Motzkin numbers, central trinomial coefficients and hybrid polynomials

Motzkin n um b ers, cen tral trinomial co efficients and h ybrid p olynomials P .Blasi ak a , G. Dattoli b , A.Hor zela a , K.A. P enson c and K.Zh uko vsky b a H.Niewo dnicza ´ nski Institut e of Nuclear Ph ysics, P ol ish Academy of Sciences ul. Eliasza-Radzi ko wskieg o 152, PL 31342 Krak´ ow, Poland b ENEA, Dipartim en to Inno v azione, Divisio n e Fisica Applica t a Cen tro Ricerc he F rascat i , Via E. F ermi 45, I 0 0 044 F rascat i, Rome, Italy c Lab oratoi re de Physique Th ´ eoriqu e de l a Mati` ere Con d ens´ ee Universit ´ e Pierr e et Marie Curie, CNRS UMR 7600 T our 24 - 2i ` eme ´ et., 4 pl . Jussieu, F 7 5252 Paris C edex 05, F rance Abstract W e sho w that the formalism of hybrid p olynomials, inte rp olat- ing b et we en Hermite and Laguerre polynomials, is v ery u se fu l in the study of Motzkin num b ers and central trinomial coefficien ts. These sequences are iden tified as sp ecia l v alues of h ybr id p olynomials, a fact whic h w e use to derive their generalized forms and n ew id e ntitie s sat- isfied b y them. Keyw ords: cen tral t rin o mial co e fficients, Motzkin n umber s , Hermite- Kamp ´ e de F´ eri ´ et p olynomials. AMS 2000 M SC Sc heme: 11B83 , 05A19, 33C45. 1 1 In tro ducti on The cen tral trinomial co effic ients (CTC) c n are defined as the co efficien ts o f x n in the expansion of (1 + x + x 2 ) n . V arious expressions ha ve b een giv en for these co effic ients (see, for example, [2, 11]); here w e will refer to the follo wing form, see A00 2426 and A0 0 1006 in [1 3]: c n = [ n 2 ] X k =0 n ! ( k !) 2 ( n − 2 k )! , (1) whic h is the most useful for our purp oses. An alternative approac h according to whic h one can define the cen tral trinomial co efficien ts is to follow [5] and to consider t he Lauren t p olynomial (1 + x + x − 1 ) n = n X j = − n  n j  2 x j , (2) where the appropriate trinomial coefficien ts  n j  2 are giv en b y:  n m  2 = X j ≥ 0 n ! j !( m + j )!( n − 2 j − m )! . (3) Comparing Eqs.(1) a nd (3) one immediately deriv es c n =  n 0  2 . The Motzkin num b e rs (MN) are connected to the num b e r of planar pat hs asso ciated with the com binatorial interpretation of c n . They are defined as follo ws ( see [2, 1 1 ]): m n = [ n 2 ] X k =0 n ! k !( k + 1)!( n − 2 k )! . (4) Similarly to the cen tra l trinomial coefficien ts also the Motzkin num bers can b e expressed in terms of the co efficien ts  n m  2 simply as follo ws: m n = 1 n + 1  n + 1 1  2 . (5) In the next se ctions w e shall demonstrate that t he Motzkin num bers m n and the cen tral trinomial co efficien ts c n can b e treated on the same fo ot- ing and framed within the contex t of the theory of the hybrid p oly nomials, 2 (see [8]). Recalling basic properties of the h ybrid p olynomials in terpo lating b et w een standard t w o-v ariable Hermite and La guerre p olynomials w e shall sho w that the central trinomial co effi cien ts and the Motzkin n um b ers sat- isfy a simple recurrence whic h relates c n +1 , c n and m n − 1 . Moreo v er, the metho ds dev elop ed on the base on the h ybrid polynomials formalism allow natural generalization of the not ions of the cen tra l tr ino m ial co efficie nts and the Motzkin num bers which is useful for the inv es tiga tion of their prop erties. Definition 1. The Hermite-Kamp´ e de F´ eri ´ et (HKdF) p olynomials are defined by the fol- lo wing e xpression H n ( x, y ) = n ! [ n 2 ] X k =0 x n − 2 k y k k !