0,1,1, 2,3,5,8,13, 21,

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1 Catala umbers, Hael determiats ad Fiboacci polyomials Joha Cigler Faultät für Mathemati Uiversität Wie Abstract I this (partly expository) paper we cosider some Hael determiats of siged Catala umbers whose values are multiples of Fiboacci umbers ad some Hael determiats of siged cetral biomial coefficiets whose values are multiples of Lucas umbers. These results are related to the fact that Catala umbers are momets of Fiboacci polyomials ad cetral biomial coefficiets are momets of Lucas polyomials. Some of these results also follow from Hael determiats of sums of Narayaa polyomials. Most proofs are computatioal but we also iclude a combiatorial oe due to Christia Krattethaler. 0. Itroductio C,,,5,4, 4,3, be the sequece of Catala umbers C 0 B,,6,0,70,5, be the sequece of cetral biomial coefficiets 0. Let also F 0,,,,3,5,8,3,, 0 be the sequece of Fiboacci umbers L,,3, 4,7,,8, 9, be the sequece of Lucas umbers. Let ad B ad A. Cvetovic, P. Rajovic, M. Ivovic [6] proved that [], [3] ad [7]). A closely related result is det C C F. (See also i j i j i, j0 i j i j i, j0 det B B L. I this ote we show that similar results also hold for the Hael determiats of the sequeces C, C, C, C, C, C, C,, C, C, C, C, C, C,, B, B, B, B, B, B, B0, B, B, B, B, B3, B3,, 3 3 ad aerated versios of these sequeces. We first give a direct proof with the method of orthogoal polyomials applied to these sequeces. The we observe that these sequeces are related to the Narayaa polyomials C () t t 0 ad B () t t. 0 More precisely C0, C, C, C, C, C3, C3, C( ) C ( ) B, B, B, B, B, B, B, B ( ) B ( ) ad sice C ( ),,0,,0,,0, 5,0,4, 0 B ( ),0,,0,6,0, 0,0,70,. 0 ad Therefore the metioed results also follow from Hael determiats of sums of Narayaa polyomials. We give a computatioal proof ad also a combiatorial oe due to Christia Krattethaler.

2 . Some bacgroud material.. Let me recall some well-ow facts about the orthogoal polyomial approach to Hael determiats (cf. e.g. []). If all Hael determiats det( ai ( j)) 0 the the polyomials i, j0 a(0) a() a( ) a() a() a( ) x px (, ) det a() a(3) a( ) x det( ai ( j)) i, j0 a ( ) a ( ) a() x are orthogoal with respect to the liear fuctioal F defied by (.) For 0 we let det( ai ( j)) by defiitio. i, j0 F( x ) a( ). (.) Orthogoality with respect to F meas that F p( m, x) p(, x) 0 if m ad F p x (, ) 0. I particular for m 0 we get which also characterizes the liear fuctioal F. From (.) we see that F p(, x) [ 0], (.3) i, j0 i, j0 det ai ( j) p (,0) ( ). det ai ( j) By Favard's theorem about orthogoal polyomials there exist umbers s ( ), t ( ) such that Remar Give the sequece a ( ) 0 (.4) px (, ) ( xs ( )) p (, x) t ( ) p (, x). (.5) ad computig some of the polyomials px (, ) it is ofte easy to guess the umbers s ( ) ad t ( ). I order to prove that these guesses are correct we ca use the followig Lemma (for a proof see e.g. []). Lemma. For give sequeces s ( ) ad t ( ) let a (, j ) be defied by a(0, j) [ j 0] a (,0) s(0) a (,0) t(0) a (,) a(, j) a(, j) s( j) a(, j) t( j) a(, j). (.6)

