q-lucas polynomials and associated Rogers-Ramanujan type identities

Transcription

1 -Lucas polyomials associated Rogers-Ramauja type idetities Joha Cigler Faultät für Mathemati, Uiversität Wie Abstract We prove some properties of aalogues of the Fiboacci Lucas polyomials, apply these to derive some idetities due to L Carlitz ud H Prodiger fially give a easy approach to L Slater s Bailey pairs A()-A(8) usig Lucas polyomials Itroductio After recallig some properties of the Fiboacci Lucas polyomials which I have itroduced i [4] [5] I apply these to derive some idetities due to L Carlitz [] ud H Prodiger [8] Fially I show that the Lucas polyomials allow a easy approach to L Slater s Bailey pairs A()-A(8) some related Rogers-Ramauja type idetities I wat to tha Adrew Sills for commetig o a previous versio poitig out to me the papers [] [] of L Slater Let () F ( x, s) s x () L ( x, s) s x be the classical Fiboacci Lucas polyomials They satisfy the recurrece F ( x, s) xf ( x, s) + sf ( x, s) (3) with iitial values F( x, s), F( x, s) L ( x, s) xl ( x, s) + sl ( x, s) (4) with iitial values L ( x, s ) L( x, s) x It will be coveiet to defie a variat L * ( x, s ) by L * ( x, s ) L * ( x, s) L ( x, s) for > x + x + 4s Let α x x + 4s β be the roots of the euatio z xz s

2 α β The it is well-ow easily verified that F ( x, s) α β This implies the well-ow formula L ( x, s) α + β for > because s αβ L ( x, s) F ( x, s) + sf ( x, s) (5) α β αβ( α β ) α ( α β) + β ( α β) Aother ow formula is ( s) L ( x, s) x (6) For the proof it is coveiet to cosider this idetity for odd eve separately For odd the left-h side is x αβ ( α β ) ( α β α β ) α β ( α β ) For m the same holds because for m the coefficiet of Usig (5) we see that (6) is euivalet with m ( s) m m is L ( x, s) ( s) F+ ( x, s) x (7) Let us first review the simpler case of aalogues of the biomial theorem ( x+ s) x s ( x+ s)( x+ s) It is well ow that there are two importat oes, (, ) ( + )( + ) ( + ), p xs x s x s x s x s (8) which satisfy the recurrece relatio p xs x sp xs (, ) ( + ) (, ) the Rogers-Szegö polyomials r ( x, s) x s, (9) which have o closed formula but satisfy the recursio

3 ( ; ) Here ( ; ) ( ; ) We also use [ ] istead of r( xs, ) ( x+ sr ) ( xs, ) + sxr ( xs, ) () j with ( x; ) ( x) deotes the biomial coefficiet A similar situatio occurs with aalogues of the Fiboacci polyomials There are the polyomials studied by L Carlitz j () f( xs,, ) x s, which satisfy the recursio f( xs,, ) xf ( xs,, ) + sf ( xs,, ) the polyomials F ( x, s, ) with which I am cocered i this paper Defiitio simple properties Defie the Fiboacci polyomials F ( x, s, ) by for + F ( x, s, ) s x () The first polyomials are 3 4 3,, x, x s, x ( ) sx, x s[3] x s, Let us recall that these Fiboacci polyomials satisfy each of the recurreces F( xs,, ) xf ( xs,, ) + sf ( xs,, ), () s F( x, s, ) xf ( x, s, ) + sf ( x,, ) (3) F( xs,, ) xf ( xs,, ) + sxf ( xs,, ) + sf ( xs,, ) (4) 3 4 3