( n − 2 k )! , (6) where x, y ∈ C . F or sp ecial v alues of x and y the HKdF p olynomials reduce to the we ll kno wn ordinary Hermite polynomials [1] H n ( x, − 1 2 ) = H e n ( x ) , H n (2 x, − 1) = H n ( x ) , (7) H n ( x ) = 2 n 2 H e n ( √ 2 x ). Remark 2. The HKdF p olynomials can b e also defined through the f ollo wing op erational rules: H n ( x, y ) = exp( y ∂ 2 ∂ x 2 ) · x n , (8) H n ( x, y ) = ( x + 2 y ∂ ∂ x ) n · 1 , (9) and the relev a n t expo nential generating function: ∞ X n =0 t n n ! H n ( x, y ) = exp( xt + y t 2 ) . (10) Other prop erties of the HKdF p olynomials can be found in the review [7]. Definition 3. The t w o- v ariable Laguerre polynomials are defined as follo ws (see [6]): L n ( x, y ) = n ! n X k =0 ( − 1) k y n − k x k ( k !) 2 ( n − k )! . (11) 3 They reduce to the ordinary Laguerre p olynomials for the v alue of the argu- men t y = 1. Remark 4. The t w o-v ariable Laguerre p olynomials (11) are also defined by t h e op era- tional rule L n ( x, y ) = ( y − b D − 1 x ) n 1 = n X k =0  n k  ( − 1) k y n − k b D − k x 1 , (12) where b D − 1 x is t he inv erse deriv ativ e op erator whose action on the unity is giv en as follo ws: b D − k x 1 = x k k ! . (13) Indeed, substituting Eq.(13) in to Eq.(12) w e immediately recov er Eq.(11) in the follo wing f o rm: L n ( x, y ) = n X k =0  n k  ( − 1) k y n − k x k k ! . (14) Hereb y , w e note tha t according to [8] the in v erse deriv ative op erator action on a f unction f ( x ) is sp ecified as f ollo ws: b D − k x f ( x ) = 1 ( k − 1) ! x Z 0 ( x − ξ ) k − 1 f ( ξ ) dξ , ( k = 1 , 2 , 3 , ... ) , (15) and w e sp ecify its zeroth order a c tion on the func tion f ( x ) b y the function itself: b D 0 x · f ( x ) = f ( x ) . (16) Next w e will in t r o duce the h ybrid Hermite-Laguerre p olynomials com bining the individual c haracteristics of b oth Laguerre and Hermite p olynomials and explore their prop erties in the contex t of the cen tral trinomial co efficien ts and Motzkin num b ers. Definition 5. The hybrid Hermite-Laguerre p olynomials Π n ( x, y ) are defined b y the fo l- lo wing e xpression: Π n ( x, y ) = H n ( y , b D − 1 x ) 1 . (17) 4 Prop osition 6. The c entr al trinomial c o effici e nt s ar e the p articular c ase of the hybrid Hermite-L aguerr e p olynomials: c n = Π n (1 , 1) . (18) Pr o of. Note that from the definition of HKdF and fro m Eq.(13), w e find Π n ( x, y ) = n ! [ n 2 ] X k =0 y n − 2 k b D − k x k !( n − 2 k )! 1 = n ! [ n 2 ] X k =0 y n − 2 k x k ( k !) 2 ( n − 2 k )! (19) and therefore the comparison of Eq.(19) with Eq.(1) yields Eq.(18). 2 Cen tral trinomial co efficien ts and sp e cial functions In this Section w e will fo cus our atten tion on some prop erties of the ce ntral trinomial co efficien ts a n d the calculatio n of the ir generating function. Definition 7. I 0 denotes the z eroth order mo dified Bessel function of the first kind. I n ( x ) is defined a s (see [4]): I n ( x ) = ∞ X r =0  x 2  n +2 r r !( n + r )! , (20) whic h is a particular case of the T ricomi function of α th order where the parameter α is not nece ssarily an in teger: C α ( x ) = ∞ X r =0 x r r !