3 The the Hael determiat det ai ( j,0) is give by i j, 0 i, j0 i If we verify that a (,0) a ( ) our guesses are correct. Fially observe that 0 det ai ( j,0) t( j). (.7) i j0 a (, ) px (, ) a (, ) sa ( ) (, ) ta ( ) (, ) px (, ) a (, ) p (, x) s ( ) px (, ) t ( ) p (, x) x a (, px ) (, ) ad thus 0 If we apply the liear fuctioal F the we get We shall also us the fact that which follows from (.4). a (, ) px (, ) x. (.8) a (,0) F a (, ) px (, ) Fxa ( ). (.9) 0 ai j p ai j i, j0 i, j0 det ( ) ( ) (,0)det ( ), (.0) Let us recall aother well-ow result (cf. e.g. []): Lemma. Let A(, ) satisfy with A(0, ) [ 0] ad A (, ) 0 for 0. The A (, ) 0 for mod. A (, ) A (, ) T( ) A (, ) (.) If we set a (, ) A(, ) ad defie s ( ) ad t ( ) by s(0) T(0), s( ) T() T( ), t( ) T( ) T() (.) The a (, ) satisfies (.6). 3

4 .. Let us first illustrate this procedure with a well-ow example. Recall that the geeratig fuctio of the sequece of Catala umbers give by C is 4z Cz () Cz (.3) z ad satisfies Cz ( ) zcz ( ). (.4) For later use defie umbers C by ( r ) r r r ( r) Cz ( ) z C z. r For techical reasos we itroduce umbers U(, r ) by the geeratig fuctio (.5) Urz (, ). (.6) rzc( z) Note that U (,) C ad U (, ) B sice () zc( z) Cz ad B z. zc( z) zc( z) 4z 4z 0 0 Cosider the aerated sequece U(0, r),0, U(, r),0, U(, r),0,. Computatios suggest that the correspodig umbers T(, r ) are give by T(0, r) r ad Tr (, ) for 0. Let U(,, r ) be the correspodig umbers defied i (.), i.e. U(,, r) U(,, r) T(, r) U(,, r) (.7) with U(0,, r) [ 0] ad U(,, r) 0 for 0. Their geeratig fuctios are zcz ( ) Urz (,, ) (.8) rz C( z ) 0 because if we defie U(,, r ) by (.8) the (.7) is equivalet with zcz ( ) r rzcz ( ) rzcz ( ), zcz ( ) zcz ( ) z Cz ( ) rz C z rz C z rz C z ( ) ( ) ( ), (.9) which is obvious by (.4). 4

5 Sice U (,0, rz ) implies U(,0, r) U(, r) ad U(,0, r) 0 our 0 rz C( z ) guesses are correct. For r we get U C (.0) ( ) (,,), because Cz () Cz zc( z) ( ). For r we get U(,,) (.) because ad. Recall that the Fiboacci polyomials are give by s x 0 ad the F ( x, s) 0 for 0 Lucas polyomials by L ( x, s) s x ad L ( x, s). They 0 satisfy F( x, s) xf ( x, s) sf( x, s) with iitial values F ( x, s) 0 ad F( x, s) ad 0 L ( x, s) xl ( x, s) sl ( x, s) with iitial values L ( x, s) ad L ( x, s) x. 0 The orthogoal polyomials for the aerated Catala umbers U(,0,) are the Fiboacci polyomials (, ) ( ). 0 F x x For arbitrary r we get Pxr (,, ) F ( x, ) ( rf ) ( x, ) for 0 ad P(0, x, r). For r we get for 0 the Lucas polyomials If r F is the liear fuctioal defied by (,0,). F x U r r (, ) ( ). 0 L x x F P(, x, r) [ 0] r F x are the the momets r 5

6 By (.7) we get for 0 Let D U i j r i, j0 D ( ): det U ( i j,0, r r. (.) 0 i, j0 ( ) det (,0, ). The D r ( ) ( ), D () 0. (.3) This follows from (.0) because P(,0, r) ( ) r ad P(,0, r) 0 for the correspodig orthogoal polyomials. By (.) the above iformatios imply for the sequece Ur (, ) 0 sr (, ) for 0, t(0, r) rad tr (, ) for 0. that s (0, r ) r, This implies that ( ) det (, ) d U i j r r ad 0 i, j0 To prove the last result observe that d ( ) det U ( i j, r ) r. i, j0 p (,0, r) p (,0, r) p (,0, r) with the iitial values p(0,0, r), p(, 0, r) r, p(,0, r) p(,0, r) rp(0,0, r) rr r ad p (,0, r) p (,0, r) p (,0, r) for. This gives ( ) p(,0, r) r.. Some Hael determiats of Catala umbers ad cetral biomial coefficiets As we shall later see (cf. (4.8)) the Hael determiats d ( ) of the sequece,,,,,5,5,4,4, 4, 4, are the values d ( ) F (, ), i.e. d ( ),, 0,,, 0,,, 0,,, 0,, of the Fiboacci polyomials F (, ). 0 x But if we isert some mius sigs we get siged Fiboacci umbers F. Theorem.. The Hael determiats d c i j ad d c i j sequece ( ) det ( ) i j 0, 0 ( ) det ( ) of the i j, 0 ( c ( ) 0 C0, C, C, C, C, C 3, C3,,,,,, 5, 5,4,4, 4, 4, (.) 0 i, j0 d ( ) det c( i j) ( ) F, (.) are thus d ( ),,, 3, 5,8, 3,,. (.3) 0 0 6