4 The simple proofs follow by comparig coefficiets usig the well-ow recurreces for the biomial coefficiets We see that () is euivalet with + (3) with + Combiig () (3) we get (4) As a coseuece we get F( x, s, ) xf ( x, s, ) sf ( x, s, ) sf ( xs,, ) + sf ( xs,, ) sf ( xs,, ) + sxf 3( xs,, ) + sf 4( xs,, ) sf ( xs,, ) s F ( xs,, ) xf ( xs,, ) sf ( xs,, ) ( 3 4 ) Iteratig this euatio observig that it holds for 3 gives / ( )( ) ( ) F( xs,, ) xf ( xs,, ) sf ( xs,, ) s F ( xs,, ) (5) Remar From () we get the followig combiatorial iterpretatio of the Fiboacci polyomials which is a aalogue of the well-ow Morse code model of the Fiboacci umbers Cosider words c cc cm of letters ci { a, b} associate with c the weight i + i + + i wc wc s sx m, if c c c b, i < < i m, all other i i i ci a The weight of the empty word ε is defied to be w( ε ) We the have wac ( s) xwc ( s), w( bc)( s) sw( c)( s), wca () s xwc ()(), s m+ wcb ( s) swc ( s) Defie the legth lc of a word c cosistig of letters b m letters a by lc () + m m+ Let ow Φ be the set of all words of legth Φ ca also be idetified with the set of all coverigs of a ( ) rectagle with moomios (ie rectagles) domios (ie rectagles) or with Morse code seueces of legth Let G ( x, s): w( c) be the weight of Φ c Φ (6) 4

5 The we get G( x, s): w( c) F( x, s, ), (7) c Φ ie F ( x, s, ) is the weight of Φ For the proof observe that by cosiderig the first letter of each word we see from (6) that G( x, s) xg ( x, s) + sg ( x, s) Thus G ( x, s ) satisfies the same recurrece as F (,, ) x s Also the iitial values coicide because G ( x, s ) G ( x, s ) For example Φ 4 { aaa, ab, ba} 3 G ( x, s) w aaa + w( ab) + w( ba) x + x s+ sx F ( x, s, ) 4 4 This iterpretatio ca also be used to obtai (3), which is euivalet with s G( x, s) xg ( x, s) + sg x, Here we cosider the last letter of each word From (6) we get wca ( s) xwc ( s), which gives the first term To obtai the secod term let us suppose that cb Φ has letters b m i+ + i The () ()() s wcb s swc s s s x swc () Sice this expressio is idepedet of, we get the secod term Let D be the differetiatio operator defied by Df ( x) show i [5] these Fiboacci polyomials also satisfy f ( x) f( x) As has bee ( x ) I order to show (8) we must verify that F ( xs,, ) F x+ ( ) sds, (8) F ( x, s, ) xf ( x, s, ) + ( ) sdf ( x, s, ) + sf ( x, s, ) (9) Comparig coefficiets this amouts to ( ) + + or ( ) which is obviously true 5

6 As i [5] we defie the Lucas polyomials by The first polyomials are L ( xs,, ) L x+ ( ) sds, () 3 4, xx, ( sx ), [3] sxx, [4] sx ( ) s, By applyig the liear map f ( x) f( x+ ( ) sd) () to (5) we get for > L ( x, s, ) F ( x, s, ) + sf ( x, s, ) () + This implies the explicit formula [ ] L ( x, s, ) s x [ ] (3) for >, which is a very ice aalogue of () For the proof observe that [ ] + [ ] [ ] + [ ] [ ] Comparig coefficiets we also get L ( xs,, ) F xs,, + sf ( xs,, ) (4) + This follows from [ ] + [ ] [ ] + [ ] + [ ] [ ] [ ] [ ] Remar () has the followig combiatorial iterpretatio: Cosider a circle whose circumferece has legth let moomios be arcs of legth domios be arcs of legth o the circle Cosider the set Λ of all coverigs with moomios domios fix a poit P o the circumferece of the circle If P is the iitial poit of a moomio or a domio of a coverig the this coverig ca be idetified 6