Γ( r + α + 1) = x − α 2 I α (2 √ x ) . (21) Prop osition 8. T h e exp onential gener ating function for the CTC is given by: ∞ X n =0 t n n ! c n = exp( t ) I 0 (2 t ) . (22) 5 Pr o of. Using t he definition of Eq.(17) a n d the generating function (10 ) of the HKdF p olynomials w e o btain: ∞ X n =0 t n n ! Π n ( x, y ) = ∞ X n =0 t n n ! H n ( y , b D − 1 x ) 1 = exp ( y t + b D − 1 x t 2 ) 1 . (23) The exp onen tial on the r.h.s of Eq .(2 3 ) can b e disen tangled b ecause y and b D − 1 x comm ute. Thus w e get: exp( y t ) exp( b D − 1 x t 2 ) 1 = exp( y t ) ∞ X r =0 b D − r x t 2 r r ! 1 = exp( y t ) ∞ X r =0 x r t 2 r ( r !) 2 . (24) hen Eq.(22) follows from Eqs.(23), (24), (20) and (18 ) and the prop osition is pro v ed. Prop osition 9. The c en t r al trinomi a l c o efficient c an b e expr esse d in terms of L e gen dr e p olynomial s P n ( x ) : c n = i n √ 3 n P n ( − i √ 3 ) . (25) Pr o of. As it has b een sho wn in [6], h ybrid p oly nomials Π n ( x, y ) ha v e the follo wing ordinary generating function: ∞ X n =0 t n Π n ( x, y ) = 1 p 1 − 2 y t + ( y 2 − 4 x ) t 2 ,    p y 2 − 4 xt    < 1 . (26) Since Legendre p olynomials satisfy the analogous relation (see [9]) written b elo w: ∞ X n =0 t n P n ( x ) = 1 √ 1 − 2 xt + t 2 , | t | < 1 , (27) w e can eas ily rearrange the summation in (26) to o bta in Π n ( x, y ) = ( y 2 − 4 x ) n 2 P n y p y 2 − 4 x ! , (28) whic h, on accoun t of Eq.(18), yields Eq .(25 ). 6 Corollary 10. The c entr al trinomial c o effic i e nt s s at isfy the fol lowing r e cur- r enc e [3] ( n + 1) c n +1 = (2 n + 1) c n + 3 nc n − 1 . (29) Pr o of. Eq.(29) follow s from Eq.(25) and from the w ell kno wn recurrence for the Legendre p olynomials [9]: ( n + 1) P n +1 ( x ) = (2 n + 1) xP n ( x ) − nP n − 1 ( x ) . (30) So far, w e ha v e show n that the cen tra l trinomial co efficien t s can b e w ritten in terms of Legend re p olynomials. F o r alternativ e deriv ation of the results of this se ction se e [10]. In the next section w e will demonstrate that analogous relations can b e obtained for the Motzkin n um b ers to o. 3 Motzkin n um b ers and sp ecial functions In this section w e concen trate on the calculation of the generating function for the asso ciated h ybrid polynomials, whic h will b e defined b elo w, a nd w e study their properties r elat ed to the Motzkin num b ers. Definition 11. Asso c iated CTC a r e define d b y c α n = [ n 2 ] X k =0 n ! ( n − 2 k )! k !Γ( k + α + 1) . (31) and the Motzkin n um b ers can b e iden tified as a particular case of the asso- ciated CTC: m n = c 1 n . (32) Definition 12. Recall the op erator b D − 1 x,α defined in [8] via the following rule for its action o n the unit y: b D − n x,α 1 = x n Γ( n + α + 1 ) . (33) 7 Definition 13. The asso ciated h ybrid Herm ite-L a guerre p olynomials Π ( α ) n ( x, y ) are defined as follows: Π α n ( x, y ) = H n ( y , b D − 1 x,α ) 1 = n ! [ n 2 ] X k r =0 x k y n − 2 k ( n − 2 k )! k !Γ( k + α + 1) . (34) Prop osition 14. The as s o ciate d hybrid p olynomials Π ( α ) n ( x, y ) p ossess the fol lowing gener ating function: ∞ X n =0 t n n ! Π α n ( x, y ) = exp ( y t )( xt 2 ) − α 2 I α (2 t √ x ) . (35) Pr o of. Using Eq.(34) and the generating function for the HKdF p olynomials Eq.(10) w e find tha t ∞ X n =0 t n n ! Π α n ( x, y ) = exp( y t + b D − 1 x,α t 2 ) 1 = exp( y t ) ∞ X r =0 b D − r x,α t 2 r r ! 1 , (36) whic h y ields Eq .