7 ad ( ) det ( ) ( ) i, j0 d c i j F (.4) or d ( ),, 3,3,8, 8,,,55,. (.5) 0 Theorem. The Hael determiats D b i j ad D b i j sequece ( ) det ( ) i j 0, 0 ( ) det ( ) of the i j, 0 ( b ( ) 0 B0, B, B, B, B, B3, B3,.,,,6,6, 0, 0,70,70, (.6) D ( ) det b( i j) ( ) L, (.7) satisfy 0 i, j0 thus D ( ),, 6, 6,56,76,,. (.8) 0 0 ad i, j0 D ( ) det b( i j) ( ) L (.9) or D( ),, 6,3,76, 35, 856,. (.0) 0 Theorem.3 0 with c ( ) as defied i (.). Let c ( ),0,,0,,0,,0,,0, 5,0, c(0),0, c(),0, c(),0, The satisfies d0 ( ): det c( i j) (.) i, j0 7

8 d (4 ) F F, 0 d (4) F, 0 d (4) F, 0 d (43) F F. 0 3 The first terms are d 0 ( ),,,, 6,9, 9,5, 40, 64, 64,. 0 (.) The determiats d ( ): det c( i j ) (.3) i, j0 are d () 0, d (4 ) F, d (4) F (.4) or d ( ), 0,, 0, 9, 0 9, 0, 64,. 0 Theorem.4 Let b ( ),0,,0,,0,6,0, b(0),0, b(),0, b(),0, (.6). The satisfies 0 with b ( ) as defied i D0 ( ): det b( i j) (.5) i j, 0 D (4 ) L L, 4 0 D (4) L, 4 0 D (4) L, 4 0 D (43) L L. 4 0 The first terms are D ( ),,,,96, 56,. 0 0 (.6) The determiats D ( ): det b( i j ) (.7) i j, 0 are 8

9 or,0, 4,0, 56,0, 04,. D () 0, D (4 ) L, 4 D (4) L 4 (.8) 3. Proof of Theorems Proof of Theorems.. ad.. Defie umbers f ( r, ) by the recursio f ( r, ) f(, r) f(, r) with iitial values f (0, r) r ad f (, r). Note that f (,) F ad f (, ) L. We shall eed Cassii s formula which ca easily be verified. Cosider the sequece f(, r) f(, r) f(, r) ( ) r r (3.) v r U r U r U r U r U r (, ) (0,0, ), (,0, ), (,0, ), (,0, ), (,0, ),. (3.) 0 Let us ote that the geeratig fuctio of the sequece vr (,) is 0 because rzc z vrz (, ) ( ) U (,0, rz ) ( ) U (,0, rz ) 0 0 z rzc z. rz C z z rz C z rz C z vrz (, ) 0 rz C z, (3.3) Computatios suggest that the umber sr (, ) ad tr (, ) correspodig to the sequece vr (, ) are s(0, r) r, r r f(, r) f(, r) (, ) ( ) for 0, sr f(, r) f(, r) tr (, ). f(, r) (3.4) 9

