf( s)) = λ( F+ ( s)) λ( F+ ( s)) e get λ( sf ( s)) = λ( F ( s)) λ( F ( s)) = λ( F ( s)) for > ad λ( sf ( s)) = λ( F ( s)) λ( F ( s)) = λ( F ( s)) λ * ( F ( s)) = λ( sf ( s)). (4.36) ( () ) * The sequece ( F s ) λ begis ith,,,, 3,3,7, 7, 55,55,73, 73, 387. Therefore ad ( F ( s) + F ( s)) = [ = ] (4.37) * λ + 3 * λ F s F s G G+ ( + ) = ( ) +. (4.38)

31 Thus e are led to cosider the basis cosistig of the polyomials F ( s) + F ( s). Here e get Theorem 4.8 Let The The first etries of ( a (, ) ) are Proof We o that a (, ) = [ ] a(, ). (4.39) = + = + + = (4.4) F () s a (, ) F () s F (). s F () s = a(, ) F (). s Therefore + = This implies that i (4.4) the matrix ( a (, i ) ) F () s + F () s = a(, ) F () s + F (). s = is the iverse of I + B, here I deotes the idetity matrix ad b (, ) = a (, ) for <. We have to sho that a (, ) is give by (4.39). It suffices to sho that ( I + B) [ i] a( i, ) = I. = This meas that for each ( ) = a (, ) a(,) + a(,) + a(, ) + a (,) + + a (, ) = [ = ]. 3

32 Let <. By defiitio aa(, ) = a(, ) a(, ). Therefore = = = aa (,) i a (, a ) (,) i = a (, a ) (,). i Usig (4.35) ad (4.) the left-had side is equivalet ith aa(, ) a(, ) a(, ) + aa(,) a(, ) a(,) + = = + aa (, ) a (, ) a (, ) + aa (, ) + a (, ) + a (,) + + a (, ) = aa(,) + a(,) + aa(,) + a(,) + + aa(, ) + a(, ) a(, i) a( i, ) = a (,) + a (,) + + a (, ) a (, ) =. For = the first sum vaishes ad a (,) + + a (, ) =. Corollary 4.9 Let The Proof By Theorem 4.7 e have = i= = a (, ) = [ ] a (, ) a ( +, ). (4.4) (4.4) F () s + F () s = a (, ) F () s + F (). s = F () s = a (, ) F () s + F () s = a (, ) F () s + F () s + F () s + F (). s = = Therefore F () s = a ( +, ) F () s + F () s + + = ad thus F s + F s = a ( F s + F s ) () () (, ) () () = i 3

33 The first terms of ( a (, ) ) are Sice a (, ) = T ( +, + )( + ) t( +, + ) e see that a (, ) F(, ) = if i Lemma 3. e choose = = + ad F () = +. (, ) (, ) ˆ (, ). = ˆ Therefore = ( + ) This gives Theorem 4. a S F s Let = + ad thus ˆ ( ). = + The ˆ ˆ ( a i ) ( S i ) ( i i ) ( s i ) (, ) = (, ) [ = ]( + ) (, ). (4.43) i, = i, = i, = i, = 4.5. Next e cosider the liear fuctioal μ defied by μ ( F + ( s)) = [ = ]. From (4.) e see that This implies ad μ( F + ( s)) e + μ( F ( s)) e +.! e! e ( )! + + = = = + B + + (4.44) μ F + ( s) = (+ ) B (4.45) + B F+ () s = F + (). s = + + (4.46) Comparig (4.6) ith (4.46) e see that 33

34 A + B = a(, ). i, = + + i, (4.47) The first terms of the sequece B + are,,,,,,,,. Let + B (, ) = [ ]. + + By (4.5) e have i + i + [ i] A i i + = i i, i, [ ]. Cosiderig the first colum e get + + ( + ) B = ( + ) B = + = + + = ( + + ) B+ = [ = ]. = Sice B i + = for i > this is equivalet ith + + ( + i+ ) B+ i= i= i for >. But it also holds for = ad =. Therefore e get Kaeo's idetity ([7],[],[3]) + + ( + i+ ) B+ i=. i= i (4.48) This idetity has first bee proved by A.v. Ettigshause [7] ad has bee rediscovered by L. Seidel [3],VIII, ad by M. Kaeo []. 34

35 (4.47) implies that (, ) (, ) [ ] (, ). + ( ) = ( T i+ + ) i = ( t i+ + ), = i, = i, = i, = (4.49) ˆ ˆ The iverse of ( a ( i, ) ) ( S ( i, ) ) ([ i ]( i ) ) ( s ( i, ) ) = = + is i, = i, = i, = i, = (, ) (, ) [ ] (, ). i + ˆ ˆ ( i ) = ( S i ) i= ( s i ) i, = i, = i, = i, = By Lemma 3. this implies that (, ) = ( (, ) ( +, ) ) else. So e have for ad (, ) = = (4.5) F () s + F () s = (, ) F () s + F (). s = 5. Aalogous results for Lucas polyomials 5.. For the Lucas polyomials e get L+ () s + e L() s =. (5.) (+ )! e +! This follos from α β ( β) ( α) α β e + e = e + e = e e + e. (5.) We rite the series expasio of e e + i the form e T = e + ( + )! + +, (5.3) + here T,, are the taget umbers,,6,7,7936,. Therefore 35