(3 5 ) with accoun t of Eq.(33). Corollary 15. The MN c an b e identifie d as the p articular c a se of the asso- ciate d hybrid Hermite-L aguerr e p olynomia l s Π ( α ) n ( x, y ) m n = Π 1 n (1 , 1) , (37) and satisfy the fol lowing iden tity: ∞ X n =0 t n n ! Π 1 n (1 , 1) = exp( t ) t I 1 (2 t ) . (38) It is now eviden t tha t man y of the prop erties o f the CTC and of the MN can b e derive d from those of the hybrid p olynomials. Theorem 16. The MN a n d the CTC ar e linke d by the r e curr enc e [3] c n +1 = c n + 2 n · m n − 1 . (39) 8 Pr o of. The HKdF p olynomials satisfy the following recurrence relation [8]: H n +1 ( x, y ) = H n ( x, y ) + 2 y nH n − 1 ( x, y ) . (40) The s ame recurrence, w ritten in opera t io nal fo rm for the h ybrid case, reads as follows: H n +1 ( y , b D − 1 x ) 1 = h H n ( y , b D − 1 x ) + 2 b D − 1 x nH n − 1 ( y , b D − 1 x ) i 1 . (41) Then, emplo ying the result of the a c tion o f the in v erse de riv ativ e on the H n ( y , b D − 1 x ) 1 as written b elo w b D − 1 x H n ( y , b D − 1 x ) 1 = x Π 1 n ( x, y ) , (42) w e find from (40) the f o llo wing recurrenc e: Π n +1 ( x, y ) = Π n ( x, y ) + 2 nx Π 1 n − 1 ( x, y ) . (43) Hence, w e hav e prov ed also the particular case of this iden tity , giv en b y Eq.(39). Corollary 17. The MN c an b e expr esse d in terms of the c entr al trinomial c o efficients as fol lows: m n = c n +2 − c n +1 2( n + 1) . (44) Corollary 18. Define the p-asso ciate d CTC ( p is an inte g e r) in the fol lowing way: c p n = n ! [ n 2 ] X k =0 1 ( n − 2 k )! k !( k + p )! . (45) Then, with help of i d entit ies Eqs.(41) and (43), we ob t ain the gener alize d form of t he formula Eq.(44): c p +1 n = c p n +2 − c p n +1 2( n + 1) . (46) Note that for p > 1, the p-asso ciated CTC c p n are not in tegers. F or example, the first 1 1 c p n n um b ers ( n = 0 . . . 10) for p = 0 , 1 , 2 are listed in T able 1. 9 n c 0 n c 1 n 6 · c 2 n 0 1 1 3 1 1 1 3 2 3 2 5 3 7 4 9 4 19 9 18 5 51 21 38 6 141 51 84 7 393 127 192 8 1107 323 451 9 3139 835 1083 10 8953 2188 2649 T able 1. The p-asso c iated CT C c p n for n = 0 , 1 , 2 , . . . , 10 and p = 0 , 1 , 2 . In the second c olumn, i.e. , for p = 1w e ha v e the usual Motzkin n um b ers . Before concluding this pap er, w e will add the follo wing note on the further generalization of the CTC and MN as a consequenc e of the approac h deve l- op ed in the presen t w ork. Definition 19. The m th order p-a sso ciated CTC a re defined as fo llo ws: m c p n = n ! [ n m ] X k =0 1 ( n − mk )! k !( k + p )! . (47) The a bov e defined family of cen tra l trinomial co efficien ts is link ed to the higher order hybrid p olynomials . Their prop erties can b e explored alo ng the lines dev elop ed ab o ve. W e jus t note, that they satisfy the follo wing recurrence: m c p n +1 = m c p n + m n ! ( n − m + 1)! m c p n − m +1 , (48) whic h is a straighforward generalization of Eq.(39). Observ e that Eqs.(44), (46) and (48) are simple recurrences that clearly share common structure rev ealing inhere nt connection b et w een c n , c p n and m c p n . 10 Discuss ion In the prese nt w ork w e hav e rein terpreted the cen tral trinomial coefficien ts and Motzkin n umbers employin g the general formalism, whic h underlie s the theory of the h ybrid p olynomials. Th e analo gous results could be ac hiev ed, using prop erties of the h yp ergeometric functions. In fact, using Eq .