10 Defie ow ar (,, ) with the above values sr (, ) ad tr (, ) by (.6). Computatios idicate that zc z f(, r) (,, ). arz zc z 0 rz C z f(, r) (3.5) I order to show that our guesses are correct we have oly to verify that (.6) holds. The idetity a (, 0, r) s(0, ra ) (, 0, r) t(0, ra ) (,, r) ra (, 0, r) rr ( ) a (,, r) is equivalet with rzc z zc z rz r( r ) z zc z rz C z rz C z r or 3 rz rzc z rz C z r( r ) z C z rz C z which reduces to C z z C z It remais to verify that which is (.4). ar (,, ) a (,, r) sra (, ) (, r, ) tra (, ) (,, r). This is equivalet with the followig idetity for the geeratig fuctios: zc z rz C z r z Cz z r rz C z f(, r) f(, r) zc z z zc z zc z z C z rz C z f(, r) rz C z f(, r) r r f(, r) z( ) zcz f(, r) f(, r) f(, r) f(, r) f(, ) f(, ) By multiplyig with rz C z f(, r) zc z f(, r). this reduces to f(, r) f(, r) zc z z C z zc z z C z f(, r) f(, r) r r r r ( ) z Cz ( ) z C z f(, r) f(, r) f(, r) f(, r) f(, r) f(, r) 3 z C z z Cz. f(, r) f(, r) 0

11 By Cassii s idetity we get r r f(, r) f(, r) ( ) f(, r) f(, r) ad z C z z C z z C z f(, r) z C z f(, r) r r f(, r) ( ) z Cz z Cz. f(, r) f(, r) f(, r) Sice zc z zc z z C z we have oly to verify that f(, r) f(, r) f(, r) z C z z C z z C z f(, r) f(, r) f(, r) 3 which is equivalet with Cz z C z By Lemma. we see that. i i, j0 i j0 det vi ( j) t( jr, ). Now we get by iductio i i f ( i, r) t( j, r) r( ) ad thus f (, ir) j0 det ui ( jr, ) ( ) r f( r, ). (3.6) If we set h ( ) : ( ) p (,0, r) the we get hr (, ) s (, rh ) (, r) t (, rh ) (, r) with iitial values h(0, r) ad f (, r) h(, r) r. This gives by iductio h(, r) r ad h(, r) r f ( r, ) sice i, j0 h(, r) s(, r) h(, r) t(, r) h(, r) r r f(, r) f(, r) f(, r) r r f (, r) f(, r) f(, r) f(, r) r f(, r) ( r r) f(, r) r f ( r, ) f(, r) f( r, ) ad

12 h(, r) s(, r) h(, r) t(, r) h(, r) r r f(, r) f(, r) f(, r) r r f(, r) f(, r) f(, r) f(, r) r (, ) (, ) r. f(, r) r r f r f r det ui ( j, r ) ( ) r f (, r ) ad By (.4) this implies i, j0 f(, r) det ui ( j, r) ( ) r f( rr, ) ( ) r f(, r). i, j0 f(, r) Thus we get i, j0 det ui ( j, r) ( ) r f, r. (3.7) For r, this gives Theorem. ad Theorem Proof of Theorem.3. ad.4. Cosider the sequece v r U r U r U r U r U r (, ) (0,0, ),0, (,0, ),0, (,0, ),0, (,0, ),0, (,0, ),0,. (3.8) 0 Let t (, r ) be the umbers defied by (.5). Computatios suggest that f(, r) f(, r) t (4, r), t (4, r), f(, r) f(, r) f (3, r) f(, r) t (4, r), t (43, r). f (, r) f(, r) (3.9) ad thus t (0, r) r. The correspodig umbers for the sequece v(, r) v(, r) are 0 0 f(, r) f(, r) s(, r) t (4, r) t (4, r) f(, r) f(, r) f(, r) f(, r) f(, r) r r, f(, r) f(, r) f(, r) f(, r) f (, r) f(3, r) s(, r) t (4, r) t (4, r) f (, r) f(, r) f (, r) f(, r) f(3, r) r r f (, r) f(, r) f(, r) f(, r)