36 Y e X (+ )! e +! + = (5.4) is equivalet ith T + + T + + Y = ( ) X. X + + = = + = T If e set (, ) G b + =, + = + the (5.4) is the same as Y = b(, ) X. (5.5) = We call the matrices B = ( b(, i ) ) = ( ) i, i T i + i + i + i, (5.6) taget-matrices. For example B 5 =. 63 If e compare (5.) ith (5.5) e see that L + () s = b(, ) L (). s This is agai (.3). = Let L () s = l(,) s s. The L () s = l(, ) s ad L + () s = l(+, ) s. = = = Therefore e get 36

37 5.. Let + = i B = l(i +, ) l( i, ). (5.7) i,, ad let U (,) = S(,) ad u (, ) = s (, ). respectively. The first values of the umbers 4 U (, ) ad 4 u (, ) are give by the folloig tables. ( 4 i Ui (, ) ) 6 i, = = ad ( 4 i ui (, ) ) 6 i, = = There is also a aalogue of Theorem 3.. To this ed e itroduce a aalogue of the Legedre-Stirlig umbers. Let ( )(+ ) = ad V(, ) = S (, ) ad v (, ) = s (, ). 4 The e get Theorem 5. l(, ) V(, ) = U(, ) (5.8) = ad l(+, ) V(, ) = ( + ) U(, ). (5.9) = 37

38 Proof We use agai iductio. Let both idetities be already proved for. The l(, ) V(, ) = l(, ) V(, ) + l(, ) V(, ) = = = = l(( ) +, ) V(, ) + l(( ), ) V( +, ) = = ( )( + ) = ( + ) U(, ) + l(( ), ) V(, ) + l(( ), ) V(, ) 4 = = ( )( + ) = ( + ) U(, ) + U(, ) + U(, ) 4 ( + ) = U (, ) + U (, ) = U (, ). 4 Ad l(+, V ) (, ) = l(, V ) (, ) + l(, ) V(, ) = = = = U (, ) + l(, V ) ( +, ) = ( )( + ) = U (, ) + l(, V ) (, ) + l(( ), V ) (, ) 4 = = ( )( + ) = U (, ) + ( ) U (, ) + (+ ) U (, ) 4 ( + ) = U (, ) + U (, ) U (, ) + 4 = U (, ) + U (, ) = (+ ) U (, ). This implies Theorem 5. The taget matrix B has the factoriatio + B= l(i+, ) l( i, ) = U( i, ) [ i = ] u( i, ), i, i, i, i, i, (5.) 38

39 here ui (, ) ad Ui (, ) are the geeralied Stirlig umbers correspodig to + =. I the same ay as i Theorem 4.5 there is a simple Seidel-array for computig V(, ) i terms of U (, ). Propositio 5.3 Let h( i,, ) = U( i, ), + h(i+,, ) = U( i, ), i hi (,, ) = hi (,, ) hi (,, ) if, i hi (,, ) = if >. (5.) The h(,, ) = V(, ). (5.) For example ( hi (,,)) begis ith. Propositio 5.4 The iverse of B is i B [ ] i =. B i + (5.3) 39

40 This follos from (5.4) ad the ell-o series e +. = B e! The from (5.) e get agai (.4). 6. Some iterestig idetities 6.. For ay sequece ( a ) the sequece colum i ( S i ) ( a i i = ) ( s i ) More precisely e have Theorem 6. (, ) ([ ] (, ). i, = i, = i, = c = ( ) i S (, ) a = i= is the first Let X = x ( i, ) = S ( i, ) a ( i )([ i = ] s ( i, ). (6.) i, i, i, i, The x(,) = ( ) S (, ) a i = i= (6.) ad s (, ) x(,) = ( ) a. (6.3) = = The sequece ( c ) [9], [], [5] ): ca be simply computed ith the Aiyama-Taigaa algorithm (cf. [], 4

41 Theorem 6. ( Aiyama-Taigaa algorithm) ( [],[9],[],[5]) Suppose that for all. Let Defie a matrix M ( m(, i ) ) i, = by m(, ) = a for ad c = ( ) i S (, ) a = i=. mi (, ) = mi (, ) mi (, + ) (6.4) The m (,) = c = ( ) i S (, ) a. = i= Proof To prove this e sho more geerally that ( ) m (, ) = sic (, ) ( i). + i= () = (6.5) This holds for = because i ( ) ( ) s ( ici, ) s ( i, ) S ( i, a ) = i i = = () () = = ( ) = i = ( ) a s (, i) S ( i, ) = a. () = No suppose that (6.5) holds for. The 4