(6) and the definition of the h yp ergeometric function p F q , see [1 2 ], the follow ing rep- resen tation is v alid H n ( x, y ) = x n 2 F 0  − n 2 , 1 − n 2 ; 4 y x  , (49) where 2 F 0 is the hypergeometric function. Most of the results of t h is pap er ma y also be deriv ed from this o bserv ation. Ev en though w e ha v e referred to the co effic ients m c p n , m > 2 , p > 0 as “cen- tral trinomial”, they do not ha v e the same in terpretation as in the case 1 p = 0 , m = 2. W e hav e noted that for p = 0 , m = 1, the CT C produce t h e Motzkin n um b ers. Thorough discussion of their combinatorial in terpretation is in tended for f u ture in v estigations. Since thro ug h Eq.(17) all the findings of this pap er are related to the HKdF p olynomials H n ( x, y ) it seems legitimate to lo ok for their com binatorial inter- pretation. W e just p oin t that for a large class of argumen ts x, y of H n ( x, y ) the resulting in teger sequenc es can b e give n a prec ise represen tation whic h ma y b e helpful in searc hing a combinatorial interpre ta t ion of CTC. W e quote t w o example s of suc h situation: a) F or x = 1, y = 1 / 2 w e hav e H n (1 , 1 / 2 ) = 2 F 0  − n 2 , 1 − n 2 ; 2  whic h generate 1 , 1 , 2 , 4 , 10 , 26 , 76 , 23 2 , ... , for n = 0 , 1 , 2 , ... . They are called inv olutio n n um- b ers, see A000085 in [1 3 ], whose classical com binatorial interpretation is the n um b er of pa r titions o f a set o f n distinguishable ob jects in to s ubsets of size one and t w o. This s equence coun ts also permu ta t io ns consisting exclus ive ly of fixed p oin ts and transp ositions. b) Another example is supplied b y the c hoice x = y = 1 / 2 ; then the quantit y 2 n H n (1 / 2 , 1 / 2) = 2 F 0  − n 2 , 1 − n 2 ; 8  furnishes the following integer sequence: 1 , 1 , 5 , 13 , 73 , 2 81 , 1741 , .. for n = 0 , 1 , 2 , ... , see A11 5329 in [13]. It counts the 1 The co efficien ts of x n of the expansion (1 + x + x m ) n are m d n = n ! [ n m ] P k =0 1 k !(( m − 1) k )!( n − mk )! and their pro pe r ties can b e also framed within the context of the prop er ties of the hybrid po lynomials. 11 n um b er of partitio ns of a set in to subsets of size one a nd tw o with additio na l feature that t he constituen ts of a set of size t wo acquire tw o colors. Man y other instances o f suc h combinatorial interpre tat io ns may b e giv en b y judicious c hoices of parameters x and y in Eq.(49). References [1] M. Abramo witz and I. A. 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( Ser. 4) 20 (2) (1997), 1 –133. [8] G. D a ttoli, H. M. Sriv asta v a and C. Cesarano, The Laguerre and Legen- dre p olynomials from an op erational p oin t of view, Applie d Math. a nd Comp. 124 (20 01), 117– 127 [9] J. Ko ndo, Inte gr al Equations , Clarendon P ress, 1991. [10] M. P etk o vsek, H. S. Wilf a nd D. Zeilb erger, A=B , A K P eters Ltd., 1996. 12 [11] D. Romik, Some form ulas for the cen tr a l trinomial co efficien ts, J. Int. Se q . 6 (2003), Art icle 03.2.4. [12] A. P . Prudnik ov, Y. A. Bryc hk ov and O. I. Maric hev, Inte gr als and Series: Mor e Sp e cial F unctions, vol. 3 , Gor do n and Breach, 1986. [13] N. J. A. Sloane, Encyclopedia of Integer Sequences (2007), h ttp://www.researc h.att.com/˜njas/sequences. 13