13 ad therefore r r sr (, ) t(, r) t( r, ) ( ). f ( r, ) f(, r) f ( r, ) f(, r) t(, r) t(4, r) t(4, r), f(, r) f (, r) f( 3, r) t(, r) t (4, r) t (4 3, r), f(, r) ad therefore f ( r, ) f(, r) tr (, ). f(, r) These are precisely the umbers obtaied above i the proof of Theorems. ad.. Therefore by Lemma. our guesses are correct. Therefore we get for the determiats D r v i j (, ) det ( ) the values i j 0, 0 D r r f r f r 4 0(4, ) (, ) (, ), D (4, r) r f(, r) 0 4 D r r f r 4 0(4, ) (, ), D r r f r f r 4 0(4 3, ) (, ) (, ). (3.0) The determiats D ( ): det u( i j ) (3.) i j, 0 are D (, r) 0, D r r f r 4 (4, ) (, ), D r r f r 4 (4, ) (, ). (3.) To this ed we must compute p (,0, r ). It satisfies p (,0, r) t(, r) p (,0, r) with iitial values p(0,0, r) ad p(, 0, r) 0. This gives by iductio p(,0, r) 0, p(4,0, r) r ad For r ad r these results reduce to Theorem.3 ad Theorem.4. f (, r) p(4,0, r) r. f ( r, ) 3

14 4. Aother approch Cosider the Narayaa polyomials C () t t 0 (4.) for 0 ad C 0 () t. The first terms of C ( t ) are The geeratig fuctio Ctz (, ) C() tz satisfies 0 For t we see that Sice C () t C t t Ctz (, ) zctz (, ) tzctz (, ) tzctz (, ). (4.) C(, z) zc(, z), which implies C(, z) C( z) by (.4). 0 (cf. e.g. [4], (.7)) we see that C C ( ) C ( ), ( ) 0. (4.3) Therefore we get C ( ),,0,,0,,0, 5,0,4,. 0 As already oted we have C C C C C C C C C,,,,,,, ( ) ( ) Our aim here is to prove Theorem 4. i j i j i, j0 det C ( t) C ( t) ( ) t F ( t, t). (4.4) A proof is implicitly cotaied i [], Theorem, (.5), but let us give a simpler proof which depeds o the followig Lemma which ca be foud i [0] (..9) (with a differet proof). 4

15 Lemma 4. Let px (, ) 0 be a sequece of moic polyomials which are orthogoal with respect to the liear fuctioal F with momets F x a( ). The Proof Let rx (, ) ax ( ) a ( ). i, j0 i, j0 det ri ( jx, ) det ai ( j ) px (, ). (4.5) Let (, ) (,0) (,) (, ) p x b b x b x x ad B x x b (,0) b (,) b (,) xb (, ) (4.6) The ri ( jx, ) Bai ( j) (4.7) i, j0 i, j0 because biai (, ) ( m) xa ( m) biai (, ) ( m) a ( m) xa ( m) a ( m) i0 i0 m L p(, x) x xa( m) a( m) xa( m) a( m) Now observe that by developpig with respect to the last colum we get det B px (, ). Let us apply this to the sequece C (). t I the followig we shall freely use results of 0 [4]. We get s ( ) tad t ( ) tad the correspodig orthogoal polyomials p( xt,, ) F ( x t, t), where This gives t x 0 is a Fiboacci polyomial. F ( x, t) i j i j i, j0 i j i, j0 det C ( t ) C ( t ) ( ) p (,, t )det C ( t ) ( ) t F ( t, t ). For t we get thus i j i j i, j0 det C ( ) C ( ) ( ) ( ) F (,) ( ) F, 5

16 which is (.). t we get the ow result For Corollary 4.3 Let ( ) det C C F (3, ) F. i j i j i, j0 c be defied by (.). The Hael determiats sequece c ( ),,,,,5,5,4,4, 0 are give by 0 d ( ) det ci ( j) of the i, j0 d ( ),, 0,,, 0,,, 0,,, 0,, (4.8) i.e. d(3 ) d(3) ( ) ad d(3) 0 or d ( ) F (, ). Proof It is easy to verify that c i ic C ( ) ( ) ( ). By Lemma this implies i j i, j0 d ( ) ( ) i det C ( ) F ( i,) ( ) i F ( i,). The sequece F ( i,), i, 0, i,, 0,, i, 0, i,, 0, 0 is periodic with period. Thus F ( i,) i,, 0,,, 0,,, 0,,, 0, 0 which fially gives (4.8). Let B () t t 0 be a Narayaa polyomial of type B. We have j j j we see that B j0 j j () B ad from B () t t ( t) ( ) ( ) ad B ( ) 0. We recall (cf. [4]) that we have s ( ) t, t(0) t ad t ( ) t. Let L ( x, t ) be a Lucas polyomial defied by L( x, t) xl ( x, t) tl( x, t) with iitial values L ( x, t) ad L ( x, t) x. 0 The the correspodig orthogoal polyomials are R ( xt, ) L( x t, t) for 0 ad R ( x, t). 0 This gives i j i j i, j0 i j i, j0 det B ( t ) B ( t ) ( ) R (, t )det B ( t ) ( ) t L ( t, t ). For t this reduces to the ow result det B B L ad for t we get (.7). i j i j 6