7 with a word c c cm We defie its weight i the same way as i the liear case Therefore the set of all those coverigs has weight F (,, ) + x s If P is the midpoit of a domio we split b ito b bb associate with this coverig the word bc cmb with c cm Φ defie its weight as sw( c c m ) Therefore w( Λ ) ( ) w Φ + + sw Φ Λ aaaa, aab, aba, baa, bb, b aab, b bb Thus Eg { } w Λ x + x s + x sx + sx + s s + sx + ss x + [4] sx + ( + ) s L ( x, s, ) 4 4 To give a combiatorial iterpretatio of (4) we cosider all words of Λ with last letter a or the two last letters ab Their weight is s F( x, s, ) + sf ( x, s, ) F+ x,, There remais the set of all words i Λ with last letter b With the same argumet as above s we see that this is sf x,, Therefore we have s s L( x, s, ) F+ x,, + sf x,,, which is euivalet with (4) We also eed the polyomials * iitial value L ( x, s, ) L * ( x, s, ) which coicide with L ( x, s, ) for >, but have Comparig () with (4) we see that L ( x, s, ) xl ( x, s, ) + ( ) sdl ( x, s, ) + sl ( x, s, ) (5) This is a recurrece for the polyomials i x but ot for idividual umbers x s I order to fid a recurrece for idividual umbers I wat to show first that for > / * * * ( )( ) * L( xs,, ) xl ( xs,, ) sl ( xs,, ) ( ) sl ( xs,, ) (6) This reduces to (4) for s It is easily verified that DL( x, s, ) [ ] F x,, Therefore s L( xs,, ) xl ( xs,, ) sl ( xs,, ) ( ) sdl ( xs,, ) ( ) sf x,, 7

8 By (4) we ow that Iteratio gives (6) s 3 s F x,, L ( x, s, ) sf 3 x,, From (6) we get L( xs,, ) xl ( xs,, ) sl ( xs,, ) ( ) sl ( xs,, ) 3 [ ] s ( L ( x, s, ) xl 3( x, s, ) sl 4( x, s, ) ) [ 3] This ca be writte as 3 ( + ) [ ] 3 [ ] L( xs,, ) xl ( xs,, ) sl ( xs,, ) sxl ( xs,, ) sl ( xs,, ) [ 3] [ 3] [ 3] This recurrece holds for 4 if L ( x, s, ) (7) 3 Iversio theorems L Carlitz [] has obtaied two aalogues of the Chebyshev iversio formulas The first oe ([],Theorem [6]) implies Theorem 3 L x s s x * (,, ) (3) the secod oe ([], Theorem 7) gives Theorem 3 F+ ( xs,, )( s) x (3) These are aalogues of (6) (7) We give aother proof of these theorems: Let Defie A x+ ( ) sd (33) A + A + 4s α( ) (34) 8

9 A A + 4s β ( ) (35) the seueces ( α ) ( β ( ) ) Sice α( ) Aα( ) s β( ) Aβ( ) s satisfy the recurrece ( ) A ( ) s ( ) ( ) A ( ) s ( ) α α α β β β for all Sice the Fiboacci the Lucas polyomials satisfy the same recurrece we get from the iitial values ( α β ) L ( x, s, ) ( ) + ( ) (36) α( ) β( ) F ( x, s, ) (37) α( ) β( ) for We ca use these idetities to exted these polyomials to egative We the get for > β( ) + α( ) L ( x, s, ) L ( x, s, ) ( α( ) + β( ) ) ( ) ( ) (38) s s F α( ) β( ) F ( x, s, ) α( ) β( ) s ( x, s, ) ( ) (39) Remar 3 It is easily verified that the idetities (), (3), (4), () (4) hold for all We wat to show that ( s) L ( x, s, ) x (3) For odd the left-h side is 9

10 ( α( ) β( )) α( ) β( ) α( ) β( ) α( ) β( ) α( ) β( ) ( + ) ( + ) For m the same holds because for m the coefficiet of L ( x, s, ) m ( s) m m is Therefore by () we see that satisfies R( xs,, ): ( s) L ( xs,, ) R( xs,, ) ( α( ) + β( )) R (, xs, ) + α( ) β( ) R (, xs, ) x+ ( ) sd R (, xs, ) s R (, xs, ) We have to show that R( xs,, ) x This is obviously true for If it holds for m < the R( xs,, ) x+ ( ) sd x s x x as asserted Remar 4 I [6] we have defied a ew aalogue of the Hermite polyomials H( x, s ) ( x sd) By applyig the liear map () to (6) (7) these ca be expressed as / ( + )/ * + (3) H ( x,( ) s ) s L ( x, s, ) s F ( x, s, ) Remar 5 ( ) The polyomials F ( x, s, ) + fuctioal L o this vector space by The from (3) we get L( x + ) are a basis of the polyomials i ( s, )[ x] Defie a liear ( ) L F + x, s, [ ] L x ( s) C ( ), where C Catala umbers This is euivalet with ( ) [ + ] is a aalogue of the ( ) C ( ) [ ] (3)