42 = = ( + ) m (, ) m (, ) + ( ) ( ) = s (, i) c( i) s (, i) c( i) i i () () = = ( ) = ( s ( i, ) + s ( +, i) ) c ( + i) i () ( ) ( ) = s (, i ) c( + i ) = s (, i) c( i) m(, ). + = i i () () = We used the fact that s (,) + s ( +,) =. For = + ad ( b ) as sho i []. a = + this reduces to the origial Aiyama-Taigaa algorithm for No e give a list of some iterestig formulas. 6.. For = +, a = + e get from (.6) ad (3.9)! S ( +, + )( ) = b. (6.6) + = Aother proof of (6.6) ca be foud i [], but I suspect that this result must be much older. Formula (6.3) gives! s ( +, + ) b = ( ). (6.7) ( + ) = 6.3. For = +, a = + e o from (4.7) ad (4.6) that c = ( ) G +. This gives = (6.8) ( ) G = ( ) T(, ) ( )! ad 4

43 ( ) tg (, ) =!( )!. (6.9) = The Aiyama-Taigaa algorithm applied to (6.8) gives aother method for computig the Geocchi umbers. Choose = ( + ) ad a = + i Theorem 7.. The the left upper part of the correspodig matrix M ( mi (, ) ) = is give by I the first colum e get c = ( ) G From (4.8) e deduce the folloig idetity (cf. [4], Exercise 5.8): + = (6.) ( ) G = ( ) T(, )!. The left upper part of the correspodig Aiyama-Taigaa matrix is (6.3) gives the compaio formula For example for = 3 e have 6.5. (4.) ad Lemma 4.3 give ( ) tg (, ) + =!. (6.) = 4G4 + 5G6 + G8 = = 36 = (3!). ( ) LS( +, + ) (( + )! ) = H + 3. (6.) = 43

44 6.6. From (4.43) ad (4.4) e deduce for = + ad a = + ( ) S (, )( + )!( + )! = G+ + G+ 4 (6.3) = ad ( ) s (, ) ( G+ + G+ 4) = ( + )!( + )!. (6.4) = 6.7. For = + ad a = + e get from (4.47) (!) (+ ) B = ( ) T( +, + ). (6.5) + Here the left upper part of the Aiyama-Taigaa matrix begis ith = From (5.6) ad (5.) e deduce ( ) 4 U (, )(+ ) (( )!! ) = T + (6.6) = 6.9. Fially Propositio 5.3 gives ( ) U (, ) (( )!! ) = B. (6.7) = ( + )4 44

45 Refereces [] S. Aiyama ad Y. Taigaa, Multiple eta values at o-positive itegers, Ramaua J. 5 (), [] G.E. Adres ad L.L. Littleoh, A combiatorial iterpretatio of the Legedre-Stirlig umbers, PAMS 37 (9), [3] J. Cigler, q-fiboacci polyomials ad q-geocchi umbers, arxiv:98.9 [4] A. de Médicis ad P. Leroux, Geeralied Stirlig umbers, covolutio formulae, ad p,qaalogues, Ca. J. Math. 47(3), 995, [5] D. Dumot ad J. Zeg, Further results o the Euler ad Geocchi umbers, Aequat. Math. 47(994), 3-4 [6] D. Dumot ad J. Zeg, Polyomes d'euler et fractios cotiues de Stieltes-Rogers, Ramaua J. (998), [7] A. v. Ettigshause, Vorlesuge über die höhere Mathemati,. Bad, Verlag Carl Gerold, Wie 87 [8] Y. Gelieau ad J. Zeg, Combiatorial iterpretatios of the Jacobi-Stirlig umbers, arxiv: [9] Y. Iaba, Hyper-sums of poers of itegers ad the Aiyama-Taigaa matrix, J. Iteger Sequeces 8 (5), Article 5..7 [] M. Kaeo, A recurrece formula for the Beroulli umbers, Proc. Japa Acad. Ser. A Math.Sci. 7 (995), 9-93 [] M. Kaeo, The Aiyama-Taigaa algorithm for Beroulli umbers, J. Iteger Sequeces 3 (), Article..9 [] J. Riorda, Combiatorial Idetities, Joh Wiley, 968 [3] L. Seidel, Über eie eifache Etstehugseise der Beroullische Zahle ud eiiger veradter Reihe, Situgsber. Müch. Aad. Math. Phys. Cl. 877, [4] R.P. Staley, Eumerative Combiatorics, Vol., Cambridge Studies i Advaced Mathematics 999 [5] J. Zeg, The Aiyama-Taigaa algorithm for Carlit's q Beroulli umbers, Itegers 6 (6), A5 45