17 I the same way as above we get for the Hael determiats of the sequece b ( ),,,6,6, 0, 0, 0 bi j det ( ),,, 8, 8,6,64,64,, (4.9) i, j0 i.e. d(3 ) d(3 ) ( ) 8 3 ad d 3 ( ) or d L ( ) (, ). 5. Related results ( r ) Origially I wated to compute the Hael determiats For r 3 we get the ow result Theorem 5. ([5], Cor. 3 for =) Dr (, ) det C for r. i j i, j0 D(3,3) D(3,3) ( ), D(3,3) 0. (5.) The method of orthogoal polyomials caot be applied sice some Hael determiats vaish. The same is true for each odd r. To obtai (5.) evertheless with the above method a ( ) C(),0, t C(),0, t C(),0, t. let us cosider the aerated sequece 0 For this sequece we get s ( ) 0, t( ) ad t( ) t. The correspodig at (,, ) satisfy a (, t, ) a (,, t) a (,, t), a (,, t) a (, t, ) ta (,, t). (5.) The first terms are We ow claim that the geeratig fuctios are give by 0 0 (,, ) (, ) (, ), a t z C t z C t z Ctz (, ) (,, ). a t z z (5.3) The first lie of (5.) is equivalet with Ctz (, ) Ctz (, ) Ctz (, ) Ctz (, ) 7

18 or Ctz (, ) Ctz (, ) which is obviously true. The secod lie of (5.) is equivalet with Ctz (, ) z or zc t z C t z tzc t z C t z (, ) (, ) (, ) (, ) Ctz zctz tzctz Ctz (, ) (, ) (, ) (, ) which is true because of (4.). Sice a t z C t z C t z 0 (,, ) (, ) (, ) for z 0 implies a (, jt, ) 0 for j 0 ad a(0,0, t) we see that at (,, ) are the uiquely determied polyomials satisfyig (.6). Thus our claims are correct. Let us ow cosider a(,, t). Note that sice by (5.3) 0 a tz Ctz Ctz C () t C () t a(,, t) t (,, ) (, ) (, ). The first terms are (5.4) Ctz (, ) zctz (, ) tz which for t reduce to By Lemma 4. we get Theorem 5. (3) C sice 3 C(, z) C(, z) C( z) C( z) zc( z) by (.4). i j () i j () C t C t det t ( ) t. (5.5) t 0 i, j0 As cojectured this gives aother proof of Theorem 5. because for t the right-had side reduces to ( ) F (, ), 0 i.e. ( ),,0,,,0, 0 which is periodic with period 6. 8

19 6. Combiatorial proof of Theorem 5. Let us reformulate Theorem 5. by usig idetity (5.4). i, j0 0 det a(i j,, t) t ( ) t. (6.) By (5.) a (, jt, ) ca be iterpreted as the umber of all lattice paths from (0,0) to (, j ) with upsteps (,) ad dowsteps (, ) which ever go below the x axis with the followig weights: Each upstep has weight ad a dowstep which eds o a eve height has weight t( ) ad a dowstep which eds o a odd height has weight t( ) t. Combiatorial proof due to Christia Krattethaler [8]. By the Lidström-Gessel-Vieot theorem the determiat a i j det (,) (6.) i, j0 P P0, P,, P of o-overlappig lattice paths with upsteps U (,) ad dowsteps D (, ) which ever go below the x axis such that a path P i goes from Ai ( i,0) to some Ej j,, i, j 0,,,. is equal to the geeratig fuctio of families EO( P) The weight of such a family is sg ( Pt ), where ( P) deotes the permutatio determied by Pi : Ai E () i ad EO( P ) deotes the umber of dowsteps of paths i the family which go from a eve height to a odd height. As a example let 6, ( P) 0345 ad EO( P) See Figure. Let a path go from A 0 to E. Figure shows a example with. This path P 0 must reach E from below. (Otherwise it would be overlappig with aother path). It is clear that P 0 is uiquely determied ad of the form UD UDUU. The also all paths P i with i are uiquely determied: P i starts with i upsteps ad eds with i dowsteps. For the paths P i with i large parts are also uiquely determied. P i starts with i upsteps ad reaches A, i, with E i E i i ad eds with ( i ) dowsteps. These joi i i,. 9