11 I the same way the liear fuctioal M defied by M ( x ) ( s) M( x + ) This is euivalet with M L x s gives * ( (,, )) [ ] for > [ ] ( ) [ ] (33) 4 Some related idetities The classical Fiboacci Lucas polyomials satisfy L ( x+ y, xy) x + y (4) x y F ( x+ y, xy) (4) x y L Carlitz [] has give aalogues of these theorems which are itimately coected with our aalogues Theorem 4 (L Carlitz[]) Let r ( x, y) x y be the Rogers-Szegö polyomials The x y x + ( xy) r ( x, y) (43) y [ ] x + y xy r x y [ ] (, ) (44) These are polyomial idetities which for ( xy, ) ( α, β ) immediately give the explicit formulae for the Fiboacci Lucas-polyomials if we defie them by (37) (36) To prove these theorems we use the idetity j+ j j ( ) j j (45)

12 for (cf Carlitz[]) To show this idetity let be fixed let U be the liear operator o the vector space of all sums x c with c x x defied by U for ` x x x x Sice ( U) by usig (8) we get the desired result j + j x j x x x ( ) ( U) ( U) ( U) ( U) j j First we prove (43) + + j j ( ) (, ) ( ) xy r x y xy x y j j i + + i i i i i xy ( ) x i i y i i i i i i x y xy xy i i x y i ( ) i For the proof of (44) observe that + + x y x y x ( x y) + y ( x y) xy x + y x y x y x y This implies + + x + y xy r ( x, y) xy xy r ( x, y) + xy r ( x, y) xy r ( x, y) [ ] ( xy) r ( x, y) ( xy) r ( x, y) [ ] For the classical Fiboacci polyomials the formula

13 ( ) x F + m ( x, s) s Fm( x, s) (46) α β holds for all m This is a easy coseuece of the Biet formula F ( x, s) α β x + x + 4s x x + 4s where α β For (46) is euivalet with + m + m m m α ( α x) β ( β x) α β s α β α β, We ow get Theorem 4 The case m gives Corollary 43 (H Prodiger [8]) m + s + m m ( ) x F ( x, s, ) s F x,, (47) Proof For this is trivially true for all m For (47) reduces to ( ) x F ( x, s, ) (48) m s Fm+ ( x, s, ) xfm+ ( x, s, ) sfm x,,, which also holds for m by (3) Remar 3 (49) Assume that (47) holds for i < all m The we get 3

14 + m + m ( ) x F ( x, s, ) ( ) x F ( x, s, ) x F + m ( x, s, ) + x F+ m + x s ( ) (,, ) + ( ) x F x F+ m + ( x, s, ) xf+ m ( x, s, ) + + m s ( ) x sf+ m x,, + m s x F( ) + m + m( ) + m + m m s x,, s s s s F x,, s Fm x,, + m ( x, s, ) For we get from (, ) ( ) + m (,, ) hm xl xs (4) L ( x, s, ) F ( x, s, ) + sf ( x, s, ) (4) + + m m s + + s m m hm (, ) sf x,, + s F x,, (4) This implies + m ( ) + L ( x, s, ) h(, m) + h(, m+ ) Because of (39) we get ( ) + L ( x, s, ) h(,) + h(,) ( ) s + + s s F x,, + s F x,, + s x+ s + s x 4