20 Figure : Figure shows those parts of the paths P3, P4, P 5 betwee the poits A i ad E for 3i 5. i Now we use Vieot s idea of dual paths: We are startig a path i E which eds i X,. We use steps, except if we would arrive ito a valley of some path P, i,,,. I this case wo go steps up i vertical directio where we arrive i a pea of some path as show i the poited path i the left side of Figure. By rotatig ad deformig we ca trasfrom this path ito a path from (0,0) to (, ) which has oly horizotal or vertical uit steps i positive directio as show i the right part of Figure. 0 i

21 Now let us determie the weights. The path P 0 has weight. I the other paths there are dowsteps oly betwee x ad x. Whe we cout all possible dowsteps from a eve height to a odd height i this regio we get 3 ( ) if we cout alog vertical strips from right to left. O the other had ot all of these steps actually occur i these paths. Those who do t occur are determied by the above costructed dual path which wids up through the holes. By the above bijectio betwee path families ad sigle paths (dual paths) the weight of a dual path is the followig oe: Horizotal steps o eve height ad vertical steps from eve to odd height have weigth, wheras we assig to horizotal steps o odd height ad vertical steps from odd to eve height the weight t because we wat to subtract their weight from the weight of all possible steps. Let ow a be the umber of the first steps with weight, b the umber of the ext steps with weight t, a the umber of the ext steps with weight ad so o. The the geeratig fuctio of the correspodig dual paths from (0,0) to (, ) is give by abab aba a,, a, b,. b, a0 If we set s bb b this sum reduces to s s because there are ( s) ( ) s Therefore we have see that Now t bb b. (6.3) ss t s, s0 (6.4) possibilities for the b i ad possibilities for the a i. ss s i, j0 0 s (6.5) det a(i j,) t ( ) t. s s s s s ss ( ) t t 0 s s0 0 s s,s ( s) t F ;. s

22 By Chu-Vadermode we have a, N ( c a) F ; c () c N for a o-egative iteger N. Therefore we fially get which is the desired result. N s s s i, j0 s0 s det a(i j,) t ( ) t (6.6) 7. Fial observatios ( r) As stated above my origial aim was to compute the determiats Dr (, ) det C. i j i, j0 For arbitrary r Mathematica gives Dr (, ) 5 0 is equal to From this it seems difficult to guess a geeral formula except that D (,) 0. Theorem 7. ( ) (, ) det i j 0. i, j0 D C (7.) Proof It suffices to show that for 0 ad 0 the followig idetity holds: j j (7.) j () (, ) ( ) C j 0. j0 j j R I wat to tha Christia Krattethaler for showig me a proof of this idetity. I will oly setch the mai steps of the proof. The left-had side of (7.) ca be writte as,,, ()! ( ) 4F3 ;.!! 3,, (7.3)

23 Usig Whipple s trasformatio formula ([9],Eq. (.4..)) together with Pfaff-Saalschütz s formula we get,,, 4F3 ; 3,, where ( a) a( a) ( a). Sice 0 for 0 we get (7.). More explicitly we get from 3 ( )!! ( ) 3 () ( )!( )! ( )!( )! 3 ( )!( 3)! that ()! ( )!! ( )!( )! R (, ) ( ) ( )!! () ( )!( )! ( )!( 3)! ( ) ( ) Thus ( ) R (, ) ( ) (7.4) ad its geeratig fuctio is 4 Rx (, ) x Cx ( ). (7.5) 0 3