15 This gives Theorem 44 (H Prodiger [8] ) ( ) + x L ( x, s, ) (43) 5 Some Rogers-Ramauja type formulas It is iterestig that the Lucas polyomials give a simple approach to the Bailey pairs A() - A(8) of Slater s paper [] Let us recall some defiitios (cf [] or [7] ) suitably modified for our purposes Two seueces a ( α ) b ( β ) are called a Bailey pair ( ab, ) m, if for some m {, } Note that α β (5) ( ; ) ( ; ) + + m ( ; ) ( )( ) ( ) To obtai Bailey pairs we start with formula (3) cosider separately eve odd umbers This gives Therefore * L ( x, s( ), ) s( ) x (5) + * + L + ( x, s( ), ) s( ) x (53) Theorem 5 is a Bailey pair with m * ( ) x s ( ; ) a L ( x, s( ), ) s( ), b (54) oe with m + * x a ( L + ( x, s( ), ) s( ) ), b s ( ; ) + (55) 5

16 If we chage with m with m we get the Bailey pairs * x ( (,, ) ), ( s ) ( ; ) a L x s s b + + * + x ( (,, ) ), + ( s ) ( ; ) a L x s s b + (56) (57) For each Bailey pair we cosider the idetity + m + m + m α β α ( ; ) ( ; ) + + m ( ; ) ( ; ) + + m (58) Here the ier sum For we have + m ca be easily computed: ( ; ) ( ; ) + + m s + s i + i (59) ( ; ) ( ; ) ( ; ) s i s i s++ i This is a easy coseuece of the Vermode formula i i s i s i i i j( i+ j+ ) + i i s i s i s i + + j j j if we let Therefore we get + m + m + m β α α ( ; ) ( ; ) + + m ( ; ) (5) I the followig formulas we set x m { } For s we get from (5), (54) (55), + m * i + mi L i+ m(,, ) (5) s ( ; ) + m ( ; ) i 6

17 For s we get + + m * + + m L m(,, ) ( ; ) ( ; ) + (5) + m I the same way we get from (56) (57) for s + m s * i + m L i+ m(,, ) * i + mi L i+ m(,, ) s ( ; ) + m s i ( ; ) i( ; ) + i+ m ( ; ) i (53) for s + m * i i+ mi + m L i+ m(,, ) * i i+ mi L i+ m ( ; ) + m i ( ; ) i( ; ) + i+ m ( ; ) i (,, ) (54) The mai advatage of these formulas derives from the fact, that the Lucas polyomials have simple values for x s or s From (4) it is easily verified (cf [5]) that (3) (3+ ) F3,,, F3 +,,, F3 +,, (55) Therefore by (4) (3) (3+ ) L3 (,, ) ( ) + for > L (3+ ) (,, ) ( ), (3) L (,, ) ( ) (56) Of course i all formulas (56) implies L (, s, ), although I shall ot state this i each case explicitly * L (,, ) L,,, L (,, ) L,, * * 6 (3) 6 (3 ) * , 6+ (,, ) (3+ ),, + L L L (,, ) L,,, L (,, ) L,, * 6 * 6 (3 ) (3+ ) + 6+ (3+ ) * 6 + +, L (,, ) L,, (57) (58) 7

18 The first terms of the seuece *,, 5, 7,, 5, 6, L (,, ) are therefore + + The sum of all these terms is Euler s petagoal umber series * The same is true for the seuece L + (,, ), which begis with ,,,,,,, This is a immediate coseuece of (54) (55) for x s, which reduce to * L (,, ) (59) ( ; ) ( ; ) ( ; ) + * L + (,, ) (5) ( ; ) ( ; ) ( ; ) If we let these formulas coverge to L (,, ) ( ; ) L + (,, ) ( ; ) respectively By () we get This implies that (3) (3 5) L3 (,, ) ( ) + for > 3+ 3, ) (3+ ) L (,, ) ( ), L (, ( )(3) ( ) (5) 6 5+ L6,, L(3 ),,, L6,, L(3 ),, + L L , 6+,, (3+ ),, (5) 6 7+ L6,, L(3 ),,, L6 3,, L(3 ),, , L6+,, L(3 ) +,, (53) 8