24 For special values of more iformatio about D (, ) ca be obtaied. Let us first cosider the determiats D (, ). It is well ow that C i j D (,) det. i, j0 (4) Let us ow compute D C (,4) det. i j i, j0 Theorem 7.. D (, 4),,,,3,3, 4, 4,. (7.6) 0 Proof We guess that s(,4) 4, s(,4) 0, t(,4) ad t(,4). Defie ow a (, ) by (.6). Computatios suggest that the geeratig fuctios are give by 0 0 a x x Cx 44 (, ) ( ), a x x Cx xcx 44 4 (, ) ( ) ( ). It remais to show that with these values (.6) holds. a (, 0) s(0, 4) a (, 0) t(0, 4) a (,) 4 a (, 0) a (,) is equivalet with Cx xcx xcx xcx ( ) 4 ( ) ( ) ( ). To prove this idetity we use 3 4 x Cx xcx x ( ) ( ) ( ) xc x ( ) C( x) to reduce the powers of C( x ). We get ad xcx x x x Cx x x x from which our idetity follows. It remais to show that a (, ) a (,) 4 a (, ) a (,). This is equivalet with ( ) ( ) 6 5, x cx ( ) x cx ( ) xcx ( ) 4 x cx ( ) x cx ( ) xcx ( ) x cx ( ) x cx ( ) x cx ( ) 4 x cx ( ) x cx ( ) x cx ( ) x c( x) x cx ( ) x cx ( ) 4 x cx ( ) x cx ( ) x cx cx xcx xcx xcx ( ) ( ) ( ) 4 ( ) ( ) 0. 4

25 For the Hael determiat we get i D(,4) t( j,4). i j0 Now we have i t( j,4) ad j0 i i t( j,4). This implies i j0 D(,4) ( ) ( ), D(,4) ( ) ( ) as asserted. Let us fially state some Cojectures: D (,6),, 9, 4, 4, 45,9,9, 6,, i.e. 0 D D (3,6) (3,6) ( ) ( ), D(3, 6) 9( ) j. j0 For D (,8) the situatio becomes more complicated. We get D (,8),, 0, 6,8,8,56, 384, 7, 7, 744,, i.e. 0 3 D(4,8) D(4,8) ( ), D 45 D 45 (4,8) ( ) ( )( 3) , (4 3,8) ( )( ) ( 3) This leads to Cojecture 7.3 For 0 we have (, ) (, ) ( ) ( ). D D (7.7) Now cosider the determiats D (, ). Here we have D (,), D (, 3),, 0,,, 0,,, 0,, 0,. 3 D (, 5),, 5, 0,5,,, 0, 0,0,, i.e. 5

26 D(5,5) D(5,5), D(5,5) 5( ), D(53,5) 0, D(54,5) 5( ). More geerally we get Cojecture 7.4. D((),) D((),) ( ), D((), ) 0, D((),) ( ) ()( ), D((),) ( ) ()( ). (7.8) Refereces [] J. Cigler, Some relatios betwee geeralized Fiboacci ad Catala umbers, Sitzugsber. OeAW. II (00) :43-54 [] J. Cigler, A simple approach to some Hael determiats, arxiv: [3] J. Cigler, Hael determiats of some polyomial sequeces, arxiv:.086 [4] J. Cigler, Some elemetary observatios o Narayaa polyomials ad related topics, arxiv: [5] J. Cigler ad C. Krattethaler, Some determiats of path geeratig fuctios, Adv. Appl. Math. 46 (0), [6] A. Cvetovic, P. Rajovic, M. Ivovic, Catala umbers, the Hael trasform, ad Fiboacci umbers, J. Iteger Sequeces 5 (00), Article 0..3 [7] C. Krattethaler, Determiats of (geeralized) Catala umbers, J. Statist. Pla. 40 (00), [8] C. Krattethaler, Persoal commuicatio, /4/07 9] L. J. Slater, Geeralized hypergeometric fuctios, Cambridge Uiv. Press, Cambridge 966 [0] G. Szegö, Orthogoal polyomials, Amer. Math. Soc. Colloquium Publ. 3, 4 th ed