19 Now it is time to harvest the Corollaries We order them so that Corollary 5i correspods to Slater's A() i Corollary 5 (cf [9], A79) ( ; ) ( ; ) (54) Proof Choose m i (5) observe that (3 i) 5i i 5i + i L6 i (,, ) +, (,, (3i+ ) 5i + i+ L6i + ) (3 ) 5 +,, i i i L6i which implies L (,, ) i i * 5 5 i thus (54) Corollary 5 (cf [9], A94) ( ) ; ; + (55) Proof We use formula (5) for m compute (3 ) (3+ ) L6 (,, ), L6+ 3,,, (3 ) L6 +,, Sice 5( ) + 6( ) we get (55) Corollary 53 (cf [9], A99) + * L ; ; ; (,, ) (56) 9

20 This follows from (5) for m (3 ) (3 ) L6,,, L6,,, + (3+ ) L6 +,, Corollary 54 (cf [9], A38) + ( ; ) ; ( ) (57) Proof This follows from (5) for m the computatio (3 ) + (3) 5 7 (3+ ) + (3+ ) L6,,, L6 3,,, + (3 ) + (3 ) L6 +,, Observe that 5( ) + 7( ) Corollary 55 (cf [9], A39) ( ) ( ; ) ( ; ) (58) Proof Here we use (53) with m (3) 7+ (3 ) + L6,,, L6,,, + (3+ ) + 7+ L6 +,,

21 Corollary 56 (cf [9], A84) + 3 ( ; ) ( ) ( ; ) ( ; ) ( ; ) ( ; ) + (59) Proof We use (54) with m (3 ) (3+ ) (+ ) (+ ) 3(+ ) + (+ ) L6,,, L6 3,,, + (3 ) L6 +,, Therefore we get i * i + i 3 i+ L (,, ) Corollary 57 (cf [9], A5) 3 ( ; ) ( ) ( ; ) ( ; ) ( ; ) ( ; ) (53) Proof This follows from (54) with m, because we get the same sums as i Corollary 56 (3 ) + (3 ) 3(+ ) (+ ) (3 ) (3 ) L6,,, L6,,, + (3+ ) (3+ ) 3(+ ) (+ ) L6 +,, The deeper reaso for the simple results (59) (53) are the formulas L L i i+ mi i+ m,, i+ m,, (53) for m {, }, which ca easily be verified

22 Corollary 58 (cf [9], A96) ( ) ( ; ) ( ; ) (53) Proof Here we use (53) with m We get (3 ) + (3) 5 (3+ ) + (3+ ) L6,,, L6 3,,, + (3 ) + (3 ) + 5 L6 +,, We have oly to verify that ( ) + ( ) Refereces [] GE Adrews, -Series: Their developmet applicatio i Aalysis, Number Theory, Combiatorics, Physics, Computer Algebra, AMS 986 [] L Carlitz, Some iversio formulas, Red Circolo Mat Palermo, (963), [3] L Carlitz, Fiboacci otes 4: -Fiboacci polyomials, Fib Quart 3 (975), 97- [4] J Cigler, Eiige -Aaloga der Lucas- ud Fiboacci-Polyome, Sitzugsber ÖAW (), 3- [5] J Cigler, A ew class of Fiboacci polyomials, Electr J Comb (3), #R9 [6] J Cigler J Zeg, A curious aalogue of Hermite polyomials, J Comb Th A 8 (), 9-6 [7] P Paule, The cocept of Bailey chais, Sémiaire Loth Comb B8f (987) [8] H Prodiger, O the expasio of Fiboacci Lucas polyomials, Joural of Iteger Seueces (9), Article 96 [9] AV Sills, Fiite Rogers-Ramauja type idetities, Electr J Comb (3), #R3 [] L Slater, A ew proof of Rogers's trasformatios of ifiite series, Proc Lodo Math Soc () 53 (95), [] L Slater, Further Idetities of the Rogers-Ramauja type, Proc Lodo Math Soc 54 (95